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<h5 class="subsubsectionHead"><a
id="x60-590002.3.1"></a><span
class="cmtt-10x-x-109">tune</span><span
class="cmtt-10x-x-109">_qnn</span><span
class="cmtt-10x-x-109">_adaptive</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">tune</span><span
class="cmbx-10x-x-109">_qnn</span><span
class="cmbx-10x-x-109">_adaptive </span>- Construct a multiplex with prescribed inter-layer
correlations.
<!--l. 3--><p class="noindent" ><span
class="cmbx-10x-x-109">SYNOPSYS</span>
<!--l. 3--><p class="indent" > <span
class="cmbx-10x-x-109">tune</span><span
class="cmbx-10x-x-109">_qnn</span><span
class="cmbx-10x-x-109">_adaptive </span><span
class="cmmi-10x-x-109"><</span><span
class="cmitt-10x-x-109">degs1</span><span
class="cmmi-10x-x-109">> <</span><span
class="cmitt-10x-x-109">degs2</span><span
class="cmmi-10x-x-109">> <</span><span
class="cmitt-10x-x-109">mu</span><span
class="cmmi-10x-x-109">> <</span><span
class="cmitt-10x-x-109">eps</span><span
class="cmmi-10x-x-109">> <</span><span
class="cmitt-10x-x-109">beta</span><span
class="cmmi-10x-x-109">></span>
<span
class="cmitt-10x-x-109">[RND|NAT|INV]</span>
<!--l. 52--><p class="noindent" ><span
class="cmbx-10x-x-109">DESCRIPTION</span>
<!--l. 52--><p class="indent" > This programs tunes the inter-layer degree correlation exponent <span
class="cmmi-10x-x-109">μ</span>. 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_doc25x.png" alt="-- ∑ ′ ′
q(k) = q P(q |k)
q′
" class="math-display" ></center></td></tr></table>
<!--l. 52--><p class="nopar" >
<!--l. 52--><p class="indent" > where <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 <span
class="cmmi-10x-x-109">k </span>on layer
1.
<!--l. 52--><p class="indent" > The program assumes that we want to set the degree correlation function
such that:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc26x.png" alt="q(k) = akμ
" class="math-display" ></center></td></tr></table>
<!--l. 52--><p class="nopar" >
<!--l. 52--><p class="indent" > where the exponent of the power-law function is given by the user
(it is indeed the parameter <span
class="cmti-10x-x-109">mu</span>), and successively adjusts the pairing
between nodes at the two layers in order to obtain a correlation function as
close as possible to the desired one. The files <span
class="cmti-10x-x-109">degs1 </span>and <span
class="cmti-10x-x-109">degs2 </span>contain,
respectively, the degrees of the nodes on the first layer and on the second
layer.
<!--l. 52--><p class="indent" > The parameter <span
class="cmti-10x-x-109">eps </span>is the accuracy of <span
class="cmti-10x-x-109">mu</span>. For instance, if <span
class="cmti-10x-x-109">mu </span>is set equal to
-0.25 and <span
class="cmti-10x-x-109">eps </span>is equal to 0.0001, the program stops when the configuration of
node pairing corresponds to a value of the exponent <span
class="cmmi-10x-x-109">μ </span>which differs from -0.25 by
less than 0.0001.
<!--l. 52--><p class="indent" > The parameter <span
class="cmti-10x-x-109">beta </span>is the typical inverse temperature of simulated
annealing.
<!--l. 52--><p class="indent" > If no other parameter is specified, or if the last parameter is <span
class="cmtt-10x-x-109">RND</span>, the program
starts from a random pairing of nodes. If the last parameter is <span
class="cmtt-10x-x-109">NAT </span>then the
program assumes that the initial pairing is the natural one, where the nodes have
the same ID on both layers. Finally, if <span
class="cmtt-10x-x-109">INV </span>is specified, the initial pairing is the
inverse pairing, i.e. the one where node 0 on layer 1 is paired with node N-1 on
layer 2, and so on.
<!--l. 62--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 62--><p class="indent" > The program prints on <span
class="cmtt-10x-x-109">stdout </span>a pairing, i.e. a list of lines in the
format:
<!--l. 62--><p class="indent" >   <span
class="cmti-10x-x-109">IDL1 IDL2</span>
<!--l. 62--><p class="indent" > where <span
class="cmti-10x-x-109">IDL1 </span>is the ID of the node on layer 1 and <span
class="cmti-10x-x-109">IDL2 </span>is the corresponding
ID of the same node 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, “Measuring and modeling correlations in multiplex
networks”, <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>
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