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\myprogram{{knn\_q\_from\_degrees.py}}
{compute the inter-layer degree-degree correlation function.}
{$<$filein$>$}
\mydescription{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
\hspace{0.5cm} \textit{ki qi}
where \textit{ki} and \textit{qi} are, respectively, the degree at
layer 1 and the degree at layer 2 of node \textit{i}.
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) = \frac{1}{N_{k}}\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 $N_k$ is the number of nodes with degree
$k$ on the first 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) = \frac{1}{N_{q}}\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 prints on \texttt{stdout} a list of lines in
the format:
\hspace{0.5cm} \textit{k $\overline{q}(k)$}
where \textit{k} is the degree on layer $1$ and
$\overline{q}(k)$ is the average degree on layer $2$ of
nodes having degree equal to $k$ on layer $1$.
The program also prints on \texttt{stderr} a list of lines in
the format:
\hspace{0.5cm} \textit{q $\overline{k}(q)$}
where \textit{q} is the degree on layer $2$ and
$\overline{k}(q)$ is the average degree on layer $1$ of
nodes having degree equal to $q$ on layer $2$.
}
\myreference{\refcorrelations
\refgrowth
\refnonlinear
}
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