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Metrics and Models for Multi-layer Networks
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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<head><title>1.2.3.0 fit_knn</title>
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<h5 class="subsubsectionHead"><a
id="x40-390001.2.3"></a><span
class="cmtt-10x-x-109">fit</span><span
class="cmtt-10x-x-109">_knn</span></h5>
<!--l. 4--><p class="noindent" ><span
class="cmbx-10x-x-109">NAME</span>
<!--l. 4--><p class="indent" > <span
class="cmbx-10x-x-109">fit</span><span
class="cmbx-10x-x-109">_knn </span>- power-law fit of the inter-layer degree correlation function.
<!--l. 4--><p class="noindent" ><span
class="cmbx-10x-x-109">SYNOPSYS</span>
<!--l. 4--><p class="indent" > <span
class="cmbx-10x-x-109">fit</span><span
class="cmbx-10x-x-109">_knn </span><span
class="cmmi-10x-x-109">&#x003C;</span><span
class="cmitt-10x-x-109">filein</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">alpha</span><span
class="cmmi-10x-x-109">&#x003E;</span>
<!--l. 37--><p class="noindent" ><span
class="cmbx-10x-x-109">DESCRIPTION</span>
<!--l. 37--><p class="indent" > Perform a power-law fit of the inter-layer degree correlation function:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc11x.png" alt="-- 1 &sum;
q(k) = --- q&prime;P (q&prime;|k)
Nq q&prime;
" class="math-display" ></center></td></tr></table>
<!--l. 37--><p class="nopar" >
<!--l. 37--><p class="indent" > where <span
class="cmmi-10x-x-109">k </span>is the degree of a node on layer 1, <span
class="cmmi-10x-x-109">q </span>is the degree on layer 2 and
<span
class="cmmi-10x-x-109">P</span>(<span
class="cmmi-10x-x-109">q</span><span
class="cmsy-10x-x-109">|</span><span
class="cmmi-10x-x-109">k</span>) is the probability that a node with degree <span
class="cmmi-10x-x-109">k </span>on layer 1 has degree <span
class="cmmi-10x-x-109">q </span>on
layer 2. The program assumes that <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>) can be written in the form <span
class="cmmi-10x-x-109">ak</span><sup><span
class="cmmi-8">b</span></sup>, and
computes the two parameters <span
class="cmmi-10x-x-109">a </span>and <span
class="cmmi-10x-x-109">b </span>through a linear fit of the log-log plot of
<span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>).
<!--l. 37--><p class="indent" > The input file <span
class="cmti-10x-x-109">filein </span>contains a list of lines in the format:
<!--l. 37--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">ki qi</span>
<!--l. 37--><p class="indent" > where <span
class="cmti-10x-x-109">ki </span>is the degree of node <span
class="cmmi-10x-x-109">i </span>at layer 1 and <span
class="cmti-10x-x-109">qi </span>is the degree of node <span
class="cmmi-10x-x-109">i </span>at
layer 2.
<!--l. 37--><p class="indent" > The second parameter <span
class="cmti-10x-x-109">alpha </span>is the ratio of the progression used to generate
the exponentially-distributed bins for the log-log plot. Typical values of <span
class="cmti-10x-x-109">alpha </span>are
between 1<span
class="cmmi-10x-x-109">.</span>1 and 2<span
class="cmmi-10x-x-109">.</span>0.
<!--l. 37--><p class="indent" > N.B.: The exponent <span
class="cmmi-10x-x-109">b </span>computed with this method is known to be
inaccurate.
<!--l. 43--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 43--><p class="indent" > The program prints on <span
class="cmtt-10x-x-109">stdout </span>the values of the parameters <span
class="cmmi-10x-x-109">a </span>and <span
class="cmmi-10x-x-109">b </span>of the
power-law fit <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>) = <span
class="cmmi-10x-x-109">ak</span><sup><span
class="cmmi-8">b</span></sup>.
<!--l. 50--><p class="noindent" ><span
class="cmbx-10x-x-109">REFERENCE</span>
<!--l. 50--><p class="indent" > V. Nicosia, V. Latora, &#8220;Measuring and modeling correlations in multiplex
networks&#8221;, <span
class="cmti-10x-x-109">Phys. Rev. E </span><span
class="cmbx-10x-x-109">92</span>, 032805 (2015).
<!--l. 50--><p class="indent" > Link to paper: <a
href="http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805" class="url" ><span
class="cmtt-10x-x-109">http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805</span></a>
<!--l. 50--><p class="indent" > V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, &#8220;Growing multiplex
networks&#8221;, <span
class="cmti-10x-x-109">Phys. Rev. Lett. </span><span
class="cmbx-10x-x-109">111</span>, 058701 (2013).
<!--l. 50--><p class="indent" > Link to paper: <a
href="http://prl.aps.org/abstract/PRL/v111/i5/e058701" class="url" ><span
class="cmtt-10x-x-109">http://prl.aps.org/abstract/PRL/v111/i5/e058701</span></a>
<!--l. 50--><p class="indent" > V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, &#8220;Non-linear growth and
condensation in multiplex networks&#8221;, <span
class="cmti-10x-x-109">Phys. Rev. E </span><span
class="cmbx-10x-x-109">90</span>, 042807 (2014).
<!--l. 50--><p class="indent" > Link to paper: <a
href="http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807" class="url" ><span
class="cmtt-10x-x-109">http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807</span></a>
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