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<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>
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<ol class='man-decor man-head man head'>
<li class='tl'>fitmle(1)</li>
<li class='tc'>www.complex-networks.net</li>
<li class='tr'>fitmle(1)</li>
</ol>
<h2 id="NAME">NAME</h2>
<p class="man-name">
<code>fitmle</code> - <span class="man-whatis">Fit a set of values with a power-law distribution</span>
</p>
<h2 id="SYNOPSIS">SYNOPSIS</h2>
<p><code>fitmle</code> <var>data_in</var> [<var>tol</var> [TEST [<var>num_test</var>]]]</p>
<h2 id="DESCRIPTION">DESCRIPTION</h2>
<p><code>fitmle</code> fits the data points contained in the file <var>data_in</var> with a
power-law function P(k) ~ k<sup>-gamma</sup>, using the Maximum-Likelihood
Estimator (MLE). In particular, <code>fitmle</code> finds the exponent <code>gamma</code>
and the minimum of the values provided on input for which the
power-law behaviour holds. The second (optional) argument <var>tol</var> sets
the acceptable statistical error on the estimate of the exponent.</p>
<p>If <code>TEST</code> is provided, the program associates a p-value to the
goodness of the fit, based on the Kolmogorov-Smirnov statistics
computed on <var>num_test</var> sampled distributions from the theoretical
power-law function. If <var>num_test</var> is not provided, the test is based
on 100 sampled distributions.</p>
<h2 id="PARAMETERS">PARAMETERS</h2>
<dl>
<dt class="flush"><var>data_in</var></dt><dd><p> Set of values to fit. If is equal to <code>-</code> (dash), read the set from
STDIN.</p></dd>
<dt class="flush"><var>tol</var></dt><dd><p> The acceptable statistical error on the estimation of the
exponent. If omitted, it is set to 0.1.</p></dd>
<dt class="flush">TEST</dt><dd><p> If the third parameter is <code>TEST</code>, the program computes an estimate
of the p-value associated to the best-fit, based on <var>num_test</var>
synthetic samples of the same size of the input set.</p></dd>
<dt><var>num_test</var></dt><dd><p> Number of synthetic samples to use for the estimation of the
p-value of the best fit.</p></dd>
</dl>
<h2 id="OUTPUT">OUTPUT</h2>
<p>If <code>fitmle</code> is given less than three parameters (i.e., if <code>TEST</code> is
not specified), the output is a line in the format:</p>
<pre><code> gamma k_min ks
</code></pre>
<p>where <code>gamma</code> is the estimate for the exponent, <code>k_min</code> is the
smallest of the input values for which the power-law behaviour holds,
and <code>ks</code> is the value of the Kolmogorov-Smirnov statistics of the
best-fit.</p>
<p>If <code>TEST</code> is specified, the output line contains also the estimate of
the p-value of the best fit:</p>
<pre><code> gamma k_min ks p-value
</code></pre>
<p>where <code>p-value</code> is based on <var>num_test</var> samples (or just 100, if
<var>num_test</var> is not specified) of the same size of the input, obtained
from the theoretical power-law function computed as a best fit.</p>
<h2 id="EXAMPLES">EXAMPLES</h2>
<p>Let us assume that the file <code>AS-20010316.net_degs</code> contains the degree
sequence of the data set <code>AS-20010316.net</code> (the graph of the Internet
at the AS level in March 2001). The exponent of the best-fit power-law
distribution can be obtained by using:</p>
<pre><code> $ fitmle AS-20010316.net_degs
Using discrete fit
2.06165 6 0.031626 0.17
$
</code></pre>
<p>where <code>2.06165</code> is the estimated value of the exponent <code>gamma</code>, <code>6</code> is
the minimum degree value for which the power-law behaviour holds, and
<code>0.031626</code> is the value of the Kolmogorov-Smirnov statistics of the
best-fit. The program is also telling us that it decided to use the
discrete fitting procedure, since all the values in
<code>AS-20010316.net_degs</code> are integers. The latter information is printed
to STDERR.</p>
<p>It is possible to compute the p-value of the estimate by running:</p>
<pre><code> $ fitmle AS-20010316.net_degs 0.1 TEST
Using discrete fit
2.06165 6 0.031626 0.17
$
</code></pre>
<p>which provides a p-value equal to 0.17, meaning that 17% of the
synthetic samples showed a value of the KS statistics larger than that
of the best-fit. The estimation of the p-value here is based on 100
synthetic samples, since <var>num_test</var> was not provided. If we allow a
slightly larger value of the statistical error on the estimate of the
exponent <code>gamma</code>, we obtain different values of <code>gamma</code> and <code>k_min</code>,
and a much higher p-value:</p>
<pre><code> $ fitmle AS-20010316.net_degs 0.15 TEST 1000
Using discrete fit
2.0585 19 0.0253754 0.924
$
</code></pre>
<p>Notice that in this case, the p-value of the estimate is equal to
0.924, and is based on 1000 synthetic samples.</p>
<h2 id="SEE-ALSO">SEE ALSO</h2>
<p><span class="man-ref">deg_seq<span class="s">(1)</span></span>, <span class="man-ref">power_law<span class="s">(1)</span></span></p>
<h2 id="REFERENCES">REFERENCES</h2>
<ul>
<li><p>A. Clauset, C. R. Shalizi, and M. E. J. Newman. "Power-law
distributions in empirical data". SIAM Rev. 51, (2007), 661-703.</p></li>
<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
Methods and Applications", Chapter 5, Cambridge University Press
(2017)</p></li>
</ul>
<h2 id="AUTHORS">AUTHORS</h2>
<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code><v.nicosia@qmul.ac.uk></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'>fitmle(1)</li>
</ol>
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