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\myprogram{{knn\_q\_from\_layers.py}}
{compute intra-layer and inter-layer degree-degree
correlation coefficients.} {$<$layer1$>$ $<$layer2$>$}
\mydescription{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:
\begin{equation*}
\avg{k_{nn}(k)} = \frac{1}{k N_k}\sum_{k'}k'P(k'|k)
\end{equation*}
where $P(k'|k)$ is the probability that a neighbour of a
node with degree $k$ has degree $k'$, and $N_k$ is the
number of nodes with degree $k$. The quantity
$\avg{k_{nn}(k)}$ is the average degree of the neighbours of
nodes having degree equal to $k$.
If we consider two layers of a multiplex, and we denote by
$k$ the degree of a node on the first layer and by $q$ the
degree of the same node on the second layers, the
inter-layer degree correlation function is defined as
\begin{equation*}
\overline{k}(q) = \sum_{k'} k' P(k'|q)
\end{equation*}
where $P(k'|q)$ is the probability that a node with degree
$q$ on the second layer has degree equal to $k'$ on the
first layer, and $N_q$ is the number of nodes with degree
$q$ on the second layer. The quantity $\overline{k}(q)$ is
the expected degree at layer $1$ of node that have degree
equal to $q$ on layer $2$. The dual quantity:
\begin{equation*}
\overline{q}(k) = \sum_{q'} q' P(q'|k)
\end{equation*}
is the average degree on layer $2$ of nodes having degree
$k$ on layer $1$.
}
\myreturn{The program creates two output files, respectively called
\hspace{0.5cm} \textit{file1\_file2\_k1}
and
\hspace{0.5cm} \textit{file1\_file2\_k2}
The first file contains a list of lines in the format:
\hspace{0.5cm} \textit{k $\avg{k_{nn}(k)}$ $\sigma_k$
$\overline{q}(k)$ $\sigma_{\overline{q}}$}
where $k$ is the degree at first layer, $\avg{k_{nn}(k)}$ is the
average degree of the neighbours at layer $1$ of nodes having degree
$k$ at layer $1$, $\sigma_k$ is the standard deviation associated to
$\avg{k_{nn}(k)}$, $\overline{q}(k)$ is the average degree at layer
$2$ of nodes having degree equal to $k$ at layer $1$, and
$\sigma_{\overline{q}}$ is the standard deviation associated to
$\overline{q}(k)$.
The second file contains a similar list of lines, in the format:
\hspace{0.5cm} \textit{q $\avg{q_{nn}(q)}$ $\sigma_q$
$\overline{k}(q)$ $\sigma_{\overline{k}}$}
with obvious meaning.
}
\myreference{\refcorrelations
\refgrowth
\refnonlinear
}
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