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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"
"http://www.w3.org/TR/html4/loose.dtd">
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<head><title>2.2.1.0 nibilab_linear_random_times</title>
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<h5 class="subsubsectionHead"><a
id="x53-520002.2.1"></a><span
class="cmtt-10x-x-109">nibilab</span><span
class="cmtt-10x-x-109">_linear</span><span
class="cmtt-10x-x-109">_random</span><span
class="cmtt-10x-x-109">_times</span></h5>
<!--l. 6--><p class="noindent" ><span
class="cmbx-10x-x-109">NAME</span>
<!--l. 6--><p class="indent" > <span
class="cmbx-10x-x-109">nibilab</span><span
class="cmbx-10x-x-109">_linear</span><span
class="cmbx-10x-x-109">_random</span><span
class="cmbx-10x-x-109">_times </span>- Multiplex linear preferential attachment
model &#8211; Asynchronous arrival with randomly sampled arrival times on layer
2.
<!--l. 6--><p class="noindent" ><span
class="cmbx-10x-x-109">SYNOPSYS</span>
<!--l. 6--><p class="indent" > <span
class="cmbx-10x-x-109">nibilab</span><span
class="cmbx-10x-x-109">_linear</span><span
class="cmbx-10x-x-109">_random</span><span
class="cmbx-10x-x-109">_times </span><span
class="cmmi-10x-x-109">&#x003C;</span><span
class="cmitt-10x-x-109">N</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">m</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">m0</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">outfile</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">a</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">b</span><span
class="cmmi-10x-x-109">&#x003E;</span>
<span
class="cmmi-10x-x-109">&#x003C;</span><span
class="cmitt-10x-x-109">c</span><span
class="cmmi-10x-x-109">&#x003E; &#x003C;</span><span
class="cmitt-10x-x-109">d</span><span
class="cmmi-10x-x-109">&#x003E;</span>
<!--l. 50--><p class="noindent" ><span
class="cmbx-10x-x-109">DESCRIPTION</span>
<!--l. 50--><p class="indent" > Grow a two-layer multiplex network using the multiplex linear preferential
attachment model by Nicosia, Bianconi, Latora, Barthelemy (NiBiLaB).
<!--l. 50--><p class="indent" > The probability for a newly arrived node <span
class="cmmi-10x-x-109">i </span>to create a link to node <span
class="cmmi-10x-x-109">j </span>on layer
1 is:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc21x.png" alt="&Pi;1i&rarr;j &prop; ak[1j]+ bk[j2]
" class="math-display" ></center></td></tr></table>
<!--l. 50--><p class="nopar" >
<!--l. 50--><p class="indent" > and the dual probability for <span
class="cmmi-10x-x-109">i </span>to create a link to <span
class="cmmi-10x-x-109">j </span>on layer 2 is:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc22x.png" alt=" 2 [1] [2]
&Pi; i&rarr;j &prop; ck j + dkj
" class="math-display" ></center></td></tr></table>
<!--l. 50--><p class="nopar" >
<!--l. 50--><p class="indent" > Each new node arrives on layer 1, but its replica on the other layer appears at
a uniformly chosen random time in [<span
class="cmmi-10x-x-109">m</span>0 + 1;<span
class="cmmi-10x-x-109">N</span>].
<!--l. 50--><p class="indent" > The (mandatory) parameters are as follows:
<ul class="itemize1">
<li class="itemize"><span
class="cmbx-10x-x-109">N </span>number of nodes in the final graph
</li>
<li class="itemize"><span
class="cmbx-10x-x-109">m </span>number of new edges brought by each new node
</li>
<li class="itemize"><span
class="cmbx-10x-x-109">m0 </span>number of nodes in the initial seed graph. <span
class="cmti-10x-x-109">m0 </span>must be larger than
of equal to <span
class="cmti-10x-x-109">m</span>.
</li>
<li class="itemize"><span
class="cmbx-10x-x-109">outfile </span>the name of the file which will contain the
</li>
<li class="itemize"><span
class="cmbx-10x-x-109">a,b,c,d </span>the coefficients of the attaching probability function
</li></ul>
<!--l. 61--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 61--><p class="indent" > The program dumps on the file <span
class="cmtt-10x-x-109">outfile </span>the (undirected) edge list of the
resulting network. Each line of the file is in the format:
<!--l. 61--><p class="indent" > &#x00A0; <span
class="cmti-10x-x-109">src</span><span
class="cmti-10x-x-109">_ID dest</span><span
class="cmti-10x-x-109">_ID</span>
<!--l. 61--><p class="indent" > where <span
class="cmti-10x-x-109">src</span><span
class="cmti-10x-x-109">_ID </span>and <span
class="cmti-10x-x-109">dest</span><span
class="cmti-10x-x-109">_ID </span>are the IDs of the two endpoints of an
edge.
<!--l. 63--><p class="noindent" ><span
class="cmbx-10x-x-109">REFERENCE</span>
<!--l. 63--><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. 63--><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>
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