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authorKatolaZ <katolaz@freaknet.org>2017-09-27 15:06:31 +0100
committerKatolaZ <katolaz@freaknet.org>2017-09-27 15:06:31 +0100
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+ <title>johnson_cycles(1) - Enumerate the simple cycles of a graph</title>
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+ <div class='man-navigation' style='display:none'>
+ <a href="#NAME">NAME</a>
+ <a href="#SYNOPSIS">SYNOPSIS</a>
+ <a href="#DESCRIPTION">DESCRIPTION</a>
+ <a href="#PARAMETERS">PARAMETERS</a>
+ <a href="#OUTPUT">OUTPUT</a>
+ <a href="#EXAMPLES">EXAMPLES</a>
+ <a href="#SEE-ALSO">SEE ALSO</a>
+ <a href="#REFERENCES">REFERENCES</a>
+ <a href="#AUTHORS">AUTHORS</a>
+ </div>
+
+ <ol class='man-decor man-head man head'>
+ <li class='tl'>johnson_cycles(1)</li>
+ <li class='tc'>www.complex-networks.net</li>
+ <li class='tr'>johnson_cycles(1)</li>
+ </ol>
+
+ <h2 id="NAME">NAME</h2>
+<p class="man-name">
+ <code>johnson_cycles</code> - <span class="man-whatis">Enumerate the simple cycles of a graph</span>
+</p>
+
+<h2 id="SYNOPSIS">SYNOPSIS</h2>
+
+<p><code>johnson_cycles</code> <var>graph_in</var> [<var>max_length</var> [SHOW]]</p>
+
+<h2 id="DESCRIPTION">DESCRIPTION</h2>
+
+<p><code>johnson_cycles</code> enumerates all the simple cycles of the graph given
+on input, and prints the total number of cycles of each length. If
+<var>max_length</var> is provided, <code>johnson_cycles</code> ignores any cycle whose
+length is larger than <var>max_length</var>. If <code>SHOW</code> is given as third
+argument, all the found cycles are printed on STDERR as soon as they
+are found.</p>
+
+<h2 id="PARAMETERS">PARAMETERS</h2>
+
+<ul>
+<li><p><var>graph_in</var>:
+ input graph (edge list) if equal to <code>-</code> (dash), read the edge list
+ from STDIN.</p></li>
+<li><p><var>max_length</var>: The maximum length of cycles to count/print. If not
+ provided, <code>johnson_cycles</code> tries to find all the cycles of the
+ graph (see the <strong>WARNING</strong> in the section <a href="#OUTPUT" title="OUTPUT" data-bare-link="true">OUTPUT</a> below).</p></li>
+<li><p>SHOW:
+ If the third (optional) parameter is equal to <code>SHOW</code>, the program
+ will dump on the standard error all the cycles found.</p></li>
+</ul>
+
+
+<h2 id="OUTPUT">OUTPUT</h2>
+
+<p><code>johnson_cycles</code> prints on the standard output the number of cycles of
+each length, in the format:</p>
+
+<pre><code> 2 N_2
+ 3 N_3
+ 4 N_4
+ 5 N_5
+ ...
+</code></pre>
+
+<p>where 2, 3, 4, 5... is the cycle lengths and N_2, N_3, N_4, N_5... is
+(twice) the number of cycles of that length. If <code>SHOW</code> is given, each
+cycle is also printed on STDERR as soon as it is found, in the format:</p>
+
+<pre><code>node_l node_(l-1) node_(l-2) ... node_0
+</code></pre>
+
+<p>where <code>node_l</code>, <code>node_(l-1)</code>, etc. are the labels of the nodes
+belonging to the cycle which starts at node <code>node_0</code>.</p>
+
+<p><strong>WARNING</strong>: If the second parameter <var>max_length</var> is not provided,
+ <code>johnson_cycles</code> will try to enumerate <strong>all</strong> the cycles of the
+ graph. In general, this might take a time <strong>exponential</strong> in the
+ number of nodes and edges of the graph. As a consequence, specifying
+ a maximum length is <strong>highly recommended</strong> if you are not interested
+ in finding the number of cycles of any length.</p>
+
+<h2 id="EXAMPLES">EXAMPLES</h2>
+
+<p>We can count the cycle of any length in the graph of Florentine
+families using the command:</p>
+
+<pre><code> $ johnson_cycles florentine.net
+ 2 20
+ 3 6
+ 4 4
+ 5 6
+ 6 10
+ 7 20
+ 8 22
+ 9 8
+ 10 2
+ 11 0
+ 12 0
+ 13 0
+ 14 0
+ 15 0
+ 16 0
+ $
+</code></pre>
+
+<p>The output means that the graph has 20 cycles of length 2 (edges),
+6/2=3 cycles of length 3, 4/2=2 cycles of length 4, and so on. We
+could otherwise focus on the cycles of length up to 5 and have each
+cycle printed on output:</p>
+
+<pre><code> $ johnson_cycles florentine.net 5 SHOW
+ 8 0
+ 5 1
+ 8 15 6 1
+ 8 12 15 6 1
+ 6 1
+ 6 15 12 8 1
+ 6 15 8 1
+ 8 1
+ 8 12 14 4 2
+ ....
+ 15 12
+ 2 20
+ 3 6
+ 4 4
+ 5 6
+ $
+</code></pre>
+
+<p>Apart from the degenerate cycles like "8 0", "5 1", etc.,
+corresponding to the cycles obtained by traversing the same undirected
+edge in the two possible directions, we see in that list some of the
+cycles of length 4 (such as "8 15 6 1") and of length 5 (such as "8 12
+15 6 1").</p>
+
+<p>The enumeration of all the cycles is normally impractical on larger
+graphs, so it is highly recommended to limit the search to short
+sizes. For instance, the command:</p>
+
+<pre><code> $ johnson_cycles er_1000_5000.net 6
+ 2 5000
+ 3 340
+ 4 2406
+ 5 19416
+ 6 160554
+ $
+</code></pre>
+
+<p>will require less than one second on a modern desktop computer, but
+the command:</p>
+
+<pre><code> $ johnson_cycles er_1000_5000.net 7
+ 2 5000
+ 3 340
+ 4 2406
+ 5 19416
+ 6 160554
+ 7 1360104
+ $
+</code></pre>
+
+<p>will probably take about 15 seconds, while:</p>
+
+<pre><code> $ johnson_cycles er_1000_5000.net 8
+ 2 5000
+ 3 340
+ 4 2406
+ 5 19416
+ 6 160554
+ 7 1360104
+ 8 11743500
+ $
+</code></pre>
+
+<p>will run for more than 2 minutes, and larger cycle lengths will
+require exponentially more time.</p>
+
+<h2 id="SEE-ALSO">SEE ALSO</h2>
+
+<p><a class="man-ref" href="f3m.1.html">f3m<span class="s">(1)</span></a>, <a class="man-ref" href="shortest.1.html">shortest<span class="s">(1)</span></a></p>
+
+<h2 id="REFERENCES">REFERENCES</h2>
+
+<ul>
+<li><p>D. B. Johnson. "Finding All the Elementary Circuits of a Directed
+Graph". SIAM J. Comput. 4 (1975), 77-84.</p></li>
+<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
+Methods and Applications", Chapter 8, Cambridge University Press
+(2017)</p></li>
+<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
+Methods and Applications", Appendix 15, Cambridge University Press
+(2017)</p></li>
+</ul>
+
+
+<h2 id="AUTHORS">AUTHORS</h2>
+
+<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code>&lt;v.nicosia@qmul.ac.uk&gt;</code>.</p>
+
+
+ <ol class='man-decor man-foot man foot'>
+ <li class='tl'>www.complex-networks.net</li>
+ <li class='tc'>September 2017</li>
+ <li class='tr'>johnson_cycles(1)</li>
+ </ol>
+
+ </div>
+</body>
+</html>
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