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\myprogram{{avg\_edge\_overlap.py}}
{compute the average edge overlap of a multiplex.}
{$<$layer1$>$ [$<$layer2$>$...]}
\mydescription{Compute and print on output the average edge overlap
\begin{equation*} \omega^{*}
= \frac{\sum_{i}\sum_{j>i}\sum_{\alpha}a_{ij}\lay{\alpha}}{ \sum_{i}\sum_{j>i}(1
- \delta_{0,\sum_{\alpha}a_{ij}\lay{\alpha}})} \end{equation*}
\noindent i.e., the expected \textit{number} of layers on which an
edge of the multiplex exists, and the corresponding normalised
quantity:
\begin{equation*}
\omega = \frac{\sum_{i}\sum_{j>i}\sum_{\alpha}a_{ij}\lay{\alpha}}{M \sum_{i}\sum_{j>i}(1
- \delta_{0,\sum_{\alpha}a_{ij}\lay{\alpha}})}
\end{equation*}
\noindent that is the expected \textit{fraction} of layers on which
an edge of the multiplex is present.
Each input file contains the (undirected) edge list of a layer, and
each line is in the format:
\hspace{0.5cm}\textit{src\_ID} \textit{dest\_ID}
where \textit{src\_ID} and \textit{dest\_ID} are the IDs of the two
endpoints of an edge.}
\myreturn{The program prints on \texttt{stdout} a single line, in the
format:
\hspace{0.5cm} \textit{omega\_star omega}
\noindent where \textit{omega\_star} and \textit{omega} are,
respectively, the expected number and fraction of layers in which an
edge is present.}
\myreference{\refmetrics
\vspace{0.5cm}\refvisibility}
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