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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 knn_q_from_degrees.py</title>
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<h5 class="subsubsectionHead"><a
id="x39-380001.2.3"></a><span
class="cmtt-10x-x-109">knn</span><span
class="cmtt-10x-x-109">_q</span><span
class="cmtt-10x-x-109">_from</span><span
class="cmtt-10x-x-109">_degrees.py</span></h5>
<!--l. 3--><p class="noindent" ><span
class="cmbx-10x-x-109">NAME</span>
<!--l. 3--><p class="indent" > <span
class="cmbx-10x-x-109">knn</span><span
class="cmbx-10x-x-109">_q</span><span
class="cmbx-10x-x-109">_from</span><span
class="cmbx-10x-x-109">_degrees.py </span>- compute the inter-layer degree-degree
correlation function.
<!--l. 3--><p class="noindent" ><span
class="cmbx-10x-x-109">SYNOPSYS</span>
<!--l. 3--><p class="indent" > <span
class="cmbx-10x-x-109">knn</span><span
class="cmbx-10x-x-109">_q</span><span
class="cmbx-10x-x-109">_from</span><span
class="cmbx-10x-x-109">_degrees.py </span><span
class="cmmi-10x-x-109">&#x003C;</span><span
class="cmitt-10x-x-109">filein</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" > 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
<!--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>and <span
class="cmti-10x-x-109">qi </span>are, respectively, the degree at layer 1 and the degree at layer
2 of node <span
class="cmti-10x-x-109">i</span>.
<!--l. 37--><p class="indent" > If we consider two layers of a multiplex, and we denote by <span
class="cmmi-10x-x-109">k </span>the degree
of a node on the first layer and by <span
class="cmmi-10x-x-109">q </span>the degree of the same node on
the second layers, the inter-layer degree correlation function is defined
as
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc9x.png" alt="k(q) = -1-&sum; k&prime;P (k&prime;|q)
Nk &prime;
k
" class="math-display" ></center></td></tr></table>
<!--l. 37--><p class="nopar" >
<!--l. 37--><p class="indent" > where <span
class="cmmi-10x-x-109">P</span>(<span
class="cmmi-10x-x-109">k</span><span
class="cmsy-10x-x-109">&prime;|</span><span
class="cmmi-10x-x-109">q</span>) is the probability that a node with degree <span
class="cmmi-10x-x-109">q </span>on the second
layer has degree equal to <span
class="cmmi-10x-x-109">k</span><span
class="cmsy-10x-x-109">&prime; </span>on the first layer, and <span
class="cmmi-10x-x-109">N</span><sub><span
class="cmmi-8">k</span></sub> is the number of
nodes with degree <span
class="cmmi-10x-x-109">k </span>on the first layer. The quantity <span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>) is the expected
degree at layer 1 of node that have degree equal to <span
class="cmmi-10x-x-109">q </span>on layer 2. The dual
quantity:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc10x.png" alt=" &sum;
q(k) = -1- 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" > is the average degree on layer 2 of nodes having degree <span
class="cmmi-10x-x-109">k </span>on layer
1.
<!--l. 57--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 57--><p class="indent" > The program prints on <span
class="cmtt-10x-x-109">stdout </span>a list of lines in the format:
<!--l. 57--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">k </span><span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>)
<!--l. 57--><p class="indent" > where <span
class="cmti-10x-x-109">k </span>is the degree on layer 1 and <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>) is the average degree on layer 2 of
nodes having degree equal to <span
class="cmmi-10x-x-109">k </span>on layer 1.
<!--l. 57--><p class="indent" > The program also prints on <span
class="cmtt-10x-x-109">stderr </span>a list of lines in the format:
<!--l. 57--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">q </span><span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>)
<!--l. 57--><p class="indent" > where <span
class="cmti-10x-x-109">q </span>is the degree on layer 2 and <span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>) is the average degree on layer 1 of
nodes having degree equal to <span
class="cmmi-10x-x-109">q </span>on layer 2.
<!--l. 64--><p class="noindent" ><span
class="cmbx-10x-x-109">REFERENCE</span>
<!--l. 64--><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. 64--><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. 64--><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. 64--><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. 64--><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. 64--><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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