\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
  }