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Metrics and Models for Multi-layer Networks
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<head><title>1.2.3.0 knn_q_from_layers.py</title>
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<h5 class="subsubsectionHead"><a
id="x38-370001.2.3"></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">_layers.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">_layers.py </span>- compute intra-layer and inter-layer degree-degree
correlation coefficients.
<!--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">_layers.py </span><span
class="cmmi-10x-x-109">&#x003C;</span><span
class="cmitt-10x-x-109">layer1</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">layer2</span><span
class="cmmi-10x-x-109">&#x003E;</span>
<!--l. 43--><p class="noindent" ><span
class="cmbx-10x-x-109">DESCRIPTION</span>
<!--l. 43--><p class="indent" > 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:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc6x.png" alt=" --1- &sum; &prime; &prime;
&#x27E8;knn(k)&#x27E9; = kNk k P(k |k )
k&prime;
" class="math-display" ></center></td></tr></table>
<!--l. 43--><p class="nopar" >
<!--l. 43--><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">k</span>) is the probability that a neighbour of a node with degree <span
class="cmmi-10x-x-109">k </span>has
degree <span
class="cmmi-10x-x-109">k</span><span
class="cmsy-10x-x-109">&prime;</span>, 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>. The quantity <span
class="cmsy-10x-x-109">&#x27E8;</span><span
class="cmmi-10x-x-109">k</span><sub><span
class="cmmi-8">nn</span></sub>(<span
class="cmmi-10x-x-109">k</span>)<span
class="cmsy-10x-x-109">&#x27E9; </span>is
the average degree of the neighbours of nodes having degree equal to
<span
class="cmmi-10x-x-109">k</span>.
<!--l. 43--><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_doc7x.png" alt="-- &sum; &prime; &prime;
k(q) = k P(k |q)
k&prime;
" class="math-display" ></center></td></tr></table>
<!--l. 43--><p class="nopar" >
<!--l. 43--><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">q</span></sub> is the number of nodes
with degree <span
class="cmmi-10x-x-109">q </span>on the second 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_doc8x.png" alt="-- &sum; &prime; &prime;
q(k) = q P(q |k)
q&prime;
" class="math-display" ></center></td></tr></table>
<!--l. 43--><p class="nopar" >
<!--l. 43--><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. 73--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 73--><p class="indent" > The program creates two output files, respectively called
<!--l. 73--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">file1</span><span
class="cmti-10x-x-109">_file2</span><span
class="cmti-10x-x-109">_k1</span>
<!--l. 73--><p class="indent" > and
<!--l. 73--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">file1</span><span
class="cmti-10x-x-109">_file2</span><span
class="cmti-10x-x-109">_k2</span>
<!--l. 73--><p class="indent" > The first file contains a list of lines in the format:
<!--l. 73--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">k </span><span
class="cmsy-10x-x-109">&#x27E8;</span><span
class="cmmi-10x-x-109">k</span><sub><span
class="cmmi-8">nn</span></sub>(<span
class="cmmi-10x-x-109">k</span>)<span
class="cmsy-10x-x-109">&#x27E9; </span><span
class="cmmi-10x-x-109">&sigma;</span><sub><span
class="cmmi-8">k</span></sub> <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">&sigma;</span><sub><span class="overline"><span
class="cmmi-10x-x-109">q</span></span></sub>
<!--l. 73--><p class="indent" > where <span
class="cmmi-10x-x-109">k </span>is the degree at first layer, <span
class="cmsy-10x-x-109">&#x27E8;</span><span
class="cmmi-10x-x-109">k</span><sub><span
class="cmmi-8">nn</span></sub>(<span
class="cmmi-10x-x-109">k</span>)<span
class="cmsy-10x-x-109">&#x27E9; </span>is the average degree of the
neighbours at layer 1 of nodes having degree <span
class="cmmi-10x-x-109">k </span>at layer 1, <span
class="cmmi-10x-x-109">&sigma;</span><sub><span
class="cmmi-8">k</span></sub> is the standard
deviation associated to <span
class="cmsy-10x-x-109">&#x27E8;</span><span
class="cmmi-10x-x-109">k</span><sub><span
class="cmmi-8">nn</span></sub>(<span
class="cmmi-10x-x-109">k</span>)<span
class="cmsy-10x-x-109">&#x27E9;</span>, <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>) is the average degree at layer 2 of nodes
having degree equal to <span
class="cmmi-10x-x-109">k </span>at layer 1, and <span
class="cmmi-10x-x-109">&sigma;</span><sub><span class="overline"><span
class="cmmi-10x-x-109">q</span></span></sub> is the standard deviation associated
to <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>).
<!--l. 73--><p class="indent" > The second file contains a similar list of lines, in the format:
<!--l. 73--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">q </span><span
class="cmsy-10x-x-109">&#x27E8;</span><span
class="cmmi-10x-x-109">q</span><sub><span
class="cmmi-8">nn</span></sub>(<span
class="cmmi-10x-x-109">q</span>)<span
class="cmsy-10x-x-109">&#x27E9; </span><span
class="cmmi-10x-x-109">&sigma;</span><sub><span
class="cmmi-8">q</span></sub> <span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>) <span
class="cmmi-10x-x-109">&sigma;</span><sub><span class="overline"><span
class="cmmi-10x-x-109">k</span></span></sub>
<!--l. 73--><p class="indent" > with obvious meaning.
<!--l. 80--><p class="noindent" ><span
class="cmbx-10x-x-109">REFERENCE</span>
<!--l. 80--><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. 80--><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. 80--><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. 80--><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. 80--><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. 80--><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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