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-0,0 +1,307 @@ + + +MAMMULT: Metrics And Models for MULTilayer networks + + + + + + + + +
+ + + + + + + + + + +

MAMMULT: Metrics And Models for MULTilayer +networks

+

+
19th October 2015
+ + + +
+ + + +
+ 1 Structural descriptors +
 1.1 Basic node, edge, and layer properties +
  1.1.1 Node and layer activity +
   node_activity.py +
   layer_activity.py +
   node_activity_vectors.py +
   layer_activity_vectors.py +
   multiplexity.py +
   hamming_dist.py +
   node_degree_vectors.py +
   degs_to_binary.py +
   degs_to_activity_overlap.py +
  1.1.2 Layer aggregation +
   aggregate_layers_w.py +
   intersect_layers.py +
  1.1.3 Node degree, participation coefficient, cartography +
   overlap_degree.py +
   cartography_from_layers.py +
   cartography_from_deg_vectors.py +
   cartography_from_columns.py +
  1.1.4 Edge overlap, reinforcement +
   edge_overlap.py +
   avg_edge_overlap.py +
   reinforcement.py +
 1.2 Inter-layer degree correlations +
  1.2.1 Node ranking +
   rank_nodes.py +
   rank_nodes_thresh.py +
   rank_occurrence.py +
  1.2.2 Interlayer degree correlation coefficients +
   compute_pearson.py +
   compute_rho.py +
   compute_tau.py +
  1.2.3 Interlayer degree correlation functions +
   dump_k_q +
   knn_q_from_layers.py +
   knn_q_from_degrees.py +
   fit_knn +
2 Models of multi-layer networks +
 2.1 Null models +
  2.1.1 Null-models of node and layer activity +
   model_hypergeometric.py +
   model_MDM.py + + + +
   model_MSM.py +
   model_layer_growth.py +
 2.2 Growing multiplex networks +
  2.2.1 Linear preferential attachment +
   nibilab_linear_delta +
   nibilab_linear_delay +
   nibilab_linear_delay_mix +
   nibilab_linear_random_times +
  2.2.2 Non-linear preferential attachment +
   nibilab_nonlinear +
  2.2.3 Utilities +
   node_deg_over_time.py +
 2.3 Multiplex networks with inter-layer degree correlations +
  2.3.1 Models based on simulated annealing +
   tune_rho +
   tune_qnn_adaptive +
3 Dynamics on multi-layer networks +
 3.1 Interacting opinions - Multilayer ising model +
   multiplex_ising +
 3.2 Biased random walks +
  3.2.1 Stationary distribution +
   statdistr2 +
  3.2.2 Entropy rate +
   entropyrate2add +
   entropyrate2mult +
   entropyrate2int +
+ + + + + + + + +

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b/doc/html/mammult_docch1.html @@ -0,0 +1,57 @@ + + +1 Structural descriptors + + + + + + + + +

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+

Chapter 1
Structural descriptors

+
+  1.1 Basic node, edge, and layer properties +
  1.1.1 Node and layer activity +
  1.1.2 Layer aggregation +
  1.1.3 Node degree, participation coefficient, cartography +
  1.1.4 Edge overlap, reinforcement +
 1.2 Inter-layer degree correlations +
  1.2.1 Node ranking +
  1.2.2 Interlayer degree correlation coefficients +
  1.2.3 Interlayer degree correlation functions + + + +
+ + + + +

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+

+ diff --git a/doc/html/mammult_docch2.html b/doc/html/mammult_docch2.html new file mode 100644 index 0000000..2d7be08 --- /dev/null +++ b/doc/html/mammult_docch2.html @@ -0,0 +1,53 @@ + + +2 Models of multi-layer networks + + + + + + + + +

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+

Chapter 2
Models of multi-layer networks

+
+  2.1 Null models +
  2.1.1 Null-models of node and layer activity +
 2.2 Growing multiplex networks +
  2.2.1 Linear preferential attachment +
  2.2.2 Non-linear preferential attachment +
  2.2.3 Utilities +
 2.3 Multiplex networks with inter-layer degree correlations +
  2.3.1 Models based on simulated annealing +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docch3.html b/doc/html/mammult_docch3.html new file mode 100644 index 0000000..e392f28 --- /dev/null +++ b/doc/html/mammult_docch3.html @@ -0,0 +1,44 @@ + + +3 Dynamics on multi-layer networks + + + + + + + + +

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+

Chapter 3
Dynamics on multi-layer networks

+
+  3.1 Interacting opinions - Multilayer ising model +
 3.2 Biased random walks +
  3.2.1 Stationary distribution +
  3.2.2 Entropy rate +
+ + + + +

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+

+ diff --git a/doc/html/mammult_docse1.html b/doc/html/mammult_docse1.html new file mode 100644 index 0000000..776bf28 --- /dev/null +++ b/doc/html/mammult_docse1.html @@ -0,0 +1,129 @@ + + +Basic node, edge, and layer properties + + + + + + + + +

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+

1.1 Basic node, edge, and layer properties

+

+

+   1.1.1 Node and layer activity +
   node_activity.py +
   layer_activity.py +
   node_activity_vectors.py +
   layer_activity_vectors.py +
   multiplexity.py +
   hamming_dist.py +
   node_degree_vectors.py +
   degs_to_binary.py +
   degs_to_activity_overlap.py +
  1.1.2 Layer aggregation +
   aggregate_layers_w.py +
   intersect_layers.py +
  1.1.3 Node degree, participation coefficient, cartography +
   overlap_degree.py +
   cartography_from_layers.py +
   cartography_from_deg_vectors.py +
   cartography_from_columns.py +
  1.1.4 Edge overlap, reinforcement +
   edge_overlap.py +
   avg_edge_overlap.py +
   reinforcement.py +
+ + + + + + +

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+

+ diff --git a/doc/html/mammult_docse2.html b/doc/html/mammult_docse2.html new file mode 100644 index 0000000..deb5077 --- /dev/null +++ b/doc/html/mammult_docse2.html @@ -0,0 +1,90 @@ + + +Inter-layer degree correlations + + + + + + + + +

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+

1.2 Inter-layer degree correlations

+

+

+   1.2.1 Node ranking +
   rank_nodes.py +
   rank_nodes_thresh.py +
   rank_occurrence.py +
  1.2.2 Interlayer degree correlation coefficients +
   compute_pearson.py +
   compute_rho.py +
   compute_tau.py +
  1.2.3 Interlayer degree correlation functions +
   dump_k_q +
   knn_q_from_layers.py +
   knn_q_from_degrees.py +
   fit_knn +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docse3.html b/doc/html/mammult_docse3.html new file mode 100644 index 0000000..f05a21a --- /dev/null +++ b/doc/html/mammult_docse3.html @@ -0,0 +1,55 @@ + + +Null models + + + + + + + + +

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+

2.1 Null models

+

+

+   2.1.1 Null-models of node and layer activity +
   model_hypergeometric.py +
   model_MDM.py +
   model_MSM.py +
   model_layer_growth.py +
+ + + +

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+

+ diff --git a/doc/html/mammult_docse4.html b/doc/html/mammult_docse4.html new file mode 100644 index 0000000..2ab68dd --- /dev/null +++ b/doc/html/mammult_docse4.html @@ -0,0 +1,76 @@ + + +Growing multiplex networks + + + + + + + + +

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+

2.2 Growing multiplex networks

+

+

+   2.2.1 Linear preferential attachment +
   nibilab_linear_delta +
   nibilab_linear_delay +
   nibilab_linear_delay_mix +
   nibilab_linear_random_times +
  2.2.2 Non-linear preferential attachment +
   nibilab_nonlinear +
  2.2.3 Utilities +
   node_deg_over_time.py +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docse5.html b/doc/html/mammult_docse5.html new file mode 100644 index 0000000..44ac716 --- /dev/null +++ b/doc/html/mammult_docse5.html @@ -0,0 +1,47 @@ + + +Multiplex networks with inter-layer degree correlations + + + + + + + + +

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+

2.3 Multiplex networks with inter-layer degree correlations

+

+

+   2.3.1 Models based on simulated annealing +
   tune_rho +
   tune_qnn_adaptive +
+ + + +

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+

+ diff --git a/doc/html/mammult_docse6.html b/doc/html/mammult_docse6.html new file mode 100644 index 0000000..fab8160 --- /dev/null +++ b/doc/html/mammult_docse6.html @@ -0,0 +1,40 @@ + + +Interacting opinions - Multilayer ising model + + + + + + + + +

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+

3.1 Interacting opinions - Multilayer ising model

+

+

+    multiplex_ising +
+ + + +

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+

+ diff --git a/doc/html/mammult_docse7.html b/doc/html/mammult_docse7.html new file mode 100644 index 0000000..89e367b --- /dev/null +++ b/doc/html/mammult_docse7.html @@ -0,0 +1,49 @@ + + +Biased random walks + + + + + + + + +

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+

3.2 Biased random walks

+

+

+   3.2.1 Stationary distribution +
   statdistr2 +
  3.2.2 Entropy rate +
   entropyrate2add +
   entropyrate2mult +
   entropyrate2int +
+

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+

+ diff --git a/doc/html/mammult_docsu1.html b/doc/html/mammult_docsu1.html new file mode 100644 index 0000000..6790cf6 --- /dev/null +++ b/doc/html/mammult_docsu1.html @@ -0,0 +1,91 @@ + + +Node and layer activity + + + + + + + + +

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+

1.1.1 Node and layer activity

+

This section includes programs related to the computation of node and layer +activity, activity vectors, pairwise multiplexity, pairwise normalised Hamming +distance, node degree vectors. +

+

+    node_activity.py + + + +
   layer_activity.py +
   node_activity_vectors.py +
   layer_activity_vectors.py +
   multiplexity.py +
   hamming_dist.py +
   node_degree_vectors.py +
   degs_to_binary.py +
   degs_to_activity_overlap.py +
+ + + + + + + + + + + +

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+

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+
degs_to_activity_overlap.py
+

NAME +

degs_to_activity_overlap.py - compute the activity and the total +(overlapping) degree of all the nodes of a multiplex. +

SYNOPSYS +

degs_to_activity_overlap.py <degree_vectors> +

DESCRIPTION +

Take a file which contains, on the n-th line, the degrees at each layer of the +n-th node, (e.g., the result of the script node_degree_vectors.py), in the +format: +

  noden_deg_lay1 noden_deg_lay2 ... noden_deg_layM +

and compute the activity (i.e., the number of layers in which a node is not +isolated) and the total (overlapping) degree of each node. +

OUTPUT +

The program prints on stdout a list of lines, where the n-th line contains the +activity and the total degree of the n-th nodem in the format: +

  noden_activity noden_tot_deg +

As usual, the program assumes that node IDs start from zero and proceed +sequentially, without gaps, i.e., if a node ID is not present in any of the layer +files given as input, the program considers it as being isolated on all the +layers. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu11.html b/doc/html/mammult_docsu11.html new file mode 100644 index 0000000..af6ef30 --- /dev/null +++ b/doc/html/mammult_docsu11.html @@ -0,0 +1,48 @@ + + +Layer aggregation + + + + + + + + +

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+

1.1.2 Layer aggregation

+

This section includes programs to obtain various single-layer aggregated graphs +associated to a multiplex network. +

+

+    aggregate_layers_w.py +
   intersect_layers.py +
+ + + + +

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+

+ diff --git a/doc/html/mammult_docsu12.html b/doc/html/mammult_docsu12.html new file mode 100644 index 0000000..2d99541 --- /dev/null +++ b/doc/html/mammult_docsu12.html @@ -0,0 +1,104 @@ + + +1.1.2.0 aggregate_layers_w.py + + + + + + + + +

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+
aggregate_layers_w.py
+

NAME +

aggregate_layers_w.py - compute the (weighted) aggregated graph +associated to a multiplex. +

SYNOPSYS +

aggregate_layers_w.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the edge list of the weighted aggregated graph +associated to the multiplex network given on input. An edge is present in +the aggregated graph if it exists in at least one of the M layers of the +multiplex. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout the edge list of the aggregated graph +associated to the multiplex network. The edge list is a list of lines in the +format: +

  ID1 ID2 weight + + + +

where ID1 and ID2 are the IDs of the two nodes and weight is the number of +layers in which an edge between ID1 and ID2 exists. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu13.html b/doc/html/mammult_docsu13.html new file mode 100644 index 0000000..02fa0b0 --- /dev/null +++ b/doc/html/mammult_docsu13.html @@ -0,0 +1,94 @@ + + +1.1.2.0 intersect_layers.py + + + + + + + + +

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+
intersect_layers.py
+

NAME +

intersect_layers.py - compute the intersection graph associated to a +multiplex. +

SYNOPSYS +

intersect_layers.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the edge list of the intersection graph +associated to the multiplex network given on input, where an edge exists only if +it is present on all the layers of the multiplex. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout the edge list of the intersection graph +associated to the multiplex network. The edge list is a list of lines in the +format: +

  ID1 ID2 +

where ID1 and ID2 are the IDs of the two nodes. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu14.html b/doc/html/mammult_docsu14.html new file mode 100644 index 0000000..571c445 --- /dev/null +++ b/doc/html/mammult_docsu14.html @@ -0,0 +1,61 @@ + + +Node degree, participation coefficient, cartography + + + + + + + + +

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+

1.1.3 Node degree, participation coefficient, cartography

+

This section includes programs to compute the total degree and participation +coefficient of each node, and to draw the cartography diagram of a multiplex. +

+

+    overlap_degree.py +
   cartography_from_layers.py +
   cartography_from_deg_vectors.py +
   cartography_from_columns.py +
+ + + + + + +

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+

+ diff --git a/doc/html/mammult_docsu15.html b/doc/html/mammult_docsu15.html new file mode 100644 index 0000000..e1d4712 --- /dev/null +++ b/doc/html/mammult_docsu15.html @@ -0,0 +1,134 @@ + + +1.1.3.0 overlap_degree.py + + + + + + + + +

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+
overlap_degree.py
+

NAME +

overlap_degree.py - compute the total (overlapping) degree of all the nodes +of a multiplex and the corresponding Z-score. +

SYNOPSYS +

overlap_degree.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the total degree oi of each node i of a +multiplex, defined as: +
+
+ ∑ ∑ [α] +oi = aij + α j + + + +
+

+

and the corresponding Z-score: +
+
+ oi − ⟨o⟩ +z(oi) = ---σ--- + o +
+

+

where ⟨o⟩ and σo are, respectively, the mean and the standard deviation of the +total degree computed over all the active nodes of the multiplex. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  ID_n deg_n z_n +

where ID_n is the ID of the node, deg_n is its total degree, and z_n is the +corresponding Z-score. +

As usual, node IDs start from zero and proceed sequentially, without gaps, i.e., if +a node ID is not present in any of the layer files given as input, the program +considers it as being isolated on all the layers, and the node is omitted from the +output. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu16.html b/doc/html/mammult_docsu16.html new file mode 100644 index 0000000..38df999 --- /dev/null +++ b/doc/html/mammult_docsu16.html @@ -0,0 +1,144 @@ + + +1.1.3.0 cartography_from_layers.py + + + + + + + + +

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+
cartography_from_layers.py
+

NAME +

cartography_from_layers.py - compute the total degree and the multiplex +participation coefficient of all the nodes of a multiplex. +

SYNOPSYS +

cartography_from_layers.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the total degree and the multiplex +participation coefficient Pi for each node i of a multiplex. The participation +coefficient is defined as: +
+
+ [ M [α] ] + -M---- ∑ ki--2 +Pi = M − 1 1 − ( oi ) + α=1 +
+

+

Note that Pi takes values in [0,1], where Pi = 0 if and only if node i is active on +exactly one of the layers, while Pi = 1 if node i has equal degree on all the M +layers. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines in the format: + + + +

  deg_n P_n col_n +

where deg_n is the total degree of node n, P_n is the participation coefficient +of node n and col is the integer representation of the activity bitstring of node n, +which is a number between 0 and 2M − 1. The field col might be useful for the +visualisation of the multiplex cartography diagram, where it would be +possible to associate different colors to nodes having different node activity +patterns. +

As usual, node IDs start from zero and proceed sequentially, without gaps, i.e., +if a node ID is not present in any of the layer files given as input, the +program considers it as being isolated on all the layers, and is set to +zero. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu17.html b/doc/html/mammult_docsu17.html new file mode 100644 index 0000000..a118869 --- /dev/null +++ b/doc/html/mammult_docsu17.html @@ -0,0 +1,106 @@ + + +1.1.3.0 cartography_from_deg_vectors.py + + + + + + + + +

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+
cartography_from_deg_vectors.py
+

NAME +

cartography_from_deg_vectors.py - create a multiplex cartography +diagram. +

SYNOPSYS +

cartography_from_deg_vectors.py <node_deg_vectors> +

DESCRIPTION +

Compute and print on output the total degree and the multiplex +participation coefficient of all the nodes of a multiplex network whose +list of node degree vectors is provided as input. The input file is in the +format: +

  IDn_deg1 IDn_deg_2 ... IDn_degM +

where IDn_degX is the degree of node n at layer X. The input file can be +generated using the script node_degree_vectors.py. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  tot_deg part_coeff +

where tot_deg is the total degree of the node and part_coeff is the corresponding +participation coefficient. +

As usual, node IDs start from zero and proceed sequentially, without gaps, +so if one of the lines in the input files contains just zeros, the program +considers the corresponding node as being isolated on all the layers, and both +its total degree and multiplex participation coefficient are set equal to +zero. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu18.html b/doc/html/mammult_docsu18.html new file mode 100644 index 0000000..ef0ae02 --- /dev/null +++ b/doc/html/mammult_docsu18.html @@ -0,0 +1,102 @@ + + +1.1.3.0 cartography_from_columns.py + + + + + + + + +

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+
cartography_from_columns.py
+

NAME +

cartography_from_columns.py - compute total and participation +coefficient of generic structural descriptors of the nodes of a multiplex. +

SYNOPSYS +

cartography_from_columns.py <filein> <col1> <col2> +[<col3>...] +

DESCRIPTION +

Compute and print on output the sum and the corresponding participation +coefficient of a generic structural descriptor of the nodes of a multiplex. +

The input file is a generic collection of single-space-separated columns, where +each line corresponds to a node. The user must specify the IDs of the columns +which contain the node structural descriptors to be used in the cartography +diagram. Columns IDs start from ZERO. For example: +

python cartography_from_layers.py filein.txt 0 2 4 6 8 +

will create a cartography diagram assuming that the multiplex network has five +layers, and that the node structural descriptors at each layers are contained in +the first (0), third (2), fifth (4), seventh (6) and nineth (8) columns of each +row. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  tot_n P_n +

where tot_n is the sum over the layers of the considered structural descriptor +for node n, and P_n is the associated participation coefficient +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu19.html b/doc/html/mammult_docsu19.html new file mode 100644 index 0000000..9f13aa6 --- /dev/null +++ b/doc/html/mammult_docsu19.html @@ -0,0 +1,52 @@ + + +Edge overlap, reinforcement + + + + + + + + +

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+

1.1.4 Edge overlap, reinforcement

+

This section includes programs to compute the egde overlap and to evaulate the +edge reinforcement effect. +

+

+    edge_overlap.py +
   avg_edge_overlap.py +
   reinforcement.py +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docsu2.html b/doc/html/mammult_docsu2.html new file mode 100644 index 0000000..2b19954 --- /dev/null +++ b/doc/html/mammult_docsu2.html @@ -0,0 +1,86 @@ + + +1.1.1.0 node_activity.py + + + + + + + + +

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+
node_activity.py
+

NAME +

node_activity.py - compute the activity of the nodes of a multiplex, i.e. the +number of layers where each node is not isolated. +

SYNOPSYS +

node_activity.py <layer1> [<layer2> ...] +

DESCRIPTION +

Compute and print on output the activity of the nodes of a multiplex +network, whose layers are given as input in the files layer1, layer2, etc. +

Each file contains the (undirected) edge list of a layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

A list of lines, where the n-th line is the value of activity of the n-th node, +starting from 0. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu20.html b/doc/html/mammult_docsu20.html new file mode 100644 index 0000000..02f16da --- /dev/null +++ b/doc/html/mammult_docsu20.html @@ -0,0 +1,115 @@ + + +1.1.4.0 edge_overlap.py + + + + + + + + +

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+
edge_overlap.py
+

NAME +

edge_overlap.py - compute the edge overlap of all the edges of the +multiplex. +

SYNOPSYS +

edge_overlap.py <layer1> [<layer2>...] +

DESCRIPTION +

Compute and print on output the edge overlap oij of each edge of the +multiplex. Given a pair of nodes (i,j) that are directly connected on at least one +of the M layers, the edge overlap oij is defined as: +
+
+ ∑ [α] +oij = aij + α + + + +
+

+

i.e., the number of layers on which the edge (i,j) exists. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  ID_1 ID_2 overlap +

where ID_1 and ID_2 are the IDs of the end-points of the edge, and overlap is +the number of layers in which the edge exists. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu21.html b/doc/html/mammult_docsu21.html new file mode 100644 index 0000000..87045b2 --- /dev/null +++ b/doc/html/mammult_docsu21.html @@ -0,0 +1,125 @@ + + +1.1.4.0 avg_edge_overlap.py + + + + + + + + +

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+
avg_edge_overlap.py
+

NAME +

avg_edge_overlap.py - compute the average edge overlap of a multiplex. +

SYNOPSYS +

avg_edge_overlap.py <layer1> [<layer2>...] +

DESCRIPTION +

Compute and print on output the average edge overlap +
+
+ ∑ ∑ ∑ [α] + ∗ ∑--∑-i--j>i---αaij---- +ω = i j>i(1− δ ∑ [α]) + 0, α aij +
+

+

i.e., the expected number of layers on which an edge of the multiplex exists, and +the corresponding normalised quantity: +
+
+ ∑ ∑ ∑ a[α] +ω = ---∑--∑i--j>i--α--ij----- + M i j>i(1− δ0,∑α a[iαj] ) +
+

+

that is the expected fraction of layers on which an edge of the multiplex is +present. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a single line, in the format: +

  omega_star omega +

where omega_star and omega are, respectively, the expected number and fraction +of layers in which an edge is present. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 +

L. Lacasa, V. Nicosia, V. Latora, “Network structure of multivariate time +series”, accepted for publication in Scientific Reports, arxiv:1408.0925 +(2015). +

Link to paper: http://arxiv.org/abs/1408.0925 + + + +

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+

+ diff --git a/doc/html/mammult_docsu22.html b/doc/html/mammult_docsu22.html new file mode 100644 index 0000000..b28a026 --- /dev/null +++ b/doc/html/mammult_docsu22.html @@ -0,0 +1,92 @@ + + +1.1.4.0 reinforcement.py + + + + + + + + +

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+
reinforcement.py
+

NAME +

reinforcement.py - compute the probability to have a link between two +nodes in layer 1 given their weight in layer 2. +

SYNOPSYS +

reinforcement.py <layer1> <layer2> < Nbins > < minvalue > +< maxvalue > +

DESCRIPTION +

Compute and print on output the probability to have a link between two +nodes in layer 1 given their weight in layer 2. As input are given the files layer1, +layer2, the number of bins for the link weights of the second layer, the minimum +and the maximum values of the binning. +

The first file contains the binary edge list of layer 1, the second file contains +the weighted edge list of layer 2. each line is in the format: +

  bin_min bin_max freq +

where bin_min and bin_max are the minimum and maximum values of the +link weights of layer 2 in that binning, and freq is the probability to have a link +on layer 1 given such weight in layer 2. +

OUTPUT +

A list of lines, where the n-th line is the minimum and maximum values of +the weight of the links in layer 2 in the n-th bin, and the frequency to have a link +on layer 2 given that weight. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu23.html b/doc/html/mammult_docsu23.html new file mode 100644 index 0000000..113a718 --- /dev/null +++ b/doc/html/mammult_docsu23.html @@ -0,0 +1,54 @@ + + +Node ranking + + + + + + + + +

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+

1.2.1 Node ranking

+

This section includes various utilities to compute and compare node rankings +induced by any generic structural node property, including degree at different +layers. +

+

+    rank_nodes.py +
   rank_nodes_thresh.py +
   rank_occurrence.py +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docsu24.html b/doc/html/mammult_docsu24.html new file mode 100644 index 0000000..7968150 --- /dev/null +++ b/doc/html/mammult_docsu24.html @@ -0,0 +1,79 @@ + + +1.2.1.0 rank_nodes.py + + + + + + + + +

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+ + + +
rank_nodes.py
+

NAME +

rank_nodes.py - rank the nodes of a layer according to a given structural +descriptor. +

SYNOPSYS +

rank_nodes.py <prop_file> +

DESCRIPTION +

Get a file as input, whose n-th line corresponds to the value of a certain +property of the n-th node, and rank the nodes according to that property, taking +into account ranking ties properly. +

For example, if propfile contains the degrees of the nodes at a certain +layer of the multiplex, the computes the ranking induced by degrees, +where the node with the highest degree will be assigned a rank equal to 1 +(one). +

OUTPUT +

The program prints on stdout a list of lines, where the n-th line contains the +rank of the n-th node corresponding to the values of the structural descriptor +provided in the input file. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu25.html b/doc/html/mammult_docsu25.html new file mode 100644 index 0000000..43094cd --- /dev/null +++ b/doc/html/mammult_docsu25.html @@ -0,0 +1,81 @@ + + +1.2.1.0 rank_nodes_thresh.py + + + + + + + + +

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+
rank_nodes_thresh.py
+

NAME +

rank_nodes_thresh.py - rank the nodes of a layer whose value of a given +structural descriptor is above a threshold. +

SYNOPSYS +

rank_nodes_thresh.py <prop_file> <thresh> +

DESCRIPTION +

Get a file as input, whose n-th line corresponds to the value of a certain +property of the n-th node, and rank the nodes according to that property, taking +into account ranking ties properly. The rank of all the nodes whose value of the +structural descriptor is smaller than the threshold thresh specified as second +parameter is set to 0 (ZERO). +

OUTPUT +

The program prints on stdout a list of lines, where the n-th line contains the +rank of the n-th node corresponding to the values of the structural descriptor +provided in the input file, or zero if such desxriptor is below the specified +threshold thresh. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu26.html b/doc/html/mammult_docsu26.html new file mode 100644 index 0000000..c13e9a2 --- /dev/null +++ b/doc/html/mammult_docsu26.html @@ -0,0 +1,87 @@ + + +1.2.1.0 rank_occurrence.py + + + + + + + + +

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+
rank_occurrence.py
+

NAME +

rank_occurrence.py - compute the intersection of two rankings. +

SYNOPSYS +

rank_occurrence.py <rank1> <rank2> <increment> +

DESCRIPTION +

Get two rankings rank1 and rank2 and compute the size of the k-intersection, +i.e. the number of elements which are present in the first k positions of both +rankings, as a function of k. The parameter increment determines the distance +between two subsequent values of k. +

Each input file is a list of node IDs, one per line, where the first line contains +the ID of the highest ranked node. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  k num_k +

where num_k is the number of nodes which are present in the first k positions +of both rankings. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu27.html b/doc/html/mammult_docsu27.html new file mode 100644 index 0000000..2a20dd4 --- /dev/null +++ b/doc/html/mammult_docsu27.html @@ -0,0 +1,52 @@ + + +Interlayer degree correlation coefficients + + + + + + + + +

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+

1.2.2 Interlayer degree correlation coefficients

+

This section includes programs for the computation of various inter-layer degree +correlation coefficients. +

+

+    compute_pearson.py +
   compute_rho.py +
   compute_tau.py +
+ + + + + +

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+

+ diff --git a/doc/html/mammult_docsu28.html b/doc/html/mammult_docsu28.html new file mode 100644 index 0000000..05c2082 --- /dev/null +++ b/doc/html/mammult_docsu28.html @@ -0,0 +1,96 @@ + + +1.2.2.0 compute_pearson.py + + + + + + + + +

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+
compute_pearson.py
+

NAME +

compute_pearson.py - compute the Pearson’s linear correlation coefficient +between two node properties. +

SYNOPSYS +

compute_pearson.py <file1> <file2> +

DESCRIPTION +

Compute the Pearson’s linear correlation coefficient between two sets of +(either integer- or real-valued) node properties provided in the input files +file1 and file2. Each input file contains a list of lines, where the n-th line +contains the value of a node property for the n-th node. For instance, file1 +and file2 might contain the degrees of nodes at two distinct layers of a +multiplex. However, the program is pretty general and can be used to +compute the Pearson’s correlation coeffcient between any pairs of node +properties. +

OUTPUT +

The program prints on stdout the value of the Pearson’s linear correlation +coefficient between the two sets of node properties. +

REFERENCE + + + +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu29.html b/doc/html/mammult_docsu29.html new file mode 100644 index 0000000..973c83d --- /dev/null +++ b/doc/html/mammult_docsu29.html @@ -0,0 +1,100 @@ + + +1.2.2.0 compute_rho.py + + + + + + + + +

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+
compute_rho.py
+

NAME +

compute_rho.py - compute the Spearman’s rank correlation coefficient ρ +between two rankings. +

SYNOPSYS +

compute_rho.py <file1> <file2> +

DESCRIPTION +

Compute the Spearman’s rank correlation coefficient ρ between two rankings +provided in the input files file1 and file2. Each input file contains a list of lines, +where the n-th line contains the value of rank of the n-th node. For instance, file1 +and file2 might contain the ranks of nodes induced by the degree sequences of +two distinct layers of a multiplex. +

However, the program is pretty general and can be used to compute +the Spearman’s rank correlation coefficient between any generic pair of +rankings. +

N.B.: A C implementation of this program, with the same interface is also +available in the executable file compute_rho. +

OUTPUT +

The program prints on stdout the value of the Spearman’s rank correlation +coefficient ρ between the two rankings provided as input. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu3.html b/doc/html/mammult_docsu3.html new file mode 100644 index 0000000..cd582bb --- /dev/null +++ b/doc/html/mammult_docsu3.html @@ -0,0 +1,87 @@ + + +1.1.1.0 layer_activity.py + + + + + + + + +

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+
layer_activity.py
+

NAME +

layer_activity.py - compute the activity of the layers of a multiplex, i.e. the +number of active nodes on each layer. +

SYNOPSYS +

layer_activity.py <layer1> [<layer2> ...] +

DESCRIPTION +

Compute and print on output the activity of the layers of a multiplex +network, where the layers are given as input in the files layer1, layer2, +etc. +

Each file contains the (undirected) edge list of a layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

A listof lines, where the n-th line is the value of activity of the n-th layer, +starting from 0. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu30.html b/doc/html/mammult_docsu30.html new file mode 100644 index 0000000..c743f7b --- /dev/null +++ b/doc/html/mammult_docsu30.html @@ -0,0 +1,100 @@ + + +1.2.2.0 compute_tau.py + + + + + + + + +

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+
compute_tau.py
+

NAME +

compute_tau.py - compute the Kendall’s rank correlation coefficient τb +between two rankings. +

SYNOPSYS +

compute_tau.py <file1> <file2> +

DESCRIPTION +

Compute the Kendall’s rank correlation coefficient τb between two rankings +provided in the input files file1 and file2. Each input file contains a list of lines, +where the n-th line contains the value of rank of the n-th node. For instance, file1 +and file2 might contain the ranks of nodes induced by the degree sequences of +two distinct layers of a multiplex. +

However, the program is pretty general and can be used to compute +the Kendall’s rank correlation coefficient between any generic pair of +rankings. +

N.B.: This implementation takes properly into account rank ties. +

OUTPUT +

The program prints on stdout the value of the Kendall’s rank correlation +coefficient τb between the two rankings provided as input. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu31.html b/doc/html/mammult_docsu31.html new file mode 100644 index 0000000..50ae7f8 --- /dev/null +++ b/doc/html/mammult_docsu31.html @@ -0,0 +1,62 @@ + + +Interlayer degree correlation functions + + + + + + + + +

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+

1.2.3 Interlayer degree correlation functions

+

This section includes programs to compute intra-layer and inter-layer degree +correlation functions, and to fit those functions with a power-law. +

M +

+    dump_k_q +
   knn_q_from_layers.py +
   knn_q_from_degrees.py +
   fit_knn +
+ + + + + + +

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+

+ diff --git a/doc/html/mammult_docsu32.html b/doc/html/mammult_docsu32.html new file mode 100644 index 0000000..bfda1c1 --- /dev/null +++ b/doc/html/mammult_docsu32.html @@ -0,0 +1,123 @@ + + +1.2.3.0 dump_k_q + + + + + + + + +

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+
dump_k_q
+

NAME +

dump_k_q - compute the degree sequences of two layers of a multiplex. +

SYNOPSYS +

dump_k_q <layer1> <layer2> <pairing> +

DESCRIPTION +

Compute and dump on stdout the degree sequences of two layers of a +multiplex. The input files layer1 and layer2 contain the (undirected) edge lists of +the two layers, and each line is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The third file pairing is a list of lines in the format: +

  IDL1 IDL2 +

where IDL1 is the ID of a node on layer 1 and IDL2 is the ID of the same +node on layer 2. For instance, the line: +

  5 27 +

indicates that node 5 on layer 1 has ID 27 on layer 2. +

OUTPUT + + + +

The program prints on stdout the degree of each node on the two layers, in +the format: +

  ki qi +

where ki is the degree of node i on layer 1 and qi is the degree of node i on +layer 2. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu33.html b/doc/html/mammult_docsu33.html new file mode 100644 index 0000000..d9b78dd --- /dev/null +++ b/doc/html/mammult_docsu33.html @@ -0,0 +1,233 @@ + + +1.2.3.0 knn_q_from_layers.py + + + + + + + + +

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+
knn_q_from_layers.py
+

NAME +

knn_q_from_layers.py - compute intra-layer and inter-layer degree-degree +correlation coefficients. +

SYNOPSYS +

knn_q_from_layers.py <layer1> <layer2> +

DESCRIPTION +

Compute the intra-layer and the inter-layer degree correlation functions for +two layers given as input. The intra-layer degree correlation function quantifies +the presence of degree-degree correlations in a single layer network, and is defined +as: +
+
+ --1- ∑ ′ ′ +⟨knn(k)⟩ = kNk k P(k |k ) + k′ +
+

+

where P(k′|k) is the probability that a neighbour of a node with degree k has +degree k′, and Nk is the number of nodes with degree k. The quantity ⟨knn(k)⟩ is +the average degree of the neighbours of nodes having degree equal to +k. +

If we consider two layers of a multiplex, and we denote by k the degree +of a node on the first layer and by q the degree of the same node on +the second layers, the inter-layer degree correlation function is defined +as +
+ + + +
+-- ∑ ′ ′ +k(q) = k P(k |q) + k′ +
+

+

where P(k′|q) is the probability that a node with degree q on the second layer +has degree equal to k′ on the first layer, and Nq is the number of nodes +with degree q on the second layer. The quantity k(q) is the expected +degree at layer 1 of node that have degree equal to q on layer 2. The dual +quantity: +
+
+-- ∑ ′ ′ +q(k) = q P(q |k) + q′ +
+

+

is the average degree on layer 2 of nodes having degree k on layer +1. +

OUTPUT +

The program creates two output files, respectively called +

  file1_file2_k1 +

and +

  file1_file2_k2 +

The first file contains a list of lines in the format: +

  k ⟨knn(k)⟩ σk q(k) σq +

where k is the degree at first layer, ⟨knn(k)⟩ is the average degree of the +neighbours at layer 1 of nodes having degree k at layer 1, σk is the standard +deviation associated to ⟨knn(k)⟩, q(k) is the average degree at layer 2 of nodes + + + +having degree equal to k at layer 1, and σq is the standard deviation associated +to q(k). +

The second file contains a similar list of lines, in the format: +

  q ⟨qnn(q)⟩ σq k(q) σk +

with obvious meaning. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu34.html b/doc/html/mammult_docsu34.html new file mode 100644 index 0000000..ccd8b5a --- /dev/null +++ b/doc/html/mammult_docsu34.html @@ -0,0 +1,164 @@ + + +1.2.3.0 knn_q_from_degrees.py + + + + + + + + +

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+
knn_q_from_degrees.py
+

NAME +

knn_q_from_degrees.py - compute the inter-layer degree-degree +correlation function. +

SYNOPSYS +

knn_q_from_degrees.py <filein> +

DESCRIPTION +

Compute the inter-layer degree correlation functions for two layers of a +multiplex, using the degrees of the nodes specified in the input file. The format of +the input file is as follows +

  ki qi +

where ki and qi are, respectively, the degree at layer 1 and the degree at layer +2 of node i. +

If we consider two layers of a multiplex, and we denote by k the degree +of a node on the first layer and by q the degree of the same node on +the second layers, the inter-layer degree correlation function is defined +as +
+
+k(q) = -1-∑ k′P (k′|q) + Nk ′ + k +
+

+

where P(k′|q) is the probability that a node with degree q on the second +layer has degree equal to k′ on the first layer, and Nk is the number of +nodes with degree k on the first layer. The quantity k(q) is the expected +degree at layer 1 of node that have degree equal to q on layer 2. The dual + + + +quantity: +
+
+ ∑ +q(k) = -1- q′P (q′|k) + Nq q′ +
+

+

is the average degree on layer 2 of nodes having degree k on layer +1. +

OUTPUT +

The program prints on stdout a list of lines in the format: +

  k q(k) +

where k is the degree on layer 1 and q(k) is the average degree on layer 2 of +nodes having degree equal to k on layer 1. +

The program also prints on stderr a list of lines in the format: +

  q k(q) +

where q is the degree on layer 2 and k(q) is the average degree on layer 1 of +nodes having degree equal to q on layer 2. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu35.html b/doc/html/mammult_docsu35.html new file mode 100644 index 0000000..7bac046 --- /dev/null +++ b/doc/html/mammult_docsu35.html @@ -0,0 +1,140 @@ + + +1.2.3.0 fit_knn + + + + + + + + +

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+
fit_knn
+

NAME +

fit_knn - power-law fit of the inter-layer degree correlation function. +

SYNOPSYS +

fit_knn <filein> <alpha> +

DESCRIPTION +

Perform a power-law fit of the inter-layer degree correlation function: +
+
+-- 1 ∑ +q(k) = --- q′P (q′|k) + Nq q′ +
+

+

where k is the degree of a node on layer 1, q is the degree on layer 2 and +P(q|k) is the probability that a node with degree k on layer 1 has degree q on +layer 2. The program assumes that q(k) can be written in the form akb, and +computes the two parameters a and b through a linear fit of the log-log plot of +q(k). +

The input file filein contains a list of lines in the format: +

  ki qi +

where ki is the degree of node i at layer 1 and qi is the degree of node i at +layer 2. +

The second parameter alpha is the ratio of the progression used to generate +the exponentially-distributed bins for the log-log plot. Typical values of alpha are +between 1.1 and 2.0. +

N.B.: The exponent b computed with this method is known to be +inaccurate. + + + +

OUTPUT +

The program prints on stdout the values of the parameters a and b of the +power-law fit q(k) = akb. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and +condensation in multiplex networks”, Phys. Rev. E 90, 042807 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807 + + + +

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+

+ diff --git a/doc/html/mammult_docsu36.html b/doc/html/mammult_docsu36.html new file mode 100644 index 0000000..a1f0f0f --- /dev/null +++ b/doc/html/mammult_docsu36.html @@ -0,0 +1,56 @@ + + +Null-models of node and layer activity + + + + + + + + +

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+

2.1.1 Null-models of node and layer activity

+

+

+    model_hypergeometric.py +
   model_MDM.py +
   model_MSM.py +
   model_layer_growth.py +
+ + + + + + +

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+

+ diff --git a/doc/html/mammult_docsu37.html b/doc/html/mammult_docsu37.html new file mode 100644 index 0000000..7f93983 --- /dev/null +++ b/doc/html/mammult_docsu37.html @@ -0,0 +1,98 @@ + + +2.1.1.0 model_hypergeometric.py + + + + + + + + + + + +

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+
model_hypergeometric.py
+

NAME +

model_hypergeometric.py - Hypergeometric node activity null +model. +

SYNOPSYS +

model_hypergeometric.py <layer_N_file> <N> +

DESCRIPTION +

This is the hypergeometric model of node activation. In this model each layer +has exactly the same number of active node of a reference multiplex network, but +nodes on each layer are activated uniformly at random, thus destroying all +inter-layer activity correlation patterns. +

The file layer_N_file reports on the n-th line the number of active nodes on +the n-th layer (starting from zero). The second parameter N is the total number +of active nodes in the multiplex. +

OUTPUT +

The program prints on stdout a node-layer list of lines in the format: +

  node_i layer_i +

where node_i is the ID of a node and layre_i is the ID of a layer. This +list indicates which nodes are active in which layer. For instance, the +line: +

  24 3 +

indicates that the node with ID 24 is active on layer 3. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu38.html b/doc/html/mammult_docsu38.html new file mode 100644 index 0000000..1f1e918 --- /dev/null +++ b/doc/html/mammult_docsu38.html @@ -0,0 +1,106 @@ + + +2.1.1.0 model_MDM.py + + + + + + + + +

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+
model_MDM.py
+

NAME +

model_MDM.py - Multi-activity Deterministic Model. +

SYNOPSYS +

model_MDM.py <Bi_file> <M> +

DESCRIPTION +

This is the Multi-activity Deterministic Model (MDM). In this model each +node i is considered active if it was active in the reference multiplex, maintains +the same value of node activity Bi (i.e., the number of layers in which it was +active) and is associated an activity vector sampled uniformly at random from +the (M + Bi) + possible activity vectors with Bi non-null entries. +

The file Bi_file is in the format: +

  Bi N(Bi) +

where Bi is a value of node activity and N(Bi) is the number of nodes which +had node activity equaly to Bi in the reference multiplex. +

The parameter M is the number of layers in the multiplex. +

OUTPUT +

The program prints on stdout a distribution of bit-strings, in the +format: +

  Bi bitstring count +

where bitstring is the activity bitstring, Bi is the number of non-zero entries +of bitstring and count is the number of times that bitstrings appear in the null +model. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu39.html b/doc/html/mammult_docsu39.html new file mode 100644 index 0000000..a372150 --- /dev/null +++ b/doc/html/mammult_docsu39.html @@ -0,0 +1,107 @@ + + +2.1.1.0 model_MSM.py + + + + + + + + +

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+
model_MSM.py
+

NAME +

model_MSM.py - Multi-activity Stochastic Model. +

SYNOPSYS +

model_MSM.py <node_Bi_file> <M> +

DESCRIPTION +

This is the Multi-activity Stochastic Model (MSM). In this model each node i +is considered active if it was active in the reference multiplex, and is activated on +each layer with a probability equal to Bi∕M where Bi was the activity of node i +in the reference multiplex. +

The file node_Bi_file is in the format: +

  node_i Bi) +

where Bi is the value of node activity of node_i in the reference multiplex. +

The parameter M is the number of layers in the multiplex. +

OUTPUT +

The program prints on stdout a node-layer list of lines in the format: +

  node_i layer_i +

where node_i is the ID of a node and layre_i is the ID of a layer. This +list indicates which nodes are active in which layer. For instance, the +line: +

  24 3 +

indicates that the node with ID 24 is active on layer 3. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu4.html b/doc/html/mammult_docsu4.html new file mode 100644 index 0000000..bfd3aa7 --- /dev/null +++ b/doc/html/mammult_docsu4.html @@ -0,0 +1,96 @@ + + +1.1.1.0 node_activity_vectors.py + + + + + + + + +

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+
node_activity_vectors.py
+

NAME +

node_activity_vectors.py - compute the activity vectors of all the nodes of +a multiplex. +

SYNOPSYS +

node_activity_vectors.py <layer1> [<layer2> ...] +

DESCRIPTION +

Compute and print on output the activity vectors of the nodes of a +multiplex network, whose layers are given as input in the files layer1, layer2, +etc. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines, where the n-th line contains +the activity vector of the n-th node, i.e. a bit-string where each bit is +set to “1” if the node is active on the corresponding layer, and to “0” +otherwise. +

As usual, node IDs start from zero and proceed sequentially, without gaps, i.e., if +a node ID is not present in any of the layer files given as input, the program +considers it as being isolated on all the layers, and will print on output a +bit-string of zeros. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu40.html b/doc/html/mammult_docsu40.html new file mode 100644 index 0000000..5edece7 --- /dev/null +++ b/doc/html/mammult_docsu40.html @@ -0,0 +1,138 @@ + + +2.1.1.0 model_layer_growth.py + + + + + + + + +

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+
model_layer_growth.py
+

NAME +

model_layer_growth.py - Layer growth with preferential activation +model. +

SYNOPSYS +

model_layer_growth.py <layer_N_file> <N> <M0> <A> +[RND] +

DESCRIPTION +

This is the model of layer growth with preferential node activation. In this +model an entire new layer arrives at time t and a number of nodes Nt is +activated (N_t is equal to the number of nodes active on that layer in the +reference multiplex). Then, each node i of the new layer is activated with a +probability: +
+
+Pi(t) ∝ A + Bi (t) +
+

+

where Bi(t) is the activity of node i at time t (i.e., the number of +layers in which node i is active at time t) while A > 0 is an intrinsic +attractiveness. +

The file layer_N_file reports on the n-th line the number of active nodes on +the n-th layer. +

The parameter N is the number of nodes in the multiplex, M0 is the number +of layers in the initial network, A is the value of node attractiveness. +

If the user specifies RND as the last parameter, the sequence of layers +is + + + +

OUTPUT +

The program prints on stdout a node-layer list of lines in the format: +

  node_i layer_i +

where node_i is the ID of a node and layre_i is the ID of a layer. This +list indicates which nodes are active in which layer. For instance, the +line: +

  24 3 +

indicates that the node with ID 24 is active on layer 3. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu41.html b/doc/html/mammult_docsu41.html new file mode 100644 index 0000000..babf303 --- /dev/null +++ b/doc/html/mammult_docsu41.html @@ -0,0 +1,61 @@ + + +Linear preferential attachment + + + + + + + + +

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+

2.2.1 Linear preferential attachment

+

+

+    nibilab_linear_delta +
   nibilab_linear_delay +
   nibilab_linear_delay_mix +
   nibilab_linear_random_times +
+ + + + + + +

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+

+ diff --git a/doc/html/mammult_docsu42.html b/doc/html/mammult_docsu42.html new file mode 100644 index 0000000..5727bf0 --- /dev/null +++ b/doc/html/mammult_docsu42.html @@ -0,0 +1,145 @@ + + +2.2.1.0 nibilab_linear_delta + + + + + + + + +

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+
nibilab_linear_delta
+

NAME +

nibilab_linear_delta - Multiplex linear preferential attachment model – +Synchronous arrival. +

SYNOPSYS + + + +

nibilab_linear_delta <N> <m> <m0> <outfile> <a> <b> <c> +<d> +

DESCRIPTION +

Grow a two-layer multiplex network using the multiplex linear preferential +attachment model by Nicosia, Bianconi, Latora, Barthelemy (NiBiLaB). +

The probability for a newly arrived node i to create a link to node j on layer +1 is: +
+
+Π1i→j ∝ ak[1j]+ bk[j2] +
+

+

and the dual probability for i to create a link to j on layer 2 is: +
+
+Π2 ∝ ck [1]+ dk[2] + i→j j j +
+

+

Each new node arrives at the same time on both layers. +

The (mandatory) parameters are as follows: +

+

OUTPUT +

The program dumps on the file outfile the (undirected) edge list of the +resulting network. Each line of the file is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu43.html b/doc/html/mammult_docsu43.html new file mode 100644 index 0000000..d88d970 --- /dev/null +++ b/doc/html/mammult_docsu43.html @@ -0,0 +1,158 @@ + + +2.2.1.0 nibilab_linear_delay + + + + + + + + +

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+
nibilab_linear_delay
+

NAME +

nibilab_linear_delay - Multiplex linear preferential attachment model – +Asynchronous arrival. +

SYNOPSYS +

nibilab_linear_delay <N> <m> <m0> <outfile> <a> <b> <c> +<d> <beta> +

DESCRIPTION +

Grow a two-layer multiplex network using the multiplex linear preferential +attachment model by Nicosia, Bianconi, Latora, Barthelemy (NiBiLaB). +

The probability for a newly arrived node i to create a link to node j on layer +1 is: +
+
+ 1 [1] [2] +Π i→j ∝ akj + bkj +
+

+

and the dual probability for i to create a link to j on layer 2 is: +
+
+ 2 [1] [2] +Π i→j ∝ ck j + dkj +
+

+

Each new node arrives first on layer 1, and its replica on the layer 2 appears +after a time delay τ sampled from the power-law function: +
+
+P(τ) ∼ τ−β +
+

+

The (mandatory) parameters are as follows: +

+ + + +

OUTPUT +

The program dumps on the file outfile the (undirected) edge list of the +resulting network. Each line of the file is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu44.html b/doc/html/mammult_docsu44.html new file mode 100644 index 0000000..2303b84 --- /dev/null +++ b/doc/html/mammult_docsu44.html @@ -0,0 +1,163 @@ + + +2.2.1.0 nibilab_linear_delay_mix + + + + + + + + +

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+
nibilab_linear_delay_mix
+

NAME +

nibilab_linear_delay_mix - Multiplex linear preferential attachment model +– Asynchronous arrival and randomly selected first layer. +

SYNOPSYS +

nibilab_linear_delay_mix <N> <m> <m0> <outfile> <a> <b> +<c> <d> <beta> +

DESCRIPTION +

Grow a two-layer multiplex network using the multiplex linear preferential +attachment model by Nicosia, Bianconi, Latora, Barthelemy (NiBiLaB). +

The probability for a newly arrived node i to create a link to node j on layer +1 is: +
+
+ 1 [1] [2] +Π i→j ∝ akj + bkj +
+

+

and the dual probability for i to create a link to j on layer 2 is: +
+
+ 2 [1] [2] +Π i→j ∝ ck j + dkj +
+

+

Each new node arrives on one of the two layers, chosen uniformly at random, +and its replica on the other layer appears after a time delay τ sampled from the +power-law function: +
+
+ −β +P(τ) ∼ τ +
+

+

The (mandatory) parameters are as follows: +

+ + + +

OUTPUT +

The program dumps on the file outfile the (undirected) edge list of the +resulting network. Each line of the file is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu45.html b/doc/html/mammult_docsu45.html new file mode 100644 index 0000000..28adf3f --- /dev/null +++ b/doc/html/mammult_docsu45.html @@ -0,0 +1,146 @@ + + +2.2.1.0 nibilab_linear_random_times + + + + + + + + +

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+
nibilab_linear_random_times
+

NAME +

nibilab_linear_random_times - Multiplex linear preferential attachment +model – Asynchronous arrival with randomly sampled arrival times on layer +2. +

SYNOPSYS +

nibilab_linear_random_times <N> <m> <m0> <outfile> <a> <b> +<c> <d> +

DESCRIPTION +

Grow a two-layer multiplex network using the multiplex linear preferential +attachment model by Nicosia, Bianconi, Latora, Barthelemy (NiBiLaB). +

The probability for a newly arrived node i to create a link to node j on layer +1 is: +
+
+Π1i→j ∝ ak[1j]+ bk[j2] +
+

+

and the dual probability for i to create a link to j on layer 2 is: +
+
+ 2 [1] [2] +Π i→j ∝ ck j + dkj +
+

+

Each new node arrives on layer 1, but its replica on the other layer appears at +a uniformly chosen random time in [m0 + 1;N]. +

The (mandatory) parameters are as follows: +

+

OUTPUT +

The program dumps on the file outfile the (undirected) edge list of the +resulting network. Each line of the file is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu46.html b/doc/html/mammult_docsu46.html new file mode 100644 index 0000000..e3e6edc --- /dev/null +++ b/doc/html/mammult_docsu46.html @@ -0,0 +1,40 @@ + + +Non-linear preferential attachment + + + + + + + + +

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+

2.2.2 Non-linear preferential attachment

+

+

+    nibilab_nonlinear +
+ + + +

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+

+ diff --git a/doc/html/mammult_docsu47.html b/doc/html/mammult_docsu47.html new file mode 100644 index 0000000..d3d1192 --- /dev/null +++ b/doc/html/mammult_docsu47.html @@ -0,0 +1,142 @@ + + +2.2.2.0 nibilab_nonlinear + + + + + + + + +

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+
nibilab_nonlinear
+

NAME +

nibilab_nonlinear - Multiplex non-linear preferential attachment model – +Synchronous arrival. +

SYNOPSYS +

nibilab_nonlinear <N> <m> <m0> <outfile> <alpha> <beta> +

DESCRIPTION +

Grow a two-layer multiplex network using the multiplex non-linear +preferential attachment model by Nicosia, Bianconi, Latora, Barthelemy +(NiBiLaB). +

The probability for a newly arrived node i to create a link to node j on layer +1 is: +
+
+ ( [1])α + 1 kj +Πi→j ∝ (-[2])β- + kj +
+ + + +

+

and the dual probability for i to create a link to j on layer 2 is: +
+
+ ( )α + k[2j] +Π2i→j ∝ (---)β- + k[1j] +
+

+

Each node arrives simultaneously on both layers. +

The (mandatory) parameters are as follows: +

+

OUTPUT +

The program dumps on the file outfile the (undirected) edge list of the +resulting network. Each line of the file is in the format: +

  src_ID dest_ID + + + +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu48.html b/doc/html/mammult_docsu48.html new file mode 100644 index 0000000..d593bbd --- /dev/null +++ b/doc/html/mammult_docsu48.html @@ -0,0 +1,42 @@ + + +Utilities + + + + + + + + +

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+

2.2.3 Utilities

+

+

+    node_deg_over_time.py +
+ + + +

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+

+ diff --git a/doc/html/mammult_docsu49.html b/doc/html/mammult_docsu49.html new file mode 100644 index 0000000..fbe3f0c --- /dev/null +++ b/doc/html/mammult_docsu49.html @@ -0,0 +1,145 @@ + + +2.2.3.0 node_deg_over_time.py + + + + + + + + +

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+
node_deg_over_time.py
+

NAME +

node_deg_over_time.py - Time evolution of the degree of a node in a +growing graph. +

SYNOPSYS +

node_deg_over_time.py <layer> <arrival_times> <node_id> +[<node_id> ...] +

DESCRIPTION +

Compute the degree ki(t) of node i in a growing network as a function of +time. The file layer contains the edge list of the final network. Each line of the +file is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The file arrival_times is a list of node arrival times, in the format: +

  time_i node_i +

where time_i is the time at which node_i arrived in the graph. Notice that +time_i must be an integer in the range [0, N-1], where N is the total number of +nodes in the final graph. +

The third parameter node_id is the ID of the node whose degree over time +will be printed on output. If more than one node_id is provided, the degrees over +time of all the corresponding nodes are printed on output. +

OUTPUT +

The program prints on stdout a list of lines in the format: + + + +

  t kit +

where kit is the degree of node i at time t. The first line of output is in the +format: +

  #### node_id +

where node_id is the ID of node i. +

If more than one node_ids is provided as input, the program prints the degree +over time of all of them, sequentially. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 + + + +

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+

+ diff --git a/doc/html/mammult_docsu5.html b/doc/html/mammult_docsu5.html new file mode 100644 index 0000000..88c4386 --- /dev/null +++ b/doc/html/mammult_docsu5.html @@ -0,0 +1,95 @@ + + +1.1.1.0 layer_activity_vectors.py + + + + + + + + +

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+
layer_activity_vectors.py
+

NAME +

layer_activity_vectors.py - compute the activity vectors of all the layers of +a multiplex. +

SYNOPSYS +

layer_activity_vectors.py <layer1> [<layer2> ...] +

DESCRIPTION +

Compute and print on output the activity vectors of the layers of a multiplex +network, where the layers are given as input in the files layer1, layer2, +etc. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines, where the n-th line contains the +activity vector of the n-th layer, i.e. a bit-string where each bit is set to +“1” if the corresponding node is active on the n-th layer, and to “0” +otherwise. +

As usual, node IDs start from zero and proceed sequentially, without gaps, i.e., if +a node ID is not present in any of the layer files given as input, the program +considers it as being isolated on all the layers. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu50.html b/doc/html/mammult_docsu50.html new file mode 100644 index 0000000..693e7d9 --- /dev/null +++ b/doc/html/mammult_docsu50.html @@ -0,0 +1,46 @@ + + +Models based on simulated annealing + + + + + + + + +

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+

2.3.1 Models based on simulated annealing

+

+

+    tune_rho +
   tune_qnn_adaptive +
+ + + + +

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+

+ diff --git a/doc/html/mammult_docsu51.html b/doc/html/mammult_docsu51.html new file mode 100644 index 0000000..8eab351 --- /dev/null +++ b/doc/html/mammult_docsu51.html @@ -0,0 +1,113 @@ + + +2.3.1.0 tune_rho + + + + + + + + +

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+
tune_rho
+

NAME +

tune_rho - Construct a multiplex with prescribed inter-layer correlations. +

SYNOPSYS +

tune_rho <rank1> <rank2> <rho> <eps> <beta> [RND|NAT|INV] +

DESCRIPTION +

This programs tunes the inter-layer degree correlation coefficient ρ +(Spearman’s rank correlation) of two layers, by adjusting the inter-layer pairing +of nodes. The files rank1 and rank2 are the rankings of nodes in the first and +second layer, where the n-th line of the file contains the rank of the n-th node +(the highest ranked node has rank equal to 1). + + + +

The parameter rho is the desired value of the Spearman’s rank correlation +coefficient, while eps is the accuracy of rho. For instance, if rho is set equal to +-0.25 and eps is equal to 0.0001, the program stops when the configuration of +node pairing corresponds to a value of ρ which differs from -0.25 by less than +0.0001. +

The parameter beta is the typical inverse temperature of simulated +annealing. +

If no other parameter is specified, or if the last parameter is RND, the program +starts from a random pairing of nodes. If the last parameter is NAT then the +program assumes that the initial pairing is the natural one, where the nodes have +the same ID on both layers. Finally, if INV is specified, the initial pairing is the +inverse pairing, i.e. the one where node 0 on layer 1 is paired with node N-1 on +layer 2, and so on. +

OUTPUT +

The program prints on stdout a pairing, i.e. a list of lines in the +format: +

  IDL1 IDL2 +

where IDL1 is the ID of the node on layer 1 and IDL2 is the corresponding +ID of the same node on layer 2. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu52.html b/doc/html/mammult_docsu52.html new file mode 100644 index 0000000..b0e0444 --- /dev/null +++ b/doc/html/mammult_docsu52.html @@ -0,0 +1,148 @@ + + +2.3.1.0 tune_qnn_adaptive + + + + + + + + +

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+
tune_qnn_adaptive
+

NAME +

tune_qnn_adaptive - Construct a multiplex with prescribed inter-layer +correlations. +

SYNOPSYS +

tune_qnn_adaptive <degs1> <degs2> <mu> <eps> <beta> +[RND|NAT|INV] +

DESCRIPTION +

This programs tunes the inter-layer degree correlation exponent μ. If +we consider two layers of a multiplex, and we denote by k the degree +of a node on the first layer and by q the degree of the same node on +the second layers, the inter-layer degree correlation function is defined +as: +
+
+-- ∑ ′ ′ +q(k) = q P(q |k) + q′ +
+

+

where q(k) is the average degree on layer 2 of nodes having degree k on layer +1. +

The program assumes that we want to set the degree correlation function +such that: +
+ + + +
+q(k) = akμ +
+

+

where the exponent of the power-law function is given by the user +(it is indeed the parameter mu), and successively adjusts the pairing +between nodes at the two layers in order to obtain a correlation function as +close as possible to the desired one. The files degs1 and degs2 contain, +respectively, the degrees of the nodes on the first layer and on the second +layer. +

The parameter eps is the accuracy of mu. For instance, if mu is set equal to +-0.25 and eps is equal to 0.0001, the program stops when the configuration of +node pairing corresponds to a value of the exponent μ which differs from -0.25 by +less than 0.0001. +

The parameter beta is the typical inverse temperature of simulated +annealing. +

If no other parameter is specified, or if the last parameter is RND, the program +starts from a random pairing of nodes. If the last parameter is NAT then the +program assumes that the initial pairing is the natural one, where the nodes have +the same ID on both layers. Finally, if INV is specified, the initial pairing is the +inverse pairing, i.e. the one where node 0 on layer 1 is paired with node N-1 on +layer 2, and so on. +

OUTPUT +

The program prints on stdout a pairing, i.e. a list of lines in the +format: +

  IDL1 IDL2 +

where IDL1 is the ID of the node on layer 1 and IDL2 is the corresponding +ID of the same node on layer 2. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu53.html b/doc/html/mammult_docsu53.html new file mode 100644 index 0000000..bf48733 --- /dev/null +++ b/doc/html/mammult_docsu53.html @@ -0,0 +1,123 @@ + + +3.1.0.0 multiplex_ising + + + + + + + + +

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+
multiplex_ising
+

NAME +

multiplex_ising - compute the coupled ising model in a multiplex with 2 +layers. +

SYNOPSYS +

multiplex_ising <layer1> <layer2> <T> <J> < γ > < h[1] > +< h[2] > < p1 > < p2 > < numepochs > +

DESCRIPTION +

Compute and print the output of the ising dynamics on two coupled layers of +a multiplex network. Files layer1, layer2, contain the (undirected) edge list of the +two layer, and each line is in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

T is the value of thermal noise in the system, J the value of peer pressure, γ +the relative ratio between internal coupling and peer pressure, h[1] and h[2] the +external fields acting on the two layers, p1 the probability for a spin on layer 1 at +t = 0 to be up, p2 the same probability for spins on layer 2, numepochs the + + + +number of epochs for the simulation. +

OUTPUT +

One line, reporting all controlling parameter, the value of consensus in +layer 1 m[1], the value of consensus in layer 2 m[2] and the coherence +C. +

REFERENCE +

F. Battiston, A. Cairoli, V. Nicosia, A. Baule, V. Latora, “Interplay between +consensus and coherence in a model of interacting opinions”, accepted for +publication in Physica D, arxiv:1506.04544 (2015). +

Link to paper: http://arxiv.org/abs/1506.04544 + + + +

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+

+ diff --git a/doc/html/mammult_docsu54.html b/doc/html/mammult_docsu54.html new file mode 100644 index 0000000..ecdc0ad --- /dev/null +++ b/doc/html/mammult_docsu54.html @@ -0,0 +1,39 @@ + + +Stationary distribution + + + + + + + + +

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+

3.2.1 Stationary distribution

+

+

+    statdistr2 +
+ + + +

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+

+ diff --git a/doc/html/mammult_docsu55.html b/doc/html/mammult_docsu55.html new file mode 100644 index 0000000..dbe980d --- /dev/null +++ b/doc/html/mammult_docsu55.html @@ -0,0 +1,121 @@ + + +3.2.1.0 statdistr2 + + + + + + + + +

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+
statdistr2
+

NAME +

statdistr2 - compute the stationary distribution of additive, multiplicative +and intensive biased walks in a multiplex with 2 layers. +

SYNOPSYS +

statdistr2 <layer1> <layer2> < overlappingnetwork > <N> b1 +b2 +

DESCRIPTION + + + +

Compute and print the stationary distribution of additive, multiplicative and +intensive biased walks in a multiplex with 2 layers. Files layer1, layer2, +contain the (undirected) edge list of the two layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The file overlapping network has also a third column indicating the number of +times two nodes are connected across all layers. +

N is the number of nodes, b1 is the first bias exponent (the bias exponent +for layer 1 for additive and multiplicative walks, the bias exponent on +the participation coefficient for intensive walks), b2 is the second bias +exponent (the bias exponent for layer 1 for additive and multiplicative +walks, the bias exponent on the participation coefficient for intensive +walks). +

OUTPUT +

N lines. In the n-th line we report the node ID, the stationary distribution of +that node for additive walks with exponents b1 and b2, the stationary distribution +for multiplicative walks with exponents b1 and b2, the stationary distribution for +multiplicative walks with exponents b1 and b2, the values of the bias exponents b1 +and b2. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Biased random walks on multiplex +networks”, arxiv:1505.01378 (2015). +

Link to paper: http://arxiv.org/abs/1505.01378 + + + +

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+

+ diff --git a/doc/html/mammult_docsu56.html b/doc/html/mammult_docsu56.html new file mode 100644 index 0000000..b3ace9c --- /dev/null +++ b/doc/html/mammult_docsu56.html @@ -0,0 +1,43 @@ + + +Entropy rate + + + + + + + + +

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+

3.2.2 Entropy rate

+

+

+    entropyrate2add +
   entropyrate2mult +
   entropyrate2int +
+ +

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+

+ diff --git a/doc/html/mammult_docsu57.html b/doc/html/mammult_docsu57.html new file mode 100644 index 0000000..d236995 --- /dev/null +++ b/doc/html/mammult_docsu57.html @@ -0,0 +1,112 @@ + + +3.2.2.0 entropyrate2add + + + + + + + + +

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+
entropyrate2add
+

NAME +

entropyrate2add - compute the entropy rate of additive biased walks in a +multiplex with 2 layers. +

SYNOPSYS +

entropyrate2add <layer1> <layer2> < overlappingnetwork > <N> +b1 b2 +

DESCRIPTION +

Compute and print the entropy rate of an additive biased walk in a +multiplex with 2 layers and bias parameters b1 and b2. Files layer1, layer2, +contain the (undirected) edge list of the two layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The file overlapping network has also a third column indicating the number of +times two nodes are connected across all layers. +

N is the number of nodes, b1 is the degree-biased exponent for layer 1, b2 is +the degree-biased exponent for layer 2. +

OUTPUT +

One line, reporting the value of the entropy rate h of an additive biased +random walks with b1 and b2 as bias exponents, b1 and b2. + + + +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Biased random walks on multiplex +networks”, arxiv:1505.01378 (2015). +

Link to paper: http://arxiv.org/abs/1505.01378 + + + +

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+

+ diff --git a/doc/html/mammult_docsu58.html b/doc/html/mammult_docsu58.html new file mode 100644 index 0000000..6aeaf8c --- /dev/null +++ b/doc/html/mammult_docsu58.html @@ -0,0 +1,109 @@ + + +3.2.2.0 entropyrate2mult + + + + + + + + +

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+
entropyrate2mult
+

NAME +

entropyrate2mult - compute the entropy rate of multiplicative biased walks +in a multiplex with 2 layers. +

SYNOPSYS +

entropyrate2mult <layer1> <layer2> < overlappingnetwork > <N> +b1 b2 +

DESCRIPTION +

Compute and print the entropy rate of a multiplicative biased walk in a +multiplex with 2 layers and bias parameters b1 and b2. Files layer1, layer2, +contain the (undirected) edge list of the two layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The file overlapping network has also a third column indicating the number of +times two nodes are connected across all layers. +

N is the number of nodes, b1 is the degree-biased exponent for layer 1, b2 is +the degree-biased exponent for layer 2. +

OUTPUT +

One line, reporting the value of the entropy rate h of an multiplicative biased +random walks with b1 and b2 as bias exponents, b1 and b2. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Biased random walks on multiplex +networks”, arxiv:1505.01378 (2015). +

Link to paper: http://arxiv.org/abs/1505.01378 + + + +

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+

+ diff --git a/doc/html/mammult_docsu59.html b/doc/html/mammult_docsu59.html new file mode 100644 index 0000000..7baf3e6 --- /dev/null +++ b/doc/html/mammult_docsu59.html @@ -0,0 +1,103 @@ + + +3.2.2.0 entropyrate2int + + + + + + + + +

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+
entropyrate2int
+

NAME +

entropyrate2int - compute the entropy rate of intensive biased walks in a +multiplex with 2 layers. +

SYNOPSYS +

entropyrate2int <layer1> <layer2> < overlappingnetwork > <N> b1 +b2 +

DESCRIPTION +

Compute and print the entropy rate of an intensive biased walks in a +multiplex with 2 layers and bias parameters bp and bo. Files layer1, layer2, +contain the (undirected) edge list of the two layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

The file overlapping network has also a third column indicating the number of +times two nodes are connected across all layers. +

N is the number of nodes, bp is the biased exponent on the participation +coefficient, bo is the biased exponent on the overlapping degree. +

OUTPUT +

One line, reporting the value of the entropy rate h of an intensive biased +random walks with bp and bo as bias exponents, bp and bo. +

REFERENCE +

F. Battiston, V. Nicosia, V. Latora, “Biased random walks on multiplex +networks”, arxiv:1505.01378 (2015). +

Link to paper: http://arxiv.org/abs/1505.01378 +

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+

+ diff --git a/doc/html/mammult_docsu6.html b/doc/html/mammult_docsu6.html new file mode 100644 index 0000000..1c83cb0 --- /dev/null +++ b/doc/html/mammult_docsu6.html @@ -0,0 +1,101 @@ + + +1.1.1.0 multiplexity.py + + + + + + + + +

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+
multiplexity.py
+

NAME +

multiplexity.py - compute the pairwise multiplexity between all the pairs of +layers of a multiplex. +

SYNOPSYS +

multiplexity.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the pairwise multiplexity Qα,β (i.e., the +fraction of nodes active on both layers) between all pairs of layers. The layers are +given as input in the files layer1, layer2, etc. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines, in the format: +

  layer1 layer2 mult +

where layer1 and layer2 are the IDs of the layers, and mult is the value of the +multiplexity Qlayer1,layer2. Layers IDs start from zero, are are associated to the +layers in the same order in which the layer files are provided on the command +line. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu7.html b/doc/html/mammult_docsu7.html new file mode 100644 index 0000000..6d0949e --- /dev/null +++ b/doc/html/mammult_docsu7.html @@ -0,0 +1,105 @@ + + +1.1.1.0 hamming_dist.py + + + + + + + + +

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+
hamming_dist.py
+

NAME +

hamming_dist.py - compute the normalised Hamming distance between all +the pairs of layers of a multiplex. +

SYNOPSYS +

hamming_dist.py <layer1> <layer2> [<layer3>...] +

DESCRIPTION +

Compute and print on output the normalised Hamming distance Hα,β (i.e., +the fraction of nodes which are active on either of the layers, but not on both) +between all pairs of layers. The layers are given as input in the files layer1, +layer2, etc. +

Each input file contains the (undirected) edge list of a layer, and each line is +in the format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

The program prints on stdout a list of lines, in the format: +

  layer1 layer2 hamm +

where layer1 and layer2 are the IDs of the layers, and hamm is the value of the +normalised Haming distance Hlayer1,layer2. Layers IDs start from zero, are are +associated to the layers in the same order in which the layer files are provided on +the command line. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/html/mammult_docsu8.html b/doc/html/mammult_docsu8.html new file mode 100644 index 0000000..2bbfd90 --- /dev/null +++ b/doc/html/mammult_docsu8.html @@ -0,0 +1,109 @@ + + +1.1.1.0 node_degree_vectors.py + + + + + + + + +

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+
node_degree_vectors.py
+

NAME +

node_degree_vectors.py - compute the degree vectors of all the nodes of a +multiplex network +

SYNOPSYS +

node_degree_vectors.py <layer1> [<layer2> ...] +

DESCRIPTION +

Compute and print on output the degree vectors of all the nodes of a +multiplex network, whose layers are given as input in the files layer1, layer2, +etc. +

Each file contains the (undirected) edge list of a layer, and each line is in the +format: +

  src_ID dest_ID +

where src_ID and dest_ID are the IDs of the two endpoints of an +edge. +

OUTPUT +

A list of lines, where the n-th line is the vector of degrees of the n-th node, in +the format: +

  noden_deg_lay1 noden_deg_lay2 ... noden_deg_layM +

As usual, node IDs start from zero and proceed sequentially, without gaps, i.e., if +a node ID is not present in any of the layer files given as input, the program +considers it as being isolated on all the layers. +

REFERENCE +

V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex +networks”, Phys. Rev. Lett. 111, 058701 (2013). +

Link to paper: http://prl.aps.org/abstract/PRL/v111/i5/e058701 +

F. Battiston, V. Nicosia, V. Latora, “Structural measures for multiplex +networks”, Phys. Rev. E 89, 032804 (2014). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804 + + + +

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+

+ diff --git a/doc/html/mammult_docsu9.html b/doc/html/mammult_docsu9.html new file mode 100644 index 0000000..2f3a901 --- /dev/null +++ b/doc/html/mammult_docsu9.html @@ -0,0 +1,97 @@ + + +1.1.1.0 degs_to_binary.py + + + + + + + + +

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+
degs_to_binary.py
+

NAME +

degs_to_binary.py - compute the activity vectors of all the nodes of a +multiplex. +

SYNOPSYS +

degs_to_binary.py <degree_vectors> +

DESCRIPTION +

Take a file which contains, on the n-th line, the degrees at each layer of the +n-th node, (e.g., the result of the script node_degree_vectors.py), in the +format: +

  noden_deg_lay1 noden_deg_lay2 ... noden_deg_layM +

and compute the corresponding node activity bit-strings, where a ”1” +signals the presence of the node on that layer, while a zero indicates its +absence. +

OUTPUT +

The program returns on stdout a list of lines, where the n-th line is the +activity bit-string of the n-th node. Additionally, the program prints on stderr +the distribution of all activity bit-strings, in the format: +

  Bn Bit-string count +

Where B is the number of ones in the activity bit-string (i.e., the node-activity +associated to that activity bit-string), Bit-string is the activity bit-string and +count is the number of times that particular activity bit-string appears in the +multiplex. +

REFERENCE +

V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex +networks”, Phys. Rev. E 92, 032805 (2015). +

Link to paper: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805 + + + +

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+

+ diff --git a/doc/latex/latex/dynamics/Ising/multiplex_ising.tex b/doc/latex/latex/dynamics/Ising/multiplex_ising.tex new file mode 100644 index 0000000..ec87454 --- /dev/null +++ b/doc/latex/latex/dynamics/Ising/multiplex_ising.tex @@ -0,0 +1,22 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{multiplex\_ising}} + {compute the coupled ising model in a multiplex with $2$ layers.} + {$<$layer1$>$ $<$layer2$>$ $<$T$>$ $<$J$>$ $<\gamma>$ $$ $$ $$ $$ $$} + +\mydescription{Compute and print the output of the ising dynamics on two coupled layers of a multiplex network. + Files \textit{layer1}, \textit{layer2}, contain the (undirected) edge list of the two layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + $T$ is the value of thermal noise in the system, $J$ the value of peer pressure, $\gamma$ the relative ratio between internal coupling and peer pressure, $h^{[1]}$ and $h^{[2]}$ the external fields acting on the two layers, $p_1$ the probability for a spin on layer $1$ at $t=0$ to be up, $p_2$ the same probability for spins on layer $2$, $num epochs$ the number of epochs for the simulation.} + +\myreturn{One line, reporting all controlling parameter, the value of consensus in layer $1$ $m^{[1]}$, the value of consensus in layer $2$ $m^{[2]}$ and the coherence $C$.} + +\myreference{\refising} diff --git a/doc/latex/latex/dynamics/randomwalks/entropyrate2add.tex b/doc/latex/latex/dynamics/randomwalks/entropyrate2add.tex new file mode 100644 index 0000000..98a432b --- /dev/null +++ b/doc/latex/latex/dynamics/randomwalks/entropyrate2add.tex @@ -0,0 +1,24 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{entropyrate2add}} + {compute the entropy rate of additive biased walks in a multiplex with $2$ layers.} + {$<$layer1$>$ $<$layer2$>$ $$ $<$N$>$ $b_1$ $b_2$} + +\mydescription{Compute and print the entropy rate of an additive biased walk in a multiplex with $2$ layers and bias parameters $b_1$ and $b_2$. + Files \textit{layer1}, \textit{layer2}, contain the (undirected) edge list of the two layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{overlapping network} has also a third column indicating the number of times two nodes are connected across all layers. + + $N$ is the number of nodes, $b_1$ is the degree-biased exponent for layer $1$, $b_2$ is the degree-biased exponent for layer $2$.} + +\myreturn{One line, reporting the value of the entropy rate $h$ of an additive biased random walks with $b_1$ and $b_2$ as bias exponents, $b_1$ and $b_2$.} + +\myreference{\refbiased} diff --git a/doc/latex/latex/dynamics/randomwalks/entropyrate2int.tex b/doc/latex/latex/dynamics/randomwalks/entropyrate2int.tex new file mode 100644 index 0000000..8a337f4 --- /dev/null +++ b/doc/latex/latex/dynamics/randomwalks/entropyrate2int.tex @@ -0,0 +1,24 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{entropyrate2int}} + {compute the entropy rate of intensive biased walks in a multiplex with $2$ layers.} + {$<$layer1$>$ $<$layer2$>$ $$ $<$N$>$ $b_1$ $b_2$} + +\mydescription{Compute and print the entropy rate of an intensive biased walks in a multiplex with $2$ layers and bias parameters $b_p$ and $b_o$. + Files \textit{layer1}, \textit{layer2}, contain the (undirected) edge list of the two layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{overlapping network} has also a third column indicating the number of times two nodes are connected across all layers. + + $N$ is the number of nodes, $b_p$ is the biased exponent on the participation coefficient, $b_o$ is the biased exponent on the overlapping degree.} + +\myreturn{One line, reporting the value of the entropy rate $h$ of an intensive biased random walks with $b_p$ and $b_o$ as bias exponents, $b_p$ and $b_o$.} + +\myreference{\refbiased} diff --git a/doc/latex/latex/dynamics/randomwalks/entropyrate2mult.tex b/doc/latex/latex/dynamics/randomwalks/entropyrate2mult.tex new file mode 100644 index 0000000..7abfecd --- /dev/null +++ b/doc/latex/latex/dynamics/randomwalks/entropyrate2mult.tex @@ -0,0 +1,24 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{entropyrate2mult}} + {compute the entropy rate of multiplicative biased walks in a multiplex with $2$ layers.} + {$<$layer1$>$ $<$layer2$>$ $$ $<$N$>$ $b_1$ $b_2$} + +\mydescription{Compute and print the entropy rate of a multiplicative biased walk in a multiplex with $2$ layers and bias parameters $b_1$ and $b_2$. + Files \textit{layer1}, \textit{layer2}, contain the (undirected) edge list of the two layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{overlapping network} has also a third column indicating the number of times two nodes are connected across all layers. + + $N$ is the number of nodes, $b_1$ is the degree-biased exponent for layer $1$, $b_2$ is the degree-biased exponent for layer $2$.} + +\myreturn{One line, reporting the value of the entropy rate $h$ of an multiplicative biased random walks with $b_1$ and $b_2$ as bias exponents, $b_1$ and $b_2$.} + +\myreference{\refbiased} diff --git a/doc/latex/latex/dynamics/randomwalks/statdistr2.tex b/doc/latex/latex/dynamics/randomwalks/statdistr2.tex new file mode 100644 index 0000000..09c1bc2 --- /dev/null +++ b/doc/latex/latex/dynamics/randomwalks/statdistr2.tex @@ -0,0 +1,24 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{statdistr2}} + {compute the stationary distribution of additive, multiplicative and intensive biased walks in a multiplex with $2$ layers.} + {$<$layer1$>$ $<$layer2$>$ $$ $<$N$>$ $b_1$ $b_2$} + +\mydescription{Compute and print the stationary distribution of additive, multiplicative and intensive biased walks in a multiplex with $2$ layers. + Files \textit{layer1}, \textit{layer2}, contain the (undirected) edge list of the two layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{overlapping network} has also a third column indicating the number of times two nodes are connected across all layers. + + $N$ is the number of nodes, $b_1$ is the first bias exponent (the bias exponent for layer $1$ for additive and multiplicative walks, the bias exponent on the participation coefficient for intensive walks), $b_2$ is the second bias exponent (the bias exponent for layer $1$ for additive and multiplicative walks, the bias exponent on the participation coefficient for intensive walks).} + +\myreturn{N lines. In the n-th line we report the node ID, the stationary distribution of that node for additive walks with exponents $b_1$ and $b_2$, the stationary distribution for multiplicative walks with exponents $b_1$ and $b_2$, the stationary distribution for multiplicative walks with exponents $b_1$ and $b_2$, the values of the bias exponents $b_1$ and $b_2$.} + +\myreference{\refbiased} diff --git a/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex b/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex new file mode 100644 index 0000000..016b674 --- /dev/null +++ b/doc/latex/latex/models/correlations/tune_qnn_adaptive.tex @@ -0,0 +1,64 @@ +\myprogram{{tune\_qnn\_adaptive}} + {Construct a multiplex with prescribed inter-layer correlations.} + {$<$degs1$>$ $<$degs2$>$ $<$mu$>$ $<$eps$>$ $<$beta$>$ [RND|NAT|INV]} + +\mydescription{This programs tunes the inter-layer degree correlation + exponent $\mu$. If we consider two layers of a multiplex, + and we denote by $k$ the degree of a node on the first layer + and by $q$ the degree of the same node on the second layers, + the inter-layer degree correlation function is defined as: + + \begin{equation*} + \overline{q}(k) = \sum_{q'} q' P(q'|k) + \end{equation*} + + where $\overline{q}(k)$ is the average degree on layer $2$ + of nodes having degree $k$ on layer $1$. + + The program assumes that we want to set the degree + correlation function such that: + + \begin{equation*} + \overline{q}(k) = a k^{\mu} + \end{equation*} + + where the exponent of the power-law function is given by + the user (it is indeed the parameter \textit{mu}), and + successively adjusts the pairing between nodes at the two + layers in order to obtain a correlation function as close + as possible to the desired one. The files \textit{degs1} + and \textit{degs2} contain, respectively, the degrees of + the nodes on the first layer and on the second layer. + + The parameter \textit{eps} is the accuracy of \textit{mu}. + For instance, if \textit{mu} is set equal to -0.25 + and \textit{eps} is equal to 0.0001, the program stops when + the configuration of node pairing corresponds to a value of + the exponent $\mu$ which differs from -0.25 by less than + 0.0001. + + The parameter \textit{beta} is the typical inverse + temperature of simulated annealing. + + If no other parameter is specified, or if the last parameter + is \texttt{RND}, the program starts from a random pairing of + nodes. If the last parameter is \texttt{NAT} then the + program assumes that the initial pairing is the natural one, + where the nodes have the same ID on both layers. Finally, + if \texttt{INV} is specified, the initial pairing is the + inverse pairing, i.e. the one where node 0 on layer 1 is + paired with node N-1 on layer 2, and so on. + + } + + +\myreturn{The program prints on \texttt{stdout} a pairing, i.e. a list +of lines in the format: + +\hspace{0.5cm} \textit{IDL1 IDL2} + +where \textit{IDL1} is the ID of the node on layer 1 and \textit{IDL2} +is the corresponding ID of the same node on layer 2. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/correlations/tune_rho.tex b/doc/latex/latex/models/correlations/tune_rho.tex new file mode 100644 index 0000000..27b6079 --- /dev/null +++ b/doc/latex/latex/models/correlations/tune_rho.tex @@ -0,0 +1,45 @@ +\myprogram{{tune\_rho}} + {Construct a multiplex with prescribed inter-layer correlations.} + {$<$rank1$>$ $<$rank2$>$ $<$rho$>$ $<$eps$>$ $<$beta$>$ [RND|NAT|INV]} + +\mydescription{This programs tunes the inter-layer degree correlation + coefficient $\rho$ (Spearman's rank correlation) of two + layers, by adjusting the inter-layer pairing of nodes. The + files \textit{rank1} and \textit{rank2} are the rankings of + nodes in the first and second layer, where the n-th line of + the file contains the rank of the n-th node (the highest + ranked node has rank equal to 1). + + The parameter \textit{rho} is the desired value of the + Spearman's rank correlation coefficient, while \textit{eps} + is the accuracy of \textit{rho}. For instance, + if \textit{rho} is set equal to -0.25 and \textit{eps} is + equal to 0.0001, the program stops when the configuration of + node pairing corresponds to a value of $\rho$ which differs + from -0.25 by less than 0.0001. + + The parameter \textit{beta} is the typical inverse + temperature of simulated annealing. + + If no other parameter is specified, or if the last parameter + is \texttt{RND}, the program starts from a random pairing of + nodes. If the last parameter is \texttt{NAT} then the + program assumes that the initial pairing is the natural one, + where the nodes have the same ID on both layers. Finally, + if \texttt{INV} is specified, the initial pairing is the + inverse pairing, i.e. the one where node 0 on layer 1 is + paired with node N-1 on layer 2, and so on. + + } + + +\myreturn{The program prints on \texttt{stdout} a pairing, i.e. a list +of lines in the format: + +\hspace{0.5cm} \textit{IDL1 IDL2} + +where \textit{IDL1} is the ID of the node on layer 1 and \textit{IDL2} +is the corresponding ID of the same node on layer 2. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/growth/nibilab_linear_delay.tex b/doc/latex/latex/models/growth/nibilab_linear_delay.tex new file mode 100644 index 0000000..afbf083 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delay.tex @@ -0,0 +1,67 @@ +\myprogram{{nibilab\_linear\_delay}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives first on layer $1$, and its replica on + the layer $2$ appears after a time delay $\tau$ sampled from + the power-law function: + + \begin{equation*} + P(\tau) \sim \tau^{-\beta} + \end{equation*} + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \item \textbf{beta} the exponent of the power-law delay + function which determines the arrival of replicas on layer $2$ + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex b/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex new file mode 100644 index 0000000..5863951 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delay_mix.tex @@ -0,0 +1,68 @@ +\myprogram{{nibilab\_linear\_delay\_mix}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival and randomly selected first layer.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives on one of the two layers, chosen + uniformly at random, and its replica on the other layer + appears after a time delay $\tau$ sampled from the power-law + function: + + \begin{equation*} + P(\tau) \sim \tau^{-\beta} + \end{equation*} + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \item \textbf{beta} the exponent of the power-law delay + function which determines the arrival of replicas on layer $2$ + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_delta.tex b/doc/latex/latex/models/growth/nibilab_linear_delta.tex new file mode 100644 index 0000000..27ebbd0 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_delta.tex @@ -0,0 +1,57 @@ +\myprogram{{nibilab\_linear\_delta}} + {Multiplex linear preferential attachment model -- + Synchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ $<$c$>$ $<$d$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives at the same time on both layers. + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_linear_random_times.tex b/doc/latex/latex/models/growth/nibilab_linear_random_times.tex new file mode 100644 index 0000000..5f1c0fe --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_linear_random_times.tex @@ -0,0 +1,63 @@ +\myprogram{{nibilab\_linear\_random\_times}} + {Multiplex linear preferential attachment model -- + Asynchronous arrival with randomly sampled arrival times on + layer 2.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$a$>$ $<$b$>$ + $<$c$>$ $<$d$>$ } + +\mydescription{Grow a two-layer multiplex network using the multiplex linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto ak\lay{1}_j + bk\lay{2}_j + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto ck\lay{1}_j + dk\lay{2}_j + \end{equation*} + + Each new node arrives on layer $1$, but its replica on the + other layer appears at a uniformly chosen random time in + $[m0+1; N]$. + + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{a,b,c,d} the coefficients of the attaching probability + function + + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/nibilab_nonlinear.tex b/doc/latex/latex/models/growth/nibilab_nonlinear.tex new file mode 100644 index 0000000..2e48a18 --- /dev/null +++ b/doc/latex/latex/models/growth/nibilab_nonlinear.tex @@ -0,0 +1,60 @@ +\myprogram{{nibilab\_nonlinear}} + {Multiplex non-linear preferential attachment model -- + Synchronous arrival.} + {$<$N$>$ $<$m$>$ $<$m0$>$ $<$outfile$>$ $<$alpha$>$ $<$beta$>$} + +\mydescription{Grow a two-layer multiplex network using the multiplex non-linear + preferential attachment model by Nicosia, Bianconi, Latora, + Barthelemy (NiBiLaB). + + The probability for a newly arrived node $i$ to create a + link to node $j$ on layer $1$ is: + + \begin{equation*} + \Pi_{i\to j}^{1} \propto \frac{\left(k\lay{1}_j\right)^{\alpha}} + {\left(k\lay{2}_j\right)^{\beta}} + \end{equation*} + + and the dual probability for $i$ to create a link to $j$ on + layer $2$ is: + + \begin{equation*} + \Pi_{i\to j}^{2} \propto \frac{\left(k\lay{2}_j\right)^{\alpha}} + {\left(k\lay{1}_j\right)^{\beta}} + \end{equation*} + + Each node arrives simultaneously on both layers. + + + The (mandatory) parameters are as follows: + + \begin{itemize} + + \item \textbf{N} number of nodes in the final graph + + \item \textbf{m} number of new edges brought by each new node + + \item \textbf{m0} number of nodes in the initial seed + graph. \textit{m0} must be larger than of equal + to \textit{m}. + + \item \textbf{outfile} the name of the file which will contain the + + \item \textbf{alpha, beta} exponents of of the attaching probability + function + + \end{itemize} + } + + +\myreturn{The program dumps on the file \texttt{outfile} the + (undirected) edge list of the resulting network. Each line of the + file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/growth/node_deg_over_time.tex b/doc/latex/latex/models/growth/node_deg_over_time.tex new file mode 100644 index 0000000..351045e --- /dev/null +++ b/doc/latex/latex/models/growth/node_deg_over_time.tex @@ -0,0 +1,50 @@ +\myprogram{{node\_deg\_over\_time.py}} + {Time evolution of the degree of a node in a growing graph.} + {$<$layer$>$ $<$arrival\_times$>$ $<$node\_id$>$ + [$<$node\_id$>$ ...]} + +\mydescription{Compute the degree $k_{i}(t)$ of node $i$ in a growing + network as a function of time. The file \textit{layer} + contains the edge list of the final network. Each line of + the file is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The file \textit{arrival\_times} is a list of node arrival times, in + the format: + + \hspace{0.5cm} \textit{time\_i node\_i} + + where \textit{time\_i} is the time at which \textit{node\_i} arrived + in the graph. Notice that \textit{time\_i} must be an integer in the + range [0, N-1], where N is the total number of nodes in the final + graph. + + The third parameter \textit{node\_id} is the ID of the node whose + degree over time will be printed on output. If more than + one \textit{node\_id} is provided, the degrees over time of all the + corresponding nodes are printed on output. + } + + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{t kit} + + where \textit{kit} is the degree of node \textit{i} at + time \textit{t}. The first line of output is in the format: + + \hspace{0.5cm} \textit{\#\#\#\# node\_id} + + where \textit{node\_id} is the ID of node \textit{i}. + + If more than one \textit{node\_id}s is provided as input, the program + prints the degree over time of all of them, sequentially. + +} + +\myreference{\refgrowth} diff --git a/doc/latex/latex/models/nullmodels/model_MDM.tex b/doc/latex/latex/models/nullmodels/model_MDM.tex new file mode 100644 index 0000000..dd754b8 --- /dev/null +++ b/doc/latex/latex/models/nullmodels/model_MDM.tex @@ -0,0 +1,37 @@ +\myprogram{{model\_MDM.py}} + {Multi-activity Deterministic Model.} + {$<$Bi\_file$>$ $<$M$>$} + +\mydescription{This is the Multi-activity Deterministic Model (MDM). + In this model each node $i$ is considered active if it was + active in the reference multiplex, maintains the same value + of node activity $B_i$ (i.e., the number of layers in which + it was active) and is associated an activity vector sampled + uniformly at random from the $M\choose{B_i}$ possible + activity vectors with $B_i$ non-null entries. + + The file \textit{Bi\_file} is in the format: + + \hspace{0.5cm} \textit{Bi N(Bi)} + + where \textit{Bi} is a value of node activity + and \textit{N(Bi)} is the number of nodes which had node + activity equaly to \textit{Bi} in the reference multiplex. + + The parameter \textit{M} is the number of layers in the + multiplex. +} + + +\myreturn{The program prints on \texttt{stdout} a distribution of + bit-strings, in the format: + + \hspace{0.5cm} \textit{Bi bitstring count} + + where \textit{bitstring} is the activity + bitstring, \textit{Bi} is the number of non-zero entries + of \textit{bitstring} and \textit{count} is the number of + times that \textit{bitstrings} appear in the null model.} + + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/nullmodels/model_MSM.tex b/doc/latex/latex/models/nullmodels/model_MSM.tex new file mode 100644 index 0000000..e3c8abf --- /dev/null +++ b/doc/latex/latex/models/nullmodels/model_MSM.tex @@ -0,0 +1,38 @@ +\myprogram{{model\_MSM.py}} + {Multi-activity Stochastic Model.} + {$<$node\_Bi\_file$>$ $<$M$>$} + +\mydescription{This is the Multi-activity Stochastic Model (MSM). + In this model each node $i$ is considered active if it was + active in the reference multiplex, and is activated on + each layer with a probability equal to $B_i/M$ where $B_i$ + was the activity of node $i$ in the reference multiplex. + + The file \textit{node\_Bi\_file} is in the format: + + \hspace{0.5cm} \textit{node\_i Bi)} + + where \textit{Bi} is the value of node activity + of \textit{node\_i} in the reference multiplex. + + + The parameter \textit{M} is the number of layers in the + multiplex. +} + +\myreturn{The program prints on \texttt{stdout} a node-layer list of lines in the + format: + + \hspace{0.5cm} \textit{node\_i layer\_i} + + where \textit{node\_i} is the ID of a node and \textit{layre\_i} is + the ID of a layer. This list indicates which nodes are active in + which layer. For instance, the line: + + \hspace{0.5cm} \textit{24 3} + + indicates that the node with ID \textit{24} is active on + layer \textit{3}. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/nullmodels/model_hypergeometric.tex b/doc/latex/latex/models/nullmodels/model_hypergeometric.tex new file mode 100644 index 0000000..0315c7b --- /dev/null +++ b/doc/latex/latex/models/nullmodels/model_hypergeometric.tex @@ -0,0 +1,33 @@ +\myprogram{{model\_hypergeometric.py}} + {Hypergeometric node activity null model.} + {$<$layer\_N\_file$>$ $<$N$>$} + +\mydescription{This is the hypergeometric model of node activation. In + this model each layer has exactly the same number of active + node of a reference multiplex network, but nodes on each + layer are activated uniformly at random, thus destroying all + inter-layer activity correlation patterns. + + The file \textit{layer\_N\_file} reports on the n-th line + the number of active nodes on the n-th layer (starting from + zero). The second parameter \textit{N} is the total number + of active nodes in the multiplex. + } + + +\myreturn{The program prints on \texttt{stdout} a node-layer list of lines in the + format: + + \hspace{0.5cm} \textit{node\_i layer\_i} + + where \textit{node\_i} is the ID of a node and \textit{layre\_i} is + the ID of a layer. This list indicates which nodes are active in + which layer. For instance, the line: + + \hspace{0.5cm} \textit{24 3} + + indicates that the node with ID \textit{24} is active on + layer \textit{3}. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/models/nullmodels/model_layer_growth.tex b/doc/latex/latex/models/nullmodels/model_layer_growth.tex new file mode 100644 index 0000000..9c0bb12 --- /dev/null +++ b/doc/latex/latex/models/nullmodels/model_layer_growth.tex @@ -0,0 +1,46 @@ +\myprogram{{model\_layer\_growth.py}} + {Layer growth with preferential activation model.} + {$<$layer\_N\_file$>$ $<$N$>$ $<$M0$>$ $<$A$>$ [RND]} + +\mydescription{This is the model of layer growth with preferential + node activation. In this model an entire new layer arrives + at time $t$ and a number of nodes $N_t$ is activated ($N\_t$ + is equal to the number of nodes active on that layer in the + reference multiplex). Then, each node $i$ of the new layer + is activated with a probability: + + \begin{equation*} + P_i(t) \propto A + B_i(t) + \end{equation*} + + where $B_i(t)$ is the activity of node $i$ at time $t$ + (i.e., the number of layers in which node $i$ is active at + time $t$) while $A>0$ is an intrinsic attractiveness. + + The file \textit{layer\_N\_file} reports on the n-th line + the number of active nodes on the n-th layer. + + The parameter \textit{N} is the number of nodes in the + multiplex, \textit{M0} is the number of layers in the + initial network, \textit{A} is the value of + node attractiveness. + + If the user specifies \texttt{RND} as the last parameter, + the sequence of layers is } + +\myreturn{The program prints on \texttt{stdout} a node-layer list of lines in the + format: + + \hspace{0.5cm} \textit{node\_i layer\_i} + + where \textit{node\_i} is the ID of a node and \textit{layre\_i} is + the ID of a layer. This list indicates which nodes are active in + which layer. For instance, the line: + + \hspace{0.5cm} \textit{24 3} + + indicates that the node with ID \textit{24} is active on + layer \textit{3}. +} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/degs_to_activity_overlap.tex b/doc/latex/latex/structure/activity/degs_to_activity_overlap.tex new file mode 100644 index 0000000..6e1908f --- /dev/null +++ b/doc/latex/latex/structure/activity/degs_to_activity_overlap.tex @@ -0,0 +1,29 @@ +\myprogram{{degs\_to\_activity\_overlap.py}} + {compute the activity and the total (overlapping) degree of + all the nodes of a multiplex.} + {$<$degree\_vectors$>$} + +\mydescription{Take a file which contains, on the n-th line, the degrees at each + layer of the n-th node, (e.g., the result of the + script \texttt{node\_degree\_vectors.py}), in the format: + + \hspace{0.5cm}\textit{noden\_deg\_lay1 noden\_deg\_lay2 ... noden\_deg\_layM} + + \noindent and compute the activity (i.e., the number of layers in + which a node is not isolated) and the total (overlapping) degree of + each node.} + +\myreturn{The program prints on \texttt{stdout} a list of lines, where + the n-th line contains the activity and the total degree of the n-th + nodem in the format: + + \hspace{0.5cm}\textit{noden\_activity noden\_tot\_deg} + + \noindent As usual, the program assumes that node IDs start from zero + and proceed sequentially, without gaps, i.e., if a node ID is not + present in any of the layer files given as input, the program + considers it as being isolated on all the layers. + } + +\myreference{\refcorrelations} + diff --git a/doc/latex/latex/structure/activity/degs_to_binary.tex b/doc/latex/latex/structure/activity/degs_to_binary.tex new file mode 100644 index 0000000..7441b2d --- /dev/null +++ b/doc/latex/latex/structure/activity/degs_to_binary.tex @@ -0,0 +1,32 @@ +\myprogram{{degs\_to\_binary.py}} + {compute the activity vectors of all the nodes of a multiplex.} + {$<$degree\_vectors$>$} + +\mydescription{Take a file which contains, on the n-th line, the degrees at each + layer of the n-th node, (e.g., the result of the + script \texttt{node\_degree\_vectors.py}), in the format: + + \hspace{0.5cm}\textit{noden\_deg\_lay1 noden\_deg\_lay2 ... noden\_deg\_layM} + + \noindent and compute the corresponding node activity bit-strings, + where a "1" signals the presence of the node on that layer, while a + zero indicates its absence. +} + +\myreturn{The program returns on \texttt{stdout} a list of lines, + where the n-th line is the activity bit-string of the n-th + node. Additionally, the program prints on \texttt{stderr} the + distribution of all activity bit-strings, in the format: + + \hspace{0.5cm}\textit{Bn Bit-string count} + + \noindent Where \textit{B} is the number of ones in the activity + bit-string (i.e., the node-activity associated to that activity + bit-string), \textit{Bit-string} is the activity bit-string + and \textit{count} is the number of times that particular activity + bit-string appears in the multiplex.} + + + +\myreference{\refcorrelations} + diff --git a/doc/latex/latex/structure/activity/hamming_dist.tex b/doc/latex/latex/structure/activity/hamming_dist.tex new file mode 100644 index 0000000..3af188f --- /dev/null +++ b/doc/latex/latex/structure/activity/hamming_dist.tex @@ -0,0 +1,31 @@ +\myprogram{{hamming\_dist.py}} + {compute the normalised Hamming distance between all the pairs of + layers of a multiplex.} + {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the normalised Hamming distance + $H_{\alpha, \beta}$ (i.e., the fraction of nodes which are active on + either of the layers, but not on both) between all pairs of + layers. The layers are given as input in the + files \textit{layer1}, \textit{layer2}, etc. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines, in + the format: + + \hspace{0.5cm} \textit{layer1 layer2 hamm} + + \noindent where \textit{layer1} and \textit{layer2} are the IDs of + the layers, and \textit{hamm} is the value of the normalised Haming + distance $H_{layer1, layer2}$. Layers IDs start from zero, are are + associated to the layers in the same order in which the layer files + are provided on the command line.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/layer_activity.tex b/doc/latex/latex/structure/activity/layer_activity.tex new file mode 100644 index 0000000..34fcdd2 --- /dev/null +++ b/doc/latex/latex/structure/activity/layer_activity.tex @@ -0,0 +1,25 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{layer\_activity.py}} + {compute the activity of the layers of a multiplex, i.e. the + number of active nodes on each layer.} + {$<$layer1$>$ [$<$layer2$>$ ...]} + +\mydescription{Compute and print on output the activity of the layers + of a multiplex network, where the layers are given as input in the + files \textit{layer1}, \textit{layer2}, etc. + + Each file contains the (undirected) edge list of a layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{A listof lines, where the n-th line is the value of activity + of the n-th layer, starting from \textbf{0}.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/layer_activity_vectors.tex b/doc/latex/latex/structure/activity/layer_activity_vectors.tex new file mode 100644 index 0000000..f0c2cd6 --- /dev/null +++ b/doc/latex/latex/structure/activity/layer_activity_vectors.tex @@ -0,0 +1,27 @@ +\myprogram{{layer\_activity\_vectors.py}} + {compute the activity vectors of all the layers of a multiplex.} + {$<$layer1$>$ [$<$layer2$>$ ...]} + +\mydescription{Compute and print on output the activity vectors of the + layers of a multiplex network, where the layers are given as input + in the files \textit{layer1}, \textit{layer2}, etc. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines, where + the n-th line contains the activity vector of the n-th layer, i.e. a + bit-string where each bit is set to ``1'' if the corresponding node + is active on the n-th layer, and to ``0'' otherwise. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/multiplexity.tex b/doc/latex/latex/structure/activity/multiplexity.tex new file mode 100644 index 0000000..b5c5506 --- /dev/null +++ b/doc/latex/latex/structure/activity/multiplexity.tex @@ -0,0 +1,30 @@ +\myprogram{{multiplexity.py}} + {compute the pairwise multiplexity between all the pairs of + layers of a multiplex.} + {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the pairwise multiplexity + $Q_{\alpha, \beta}$ (i.e., the fraction of nodes active on both + layers) between all pairs of layers. The layers are given as + input in the files \textit{layer1}, \textit{layer2}, etc. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines, in + the format: + + \hspace{0.5cm} \textit{layer1 layer2 mult} + + \noindent where \textit{layer1} and \textit{layer2} are the IDs of + the layers, and \textit{mult} is the value of the multiplexity + $Q_{layer1, layer2}$. Layers IDs start from zero, are are associated + to the layers in the same order in which the layer files are + provided on the command line.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/node_activity.tex b/doc/latex/latex/structure/activity/node_activity.tex new file mode 100644 index 0000000..882233d --- /dev/null +++ b/doc/latex/latex/structure/activity/node_activity.tex @@ -0,0 +1,25 @@ +%%% +%%% node_activity +%%% +\myprogram{{node\_activity.py}} + {compute the activity of the nodes of a multiplex, i.e. the + number of layers where each node is not isolated.} + {$<$layer1$>$ [$<$layer2$>$ ...]} + +\mydescription{Compute and print on output the activity of the nodes + of a multiplex network, whose layers are given as input in the files + \textit{layer1}, \textit{layer2}, etc. + + Each file contains the (undirected) edge list of a layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{A list of lines, where the n-th line is the value of activity + of the n-th node, starting from \textbf{0}.} + +\myreference{\refcorrelations} + diff --git a/doc/latex/latex/structure/activity/node_activity_vectors.tex b/doc/latex/latex/structure/activity/node_activity_vectors.tex new file mode 100644 index 0000000..ca9301d --- /dev/null +++ b/doc/latex/latex/structure/activity/node_activity_vectors.tex @@ -0,0 +1,28 @@ +\myprogram{{node\_activity\_vectors.py}} + {compute the activity vectors of all the nodes of a multiplex.} + {$<$layer1$>$ [$<$layer2$>$ ...]} + +\mydescription{Compute and print on output the activity vectors of the + nodes of a multiplex network, whose layers are given as input in the + files \textit{layer1}, \textit{layer2}, etc. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines, where + the n-th line contains the activity vector of the n-th node, i.e. a + bit-string where each bit is set to ``1'' if the node is active on + the corresponding layer, and to ``0'' otherwise. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers, and will print on output a bit-string of + zeros.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/activity/node_degree_vectors.tex b/doc/latex/latex/structure/activity/node_degree_vectors.tex new file mode 100644 index 0000000..c86f083 --- /dev/null +++ b/doc/latex/latex/structure/activity/node_degree_vectors.tex @@ -0,0 +1,30 @@ +\myprogram{{node\_degree\_vectors.py}} + {compute the degree vectors of all the nodes of a multiplex network} + {$<$layer1$>$ [$<$layer2$>$ ...]} + +\mydescription{Compute and print on output the degree vectors of all + the nodes of a multiplex network, whose layers are given as + input in the files \textit{layer1}, \textit{layer2}, etc. + + Each file contains the (undirected) edge list of a layer, and each + line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{A list of lines, where the n-th line is the + vector of degrees of the n-th node, in the format: + + \hspace{0.5cm}\textit{noden\_deg\_lay1 noden\_deg\_lay2 ... noden\_deg\_layM} + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers. + +} + +\myreference{\refgrowth\\ \\ \indent \refmetrics} + diff --git a/doc/latex/latex/structure/correlations/compute_pearson.tex b/doc/latex/latex/structure/correlations/compute_pearson.tex new file mode 100644 index 0000000..8202753 --- /dev/null +++ b/doc/latex/latex/structure/correlations/compute_pearson.tex @@ -0,0 +1,28 @@ +\myprogram{{compute\_pearson.py}} + {compute the Pearson's linear correlation coefficient + between two node properties.} + {$<$file1$>$ $<$file2$>$} + +\mydescription{Compute the Pearson's linear correlation coefficient + between two sets of (either integer- or real-valued) node + properties provided in the input files \textit{file1} + and \textit{file2}. Each input file contains a list of + lines, where the n-th line contains the value of a node + property for the n-th node. For instance, \textit{file1} + and \textit{file2} might contain the degrees of nodes at two + distinct layers of a multiplex. However, the program is + pretty general and can be used to compute the Pearson's + correlation coeffcient between any pairs of node properties. + } + +\myreturn{The program prints on \texttt{stdout} the value of the + Pearson's linear correlation coefficient between the two + sets of node properties. +} + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear +} diff --git a/doc/latex/latex/structure/correlations/compute_rho.tex b/doc/latex/latex/structure/correlations/compute_rho.tex new file mode 100644 index 0000000..957b04c --- /dev/null +++ b/doc/latex/latex/structure/correlations/compute_rho.tex @@ -0,0 +1,32 @@ +\myprogram{{compute\_rho.py}} + {compute the Spearman's rank correlation coefficient $\rho$ + between two rankings.} {$<$file1$>$ $<$file2$>$} + +\mydescription{Compute the Spearman's rank correlation coefficient + $\rho$ between two rankings provided in the input + files \textit{file1} and \textit{file2}. Each input file + contains a list of lines, where the n-th line contains the + value of rank of the n-th node. For instance, \textit{file1} + and \textit{file2} might contain the ranks of nodes induced + by the degree sequences of two distinct layers of a + multiplex. + + However, the program is pretty general and can be used to + compute the Spearman's rank correlation coefficient between + any generic pair of rankings. + + N.B.: A C implementation of this program, with the same + interface is also available in the executable + file \texttt{compute\_rho}.} + + +\myreturn{The program prints on \texttt{stdout} the value of the + Spearman's rank correlation coefficient $\rho$ between the + two rankings provided as input. } + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear + } diff --git a/doc/latex/latex/structure/correlations/compute_tau.tex b/doc/latex/latex/structure/correlations/compute_tau.tex new file mode 100644 index 0000000..e6e8589 --- /dev/null +++ b/doc/latex/latex/structure/correlations/compute_tau.tex @@ -0,0 +1,31 @@ +\myprogram{{compute\_tau.py}} + {compute the Kendall's rank correlation coefficient $\tau_b$ + between two rankings.} {$<$file1$>$ $<$file2$>$} + +\mydescription{Compute the Kendall's rank correlation coefficient + $\tau_b$ between two rankings provided in the input + files \textit{file1} and \textit{file2}. Each input file + contains a list of lines, where the n-th line contains the + value of rank of the n-th node. For instance, \textit{file1} + and \textit{file2} might contain the ranks of nodes induced + by the degree sequences of two distinct layers of a + multiplex. + + However, the program is pretty general and can be used to + compute the Kendall's rank correlation coefficient between + any generic pair of rankings. + + N.B.: This implementation takes properly into account rank + ties.} + + +\myreturn{The program prints on \texttt{stdout} the value of the + Kendall's rank correlation coefficient $\tau_b$ between the + two rankings provided as input. } + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear +} diff --git a/doc/latex/latex/structure/correlations/dump_k_q.tex b/doc/latex/latex/structure/correlations/dump_k_q.tex new file mode 100644 index 0000000..35aef8d --- /dev/null +++ b/doc/latex/latex/structure/correlations/dump_k_q.tex @@ -0,0 +1,42 @@ +M\myprogram{{dump\_k\_q}} + {compute the degree sequences of two layers of a multiplex.} + {$<$layer1$>$ $<$layer2$>$ $<$pairing$>$} + +\mydescription{Compute and dump on \texttt{stdout} the degree + sequences of two layers of a multiplex. The input + files \textit{layer1} and \textit{layer2} contain the + (undirected) edge lists of the two layers, and each line is + in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge. + + The third file \textit{pairing} is a list of lines in the format: + + \hspace{0.5cm} \textit{IDL1 IDL2} + + where \textit{IDL1} is the ID of a node on layer $1$ + and \textit{IDL2} is the ID of the same node on layer $2$. For + instance, the line: + + \hspace{0.5cm} \textit{5 27} + + indicates that node $5$ on layer $1$ has ID $27$ on layer $2$. } + + +\myreturn{The program prints on \texttt{stdout} the degree of each +node on the two layers, in the format: + +\hspace{0.5cm} \textit{ki qi} + +where \textit{ki} is the degree of node \textit{i} on layer $1$ +and \textit{qi} is the degree of node \textit{i} on layer $2$.} + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear + } diff --git a/doc/latex/latex/structure/correlations/fit_knn.tex b/doc/latex/latex/structure/correlations/fit_knn.tex new file mode 100644 index 0000000..cb90c2b --- /dev/null +++ b/doc/latex/latex/structure/correlations/fit_knn.tex @@ -0,0 +1,50 @@ +\myprogram{{fit\_knn}} + {power-law fit of the inter-layer degree correlation + function.} + {$<$filein$>$ $<$alpha$>$} + +\mydescription{Perform a power-law fit of the inter-layer degree + correlation function: + + \begin{equation*} + \overline{q}(k) = \frac{1}{N_{q}}\sum_{q'} q' P(q'|k) + \end{equation*} + + where $k$ is the degree of a node on layer $1$, $q$ is the + degree on layer $2$ and $P(q|k)$ is the probability that a + node with degree $k$ on layer $1$ has degree $q$ on layer + $2$. The program assumes that $\overline{q}(k)$ can be + written in the form $a k^{b}$, and computes the two + parameters $a$ and $b$ through a linear fit of the log-log + plot of $\overline{q}(k)$. + + The input file \textit{filein} contains a list of lines in + the format: + + \hspace{0.5cm} \textit{ki qi} + + where \textit{ki} is the degree of node $i$ at layer $1$ + and \textit{qi} is the degree of node $i$ at layer $2$. + + The second parameter \textit{alpha} is the ratio of the + progression used to generate the exponentially-distributed + bins for the log-log plot. Typical values of \textit{alpha} + are between $1.1$ and $2.0$. + + + N.B.: The exponent $b$ computed with this method is known to + be inaccurate. +} + + + +\myreturn{The program prints on \texttt{stdout} the values of the + parameters $a$ and $b$ of the power-law fit $\overline{q}(k) + = a k^{b}$.} + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear + } diff --git a/doc/latex/latex/structure/correlations/knn_q_from_degrees.tex b/doc/latex/latex/structure/correlations/knn_q_from_degrees.tex new file mode 100644 index 0000000..cab0905 --- /dev/null +++ b/doc/latex/latex/structure/correlations/knn_q_from_degrees.tex @@ -0,0 +1,64 @@ +\myprogram{{knn\_q\_from\_degrees.py}} + {compute the inter-layer degree-degree correlation function.} + {$<$filein$>$} + +\mydescription{Compute the inter-layer degree + correlation functions for two layers of a multiplex, using + the degrees of the nodes specified in the input file. The + format of the input file is as follows + +\hspace{0.5cm} \textit{ki qi} + +where \textit{ki} and \textit{qi} are, respectively, the degree at +layer 1 and the degree at layer 2 of node \textit{i}. + + If we consider two layers of a multiplex, and we denote by + $k$ the degree of a node on the first layer and by $q$ the + degree of the same node on the second layers, the + inter-layer degree correlation function is defined as + + \begin{equation*} + \overline{k}(q) = \frac{1}{N_{k}}\sum_{k'} k' P(k'|q) + \end{equation*} + + where $P(k'|q)$ is the probability that a node with degree + $q$ on the second layer has degree equal to $k'$ on the + first layer, and $N_k$ is the number of nodes with degree + $k$ on the first layer. The quantity $\overline{k}(q)$ is + the expected degree at layer $1$ of node that have degree + equal to $q$ on layer $2$. The dual quantity: + + \begin{equation*} + \overline{q}(k) = \frac{1}{N_{q}}\sum_{q'} q' P(q'|k) + \end{equation*} + + is the average degree on layer $2$ of nodes having degree + $k$ on layer $1$. +} + + +\myreturn{The program prints on \texttt{stdout} a list of lines in + the format: + + \hspace{0.5cm} \textit{k $\overline{q}(k)$} + + where \textit{k} is the degree on layer $1$ and + $\overline{q}(k)$ is the average degree on layer $2$ of + nodes having degree equal to $k$ on layer $1$. + + The program also prints on \texttt{stderr} a list of lines in + the format: + + \hspace{0.5cm} \textit{q $\overline{k}(q)$} + + where \textit{q} is the degree on layer $2$ and + $\overline{k}(q)$ is the average degree on layer $1$ of + nodes having degree equal to $q$ on layer $2$. + } + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear + } diff --git a/doc/latex/latex/structure/correlations/knn_q_from_layers.tex b/doc/latex/latex/structure/correlations/knn_q_from_layers.tex new file mode 100644 index 0000000..f124e55 --- /dev/null +++ b/doc/latex/latex/structure/correlations/knn_q_from_layers.tex @@ -0,0 +1,80 @@ +\myprogram{{knn\_q\_from\_layers.py}} + {compute intra-layer and inter-layer degree-degree + correlation coefficients.} {$<$layer1$>$ $<$layer2$>$} + +\mydescription{Compute the intra-layer and the inter-layer degree + correlation functions for two layers given as input. The + intra-layer degree correlation function quantifies the + presence of degree-degree correlations in a single layer + network, and is defined as: + + \begin{equation*} + \avg{k_{nn}(k)} = \frac{1}{k N_k}\sum_{k'}k'P(k'|k) + \end{equation*} + + where $P(k'|k)$ is the probability that a neighbour of a + node with degree $k$ has degree $k'$, and $N_k$ is the + number of nodes with degree $k$. The quantity + $\avg{k_{nn}(k)}$ is the average degree of the neighbours of + nodes having degree equal to $k$. + + If we consider two layers of a multiplex, and we denote by + $k$ the degree of a node on the first layer and by $q$ the + degree of the same node on the second layers, the + inter-layer degree correlation function is defined as + + \begin{equation*} + \overline{k}(q) = \sum_{k'} k' P(k'|q) + \end{equation*} + + where $P(k'|q)$ is the probability that a node with degree + $q$ on the second layer has degree equal to $k'$ on the + first layer, and $N_q$ is the number of nodes with degree + $q$ on the second layer. The quantity $\overline{k}(q)$ is + the expected degree at layer $1$ of node that have degree + equal to $q$ on layer $2$. The dual quantity: + + \begin{equation*} + \overline{q}(k) = \sum_{q'} q' P(q'|k) + \end{equation*} + + is the average degree on layer $2$ of nodes having degree + $k$ on layer $1$. +} + + +\myreturn{The program creates two output files, respectively called + +\hspace{0.5cm} \textit{file1\_file2\_k1} + +and + +\hspace{0.5cm} \textit{file1\_file2\_k2} + +The first file contains a list of lines in the format: + +\hspace{0.5cm} \textit{k $\avg{k_{nn}(k)}$ $\sigma_k$ +$\overline{q}(k)$ $\sigma_{\overline{q}}$} + +where $k$ is the degree at first layer, $\avg{k_{nn}(k)}$ is the +average degree of the neighbours at layer $1$ of nodes having degree +$k$ at layer $1$, $\sigma_k$ is the standard deviation associated to +$\avg{k_{nn}(k)}$, $\overline{q}(k)$ is the average degree at layer +$2$ of nodes having degree equal to $k$ at layer $1$, and +$\sigma_{\overline{q}}$ is the standard deviation associated to +$\overline{q}(k)$. + +The second file contains a similar list of lines, in the format: + +\hspace{0.5cm} \textit{q $\avg{q_{nn}(q)}$ $\sigma_q$ +$\overline{k}(q)$ $\sigma_{\overline{k}}$} + +with obvious meaning. +} + +\myreference{\refcorrelations + + \refgrowth + + \refnonlinear + } diff --git a/doc/latex/latex/structure/correlations/rank_nodes.tex b/doc/latex/latex/structure/correlations/rank_nodes.tex new file mode 100644 index 0000000..b1a970a --- /dev/null +++ b/doc/latex/latex/structure/correlations/rank_nodes.tex @@ -0,0 +1,19 @@ +\myprogram{{rank\_nodes.py}} + {rank the nodes of a layer according to a given structural + descriptor.} {$<$prop\_file$>$} + +\mydescription{Get a file as input, whose n-th line corresponds to the value of a + certain property of the n-th node, and rank the nodes according to + that property, taking into account ranking ties properly. + + For example, if \textit{propfile} contains the degrees of the nodes + at a certain layer of the multiplex, the computes the ranking induced + by degrees, where the node with the highest degree will be assigned a + rank equal to \textbf{1} (one). +} + +\myreturn{The program prints on \texttt{stdout} a list of lines, where + the n-th line contains the rank of the n-th node corresponding to the + values of the structural descriptor provided in the input file.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/correlations/rank_nodes_thresh.tex b/doc/latex/latex/structure/correlations/rank_nodes_thresh.tex new file mode 100644 index 0000000..3b430d1 --- /dev/null +++ b/doc/latex/latex/structure/correlations/rank_nodes_thresh.tex @@ -0,0 +1,18 @@ +\myprogram{{rank\_nodes\_thresh.py}} + {rank the nodes of a layer whose value of a given structural + descriptor is above a threshold.} {$<$prop\_file$>$ $<$thresh$>$} + +\mydescription{Get a file as input, whose n-th line corresponds to the value of a + certain property of the n-th node, and rank the nodes according to + that property, taking into account ranking ties properly. The rank of + all the nodes whose value of the structural descriptor is smaller + than the threshold \textit{thresh} specified as second parameter is + set to \textbf{0} (ZERO). } + +\myreturn{The program prints on \texttt{stdout} a list of lines, where + the n-th line contains the rank of the n-th node corresponding to the + values of the structural descriptor provided in the input file, or + zero if such desxriptor is below the specified + threshold \textit{thresh}.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/correlations/rank_occurrence.tex b/doc/latex/latex/structure/correlations/rank_occurrence.tex new file mode 100644 index 0000000..96fb914 --- /dev/null +++ b/doc/latex/latex/structure/correlations/rank_occurrence.tex @@ -0,0 +1,24 @@ +\myprogram{{rank\_occurrence.py}} + {compute the intersection of two rankings.} + {$<$rank1$>$ $<$rank2$>$ $<$increment$>$} + +\mydescription{Get two rankings \textit{rank1} and \textit{rank2} + and compute the size of +the \textit{k}-intersection, i.e. the number of elements which are +present in the first k positions of both rankings, as a function +of \textit{k}. The parameter \textit{increment} determines the +distance between two subsequent values of \textit{k}. + +Each input file is a list of node IDs, one per line, where the first +line contains the ID of the highest ranked node. +} + +\myreturn{The program prints on \texttt{stdout} a list of lines in the +format: + + \hspace{0.5cm} \textit{k num\_k} + + where \textit{num\_k} is the number of nodes which are present in + the first \textit{k} positions of both rankings.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/metrics/aggregate_layers_w.tex b/doc/latex/latex/structure/metrics/aggregate_layers_w.tex new file mode 100644 index 0000000..8917e3e --- /dev/null +++ b/doc/latex/latex/structure/metrics/aggregate_layers_w.tex @@ -0,0 +1,30 @@ +\myprogram{{aggregate\_layers\_w.py}} + {compute the (weighted) aggregated graph associated to a + multiplex.} {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the edge list of the + weighted aggregated graph associated to the multiplex + network given on input. An edge is present in the + aggregated graph if it exists in at least one of the M + layers of the multiplex. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} the edge list of the + aggregated graph associated to the multiplex network. The edge list + is a list of lines in the format: + + + \hspace{0.5cm} \textit{ID1 ID2 weight} + + \noindent where \textit{ID1} and \textit{ID2} are the IDs of the two + nodes and \textit{weight} is the number of layers in which an edge + between \textit{ID1} and \textit{ID2} exists.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/metrics/avg_edge_overlap.tex b/doc/latex/latex/structure/metrics/avg_edge_overlap.tex new file mode 100644 index 0000000..4df2d51 --- /dev/null +++ b/doc/latex/latex/structure/metrics/avg_edge_overlap.tex @@ -0,0 +1,42 @@ +\myprogram{{avg\_edge\_overlap.py}} + {compute the average edge overlap of a multiplex.} + {$<$layer1$>$ [$<$layer2$>$...]} + +\mydescription{Compute and print on output the average edge overlap + + \begin{equation*} \omega^{*} + = \frac{\sum_{i}\sum_{j>i}\sum_{\alpha}a_{ij}\lay{\alpha}}{ \sum_{i}\sum_{j>i}(1 + - \delta_{0,\sum_{\alpha}a_{ij}\lay{\alpha}})} \end{equation*} + + \noindent i.e., the expected \textit{number} of layers on which an + edge of the multiplex exists, and the corresponding normalised + quantity: + + \begin{equation*} + \omega = \frac{\sum_{i}\sum_{j>i}\sum_{\alpha}a_{ij}\lay{\alpha}}{M \sum_{i}\sum_{j>i}(1 + - \delta_{0,\sum_{\alpha}a_{ij}\lay{\alpha}})} + \end{equation*} + + \noindent that is the expected \textit{fraction} of layers on which + an edge of the multiplex is present. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a single line, in the + format: + + \hspace{0.5cm} \textit{omega\_star omega} + + \noindent where \textit{omega\_star} and \textit{omega} are, + respectively, the expected number and fraction of layers in which an + edge is present.} + +\myreference{\refmetrics + + \vspace{0.5cm}\refvisibility} diff --git a/doc/latex/latex/structure/metrics/cartography_from_columns.tex b/doc/latex/latex/structure/metrics/cartography_from_columns.tex new file mode 100644 index 0000000..8e797af --- /dev/null +++ b/doc/latex/latex/structure/metrics/cartography_from_columns.tex @@ -0,0 +1,35 @@ +\myprogram{{cartography\_from\_columns.py}} + {compute total and participation coefficient of generic + structural descriptors of the nodes of a multiplex.} + {$<$filein$>$ $<$col1$>$ $<$col2$>$ [$<$col3$>$...]} + +\mydescription{Compute and print on output the sum and the + corresponding participation coefficient of a generic + structural descriptor of the nodes of a multiplex. + + \noindent The input file is a generic collection of + single-space-separated columns, where each line corresponds to a + node. The user must specify the IDs of the columns which contain + the node structural descriptors to be used in the cartography + diagram. Columns IDs start from ZERO. For example: + + \textbf{python cartography\_from\_layers.py filein.txt 0 + 2 4 6 8} + + \noindent will create a cartography diagram assuming that the + multiplex network has five layers, and that the node structural + descriptors at each layers are contained in the first (0), third + (2), fifth (4), seventh (6) and nineth (8) columns of each row.} + + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{tot\_n P\_n} + + where \textit{tot\_n} is the sum over the layers of the considered + structural descriptor for node $n$, and \textit{P\_n} is the + associated participation coefficient + } + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/metrics/cartography_from_deg_vectors.tex b/doc/latex/latex/structure/metrics/cartography_from_deg_vectors.tex new file mode 100644 index 0000000..848ffd1 --- /dev/null +++ b/doc/latex/latex/structure/metrics/cartography_from_deg_vectors.tex @@ -0,0 +1,32 @@ +\myprogram{{cartography\_from\_deg\_vectors.py}} + {create a multiplex cartography diagram.} + {$<$node\_deg\_vectors$>$} + +\mydescription{Compute and print on output the total degree and the + multiplex participation coefficient of all the nodes of a + multiplex network whose list of node degree vectors is + provided as input. The input file is in the format: + + \hspace{0.5cm} \textit{IDn\_deg1 IDn\_deg\_2 ... IDn\_degM} + + \noindent where \textit{IDn\_degX} is the degree of node $n$ at + layer $X$. The input file can be generated using the + script \texttt{node\_degree\_vectors.py}.} + + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{tot\_deg part\_coeff} + + \noindent where \textit{tot\_deg} is the total degree of the node + and \textit{part\_coeff} is the corresponding participation + coefficient. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, so if one of the lines in the input + files contains just zeros, the program considers the corresponding + node as being isolated on all the layers, and both its total degree + and multiplex participation coefficient are set equal to zero. } + +\myreference{\refmetrics} diff --git a/doc/latex/latex/structure/metrics/cartography_from_layers.tex b/doc/latex/latex/structure/metrics/cartography_from_layers.tex new file mode 100644 index 0000000..8958cef --- /dev/null +++ b/doc/latex/latex/structure/metrics/cartography_from_layers.tex @@ -0,0 +1,45 @@ +\myprogram{{cartography\_from\_layers.py}} + {compute the total degree and the multiplex participation + coefficient of all the nodes of a multiplex.} {$<$layer1$>$ + $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the total degree and the multiplex participation + coefficient $P_i$ for each node $i$ of a multiplex. The + participation coefficient is defined as: + + \begin{equation*} + P_i=\frac{M}{M-1}\left[1-\sum_{\alpha=1}^M\biggl(\frac{k_i^{[\alpha]}}{o_i}\biggr)^2\right] + \end{equation*} + + \noindent Note that $P_i$ takes values in $[0,1]$, where $P_i=0$ + if and only if node $i$ is active on exactly one of the layers, + while $P_i=1$ if node $i$ has equal degree on all the $M$ layers. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{deg\_n P\_n col\_n} + + where \textit{deg\_n} is the total degree of node $n$, \textit{P\_n} + is the participation coefficient of node $n$ and \textit{col} is the + integer representation of the activity bitstring of node $n$, which + is a number between $0$ and $2^{M}-1$. The field \textit{col} might + be useful for the visualisation of the multiplex cartography + diagram, where it would be possible to associate different colors to + nodes having different node activity patterns. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers, and is set to zero. + } + +\myreference{\refmetrics} diff --git a/doc/latex/latex/structure/metrics/edge_overlap.tex b/doc/latex/latex/structure/metrics/edge_overlap.tex new file mode 100644 index 0000000..64d7dbf --- /dev/null +++ b/doc/latex/latex/structure/metrics/edge_overlap.tex @@ -0,0 +1,36 @@ +\myprogram{{edge\_overlap.py}} + {compute the edge overlap of all the edges of the + multiplex.} + {$<$layer1$>$ [$<$layer2$>$...]} + +\mydescription{Compute and print on output the edge overlap $o_{ij}$ of each + edge of the multiplex. Given a pair of nodes $(i,j)$ that + are directly connected on at least one of the $M$ layers, + the edge overlap $o_{ij}$ is defined as: + + \begin{equation*} + o_{ij} = \sum_{\alpha}a_{ij}\lay{\alpha} + \end{equation*} + + \noindent i.e., the number of layers on which the edge $(i,j)$ + exists. + + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{ID\_1 ID\_2 overlap} + + \noindent where \textit{ID\_1} and \textit{ID\_2} are the IDs of the + end-points of the edge, and \textit{overlap} is the number of layers + in which the edge exists.} + +\myreference{\refmetrics} diff --git a/doc/latex/latex/structure/metrics/intersect_layers.tex b/doc/latex/latex/structure/metrics/intersect_layers.tex new file mode 100644 index 0000000..0824400 --- /dev/null +++ b/doc/latex/latex/structure/metrics/intersect_layers.tex @@ -0,0 +1,28 @@ +\myprogram{{intersect\_layers.py}} + {compute the intersection graph associated to a + multiplex.} {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the edge list of the + intersection graph associated to the multiplex network + given on input, where an edge exists only if it is + present on \textbf{all} the layers of the multiplex. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} the edge list of the + intersection graph associated to the multiplex network. The edge + list is a list of lines in the format: + + + \hspace{0.5cm} \textit{ID1 ID2} + + \noindent where \textit{ID1} and \textit{ID2} are the IDs of the two + nodes.} + +\myreference{\refcorrelations} diff --git a/doc/latex/latex/structure/metrics/overlap_degree.tex b/doc/latex/latex/structure/metrics/overlap_degree.tex new file mode 100644 index 0000000..500a76a --- /dev/null +++ b/doc/latex/latex/structure/metrics/overlap_degree.tex @@ -0,0 +1,45 @@ +\myprogram{{overlap\_degree.py}} + {compute the total (overlapping) degree of all the nodes of + a multiplex and the corresponding Z-score. } {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the total degree $o_i$ of each + node $i$ of a multiplex, defined as: + + \begin{equation*} + o_{i} = \sum_{\alpha}\sum_{j}a_{ij}\lay{\alpha} + \end{equation*} + + \noindent and the corresponding Z-score: + + \begin{equation*} + z(o_i) = \frac{o_i - \avg{o}}{\sigma_o} + \end{equation*} + + \noindent where $\avg{o}$ and $\sigma_o$ are, respectively, the mean + and the standard deviation of the total degree computed over all the + active nodes of the multiplex. + + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{ID\_n deg\_n z\_n} + + where \textit{ID\_n} is the ID of the node, \textit{deg\_n} is its + total degree, and \textit{z\_n} is the corresponding Z-score. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers, and the node is omitted from the + output.} + +\myreference{\refmetrics} diff --git a/doc/latex/latex/structure/metrics/part_coeff.tex b/doc/latex/latex/structure/metrics/part_coeff.tex new file mode 100644 index 0000000..f3afd7d --- /dev/null +++ b/doc/latex/latex/structure/metrics/part_coeff.tex @@ -0,0 +1,43 @@ +\myprogram{{part\_coeff.py}} + {compute the multiplex partifipation coefficient of all the nodes of + a multiplex.} {$<$layer1$>$ $<$layer2$>$ [$<$layer3$>$...]} + +\mydescription{Compute and print on output the multiplex participation + coefficient $P_i$ for each node $i$ of a multiplex. The + participation coefficient is defined as: + + \begin{equation*} + P_i=\frac{M}{M-1}\left[1-\sum_{\alpha=1}^M\biggl(\frac{k_i^{[\alpha]}}{o_i}\biggr)^2\right] + \end{equation*} + + \noindent Note that $P_i$ takes values in $[0,1]$, where $P_i=0$ + if and only if node $i$ is active on exactly one of the layers, + while $P_i=1$ if node $i$ has equal degree on all the $M$ layers. + + Each input file contains the (undirected) edge list of a layer, and + each line is in the format: + + \hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID} + + where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two + endpoints of an edge.} + +\myreturn{The program prints on \texttt{stdout} a list of lines in the + format: + + \hspace{0.5cm} \textit{deg\_n P\_n col\_n} + + where \textit{deg\_n} is the total degree of node $n$, \textit{P\_n} + is the participation coefficient of node $n$ and \textit{col} is the + integer representation of the activity bitstring of node $n$, which + is a number between $0$ and $2^{M}-1$. The field \textit{col} might + be useful in visualisations, where it would be possible to associate + different colors to nodes having diffrent node activity patterns. + + \noindent As usual, node IDs start from zero and proceed + sequentially, without gaps, i.e., if a node ID is not present in any + of the layer files given as input, the program considers it as being + isolated on all the layers, and is set to zero. + } + +\myreference{\refmetrics} diff --git a/doc/latex/latex/structure/reinforcement/reinforcement.tex b/doc/latex/latex/structure/reinforcement/reinforcement.tex new file mode 100644 index 0000000..a65b05e --- /dev/null +++ b/doc/latex/latex/structure/reinforcement/reinforcement.tex @@ -0,0 +1,21 @@ +%%% +%%% Layer activity +%%% + +\myprogram{{reinforcement.py}} + {compute the probability to have a link between two nodes in layer $1$ given their weight in layer $2$.} + {$<$layer1$>$ $<$layer2$>$ $$ $$ $$} + +\mydescription{Compute and print on output the probability to have a link between two nodes in layer $1$ given their weight in layer $2$. +As input are given the files \textit{layer1}, \textit{layer2}, the number of bins for the link weights of the second layer, the minimum and the maximum values of the binning. + + The first file contains the binary edge list of layer $1$, the second file contains the weighted edge list of layer $2$. each + line is in the format: + + \hspace{0.5cm}\textit{bin\_min} \textit{bin\_max} \textit{freq} + + where \textit{bin\_min} and \textit{bin\_max} are the minimum and maximum values of the link weights of layer $2$ in that binning, and \textit{freq} is the probability to have a link on layer $1$ given such weight in layer $2$.} + +\myreturn{A list of lines, where the n-th line is the minimum and maximum values of the weight of the links in layer $2$ in the n-th bin, and the frequency to have a link on layer $2$ given that weight.} + +\myreference{\refmetrics} diff --git a/doc/latex/mammult_doc.tex b/doc/latex/mammult_doc.tex new file mode 100644 index 0000000..4d93d88 --- /dev/null +++ b/doc/latex/mammult_doc.tex @@ -0,0 +1,265 @@ +\documentclass[a4paper,11pt]{book} + +\usepackage[british]{babel} +\usepackage[latin1]{inputenc} +\usepackage{hyperref} +\usepackage{amsmath, amssymb} + +\newcommand{\lay}[1]{^{[#1]}} +\newcommand{\avg}[1]{\langle #1 \rangle} + + +\newcommand{\myprogram}[3]{\subsubsection{\texttt{#1}} + \textbf{NAME} + + {\textbf{#1} - #2} + + \vspace{0.5cm} + \noindent + \textbf{SYNOPSYS} + + {\textbf{#1} { }\texttt{\textit{#3} } } +} + +\newcommand{\mydescription}[1]{ + \vspace{0.5cm} + \noindent + \textbf{DESCRIPTION} + + {#1} +} + +\newcommand{\myreturn}[1]{ + \vspace{0.5cm} + \noindent + \textbf{OUTPUT} + + {#1} +} + + +\newcommand{\myreference}[1]{ + \vspace{0.5cm} + \noindent + \textbf{REFERENCE} + + {#1} +} + +%% +%% REFERENCES +%% + +\newcommand{\refgrowth}{V. Nicosia, G. Bianconi, V. Latora, + M. Barthelemy, ``Growing multiplex networks'', + \textit{Phys. Rev. Lett.} {\bf 111}, 058701 (2013). + + Link to paper: \url{http://prl.aps.org/abstract/PRL/v111/i5/e058701} +} +\newcommand{\refnonlinear}{V. Nicosia, G. Bianconi, V. Latora, + M. Barthelemy, ``Non-linear growth and condensation in multiplex + networks'', \textit{Phys. Rev. E} {\bf 90}, 042807 (2014). + + Link to paper: \url{http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807} +} +\newcommand{\refmetrics}{F. Battiston, V. Nicosia, V. Latora, + ``Structural measures for multiplex networks'', + \textit{Phys. Rev. E} {\bf 89}, 032804 (2014). + + Link to paper: \url{http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032804} +} +\newcommand{\refcorrelations}{V. Nicosia, V. Latora, ``Measuring and + modeling correlations in multiplex networks'', \textit{Phys. Rev. E} + {\bf 92}, 032805 (2015). + + Link to paper: \url{http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805}} + +\newcommand{\refreducibility}{M. De Domenico, V. Nicosia, A. Arenas, + V. Latora, \textit{``Structural reducibility of multilayer + networks''}, Nat. Commun. {\bf 6}, 6864 (2015). + + Link to paper: \url{http://www.nature.com/ncomms/2015/150423/ncomms7864/full/ncomms7864.html} +} +\newcommand{\refvisibility}{L. Lacasa, V. Nicosia, V. Latora, + \textit{``Network structure of multivariate time series''}, accepted + for publication in Scientific Reports, arxiv:1408.0925 (2015). + + Link to paper: \url{http://arxiv.org/abs/1408.0925} +} + +\newcommand{\refbiased}{F. Battiston, V. Nicosia, V. Latora, + \textit{``Biased random walks on multiplex networks''}, arxiv:1505.01378 (2015). + + Link to paper: \url{http://arxiv.org/abs/1505.01378} +} +\newcommand{\refising}{F. Battiston, A. Cairoli, V. Nicosia, A. Baule, V. Latora, + \textit{``Interplay between consensus and coherence in a model of interacting opinions''}, accepted + for publication in Physica D, arxiv:1506.04544 (2015). + + Link to paper: \url{http://arxiv.org/abs/1506.04544} +} +\newcommand{\refcommunity}{F. Battiston, J. Iacovacci, V. Nicosia, G. Bianconi, V. Latora, + \textit{``Emergence of multiplex communities in collaboration networks''}, arxiv:1506.01280 (2015). + + Link to paper: \url{http://arxiv.org/abs/1506.01280} +} + +\newcommand{\refaxelrod}{F. Battiston, V. Nicosia, V. Latora, M. San Miguel, + \textit{``Layered social influence promotes multiculturality''}, in preparation (2015). + + %Link to paper: \url{http://arxiv.org/abs/inpreparation} +} +\newcommand{\refmotifs}{F. Battiston, M. Chavez, V. Nicosia, V. Latora, + \textit{``Multilayer motifs in brain networks''}, in preparation (2015). + + %Link to paper: \url{http://arxiv.org/abs/inpreparation} +} + +\title{MAMMULT: Metrics And Models for MULTilayer networks} + + +\begin{document} + +\maketitle + +\chapter{Structural descriptors} + +\section{Basic node, edge, and layer properties} + +\subsection{Node and layer activity} + +This section includes programs related to the computation of node and +layer activity, activity vectors, pairwise multiplexity, pairwise +normalised Hamming distance, node degree vectors. + +\input{./latex/structure/activity/node_activity.tex} +\input{./latex/structure/activity/layer_activity.tex} +\input{./latex/structure/activity/node_activity_vectors.tex} +\input{./latex/structure/activity/layer_activity_vectors.tex} +\input{./latex/structure/activity/multiplexity.tex} +\input{./latex/structure/activity/hamming_dist.tex} +\input{./latex/structure/activity/node_degree_vectors.tex} +\input{./latex/structure/activity/degs_to_binary.tex} +\input{./latex/structure/activity/degs_to_activity_overlap.tex} + +\subsection{Layer aggregation} + +This section includes programs to obtain various single-layer +aggregated graphs associated to a multiplex network. + +\input{./latex/structure/metrics/aggregate_layers_w.tex} +\input{./latex/structure/metrics/intersect_layers.tex} + +\subsection{Node degree, participation coefficient, cartography} + +This section includes programs to compute the total degree and +participation coefficient of each node, and to draw the cartography +diagram of a multiplex. + +\input{./latex/structure/metrics/overlap_degree.tex} +\input{./latex/structure/metrics/cartography_from_layers.tex} +\input{./latex/structure/metrics/cartography_from_deg_vectors.tex} +\input{./latex/structure/metrics/cartography_from_columns.tex} + +\subsection{Edge overlap, reinforcement} +This section includes programs to compute the egde overlap and to +evaulate the edge reinforcement effect. + +\input{./latex/structure/metrics/edge_overlap.tex} +\input{./latex/structure/metrics/avg_edge_overlap.tex} +\input{./latex/structure/reinforcement/reinforcement.tex} + +\section{Inter-layer degree correlations} + +\subsection{Node ranking} + +This section includes various utilities to compute and compare node +rankings induced by any generic structural node property, including +degree at different layers. + +\input{./latex/structure/correlations/rank_nodes.tex} +\input{./latex/structure/correlations/rank_nodes_thresh.tex} +\input{./latex/structure/correlations/rank_occurrence.tex} + +\subsection{Interlayer degree correlation coefficients} + +This section includes programs for the computation of various +inter-layer degree correlation coefficients. + +\input{./latex/structure/correlations/compute_pearson.tex} +\input{./latex/structure/correlations/compute_rho.tex} +\input{./latex/structure/correlations/compute_tau.tex} + + +\subsection{Interlayer degree correlation functions} + +This section includes programs to compute intra-layer and inter-layer +degree correlation functions, and to fit those functions with a +power-law. + + +\input{./latex/structure/correlations/dump_k_q.tex} +\input{./latex/structure/correlations/knn_q_from_layers.tex} +%% \input{./latex/structure/correlations/knn_q_from_layers_log.tex} +\input{./latex/structure/correlations/knn_q_from_degrees.tex} +%% \input{./latex/structure/correlations/knn_q_from_degrees_log.tex} +\input{./latex/structure/correlations/fit_knn.tex} + +\chapter{Models of multi-layer networks} + +\section{Null models} + +\subsection{Null-models of node and layer activity} + +\input{./latex/models/nullmodels/model_hypergeometric.tex} +\input{./latex/models/nullmodels/model_MDM.tex} +\input{./latex/models/nullmodels/model_MSM.tex} +\input{./latex/models/nullmodels/model_layer_growth.tex} + + +\section{Growing multiplex networks} + +\subsection{Linear preferential attachment} + +\input{./latex/models/growth/nibilab_linear_delta.tex} +\input{./latex/models/growth/nibilab_linear_delay.tex} +\input{./latex/models/growth/nibilab_linear_delay_mix.tex} +\input{./latex/models/growth/nibilab_linear_random_times.tex} +%%\input{./latex/models/growth/nibilab_semilinear.tex} + +\subsection{Non-linear preferential attachment} + +\input{./latex/models/growth/nibilab_nonlinear.tex} + + +\subsection{Utilities} + +\input{./latex/models/growth/node_deg_over_time.tex} + +\section{Multiplex networks with inter-layer degree correlations} + +\subsection{Models based on simulated annealing} + +\input{./latex/models/correlations/tune_rho.tex} +\input{./latex/models/correlations/tune_qnn_adaptive.tex} + + +\chapter{Dynamics on multi-layer networks} + +\section{Interacting opinions - Multilayer ising model} + +\input{./latex/dynamics/Ising/multiplex_ising.tex} + +\section{Biased random walks} + +\subsection{Stationary distribution} + +\input{./latex/dynamics/randomwalks/statdistr2.tex} + +\subsection{Entropy rate} + +\input{./latex/dynamics/randomwalks/entropyrate2add.tex} +\input{./latex/dynamics/randomwalks/entropyrate2mult.tex} +\input{./latex/dynamics/randomwalks/entropyrate2int.tex} + +\end{document} -- cgit v1.2.3