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diff --git a/doc/latex/latex/structure/metrics/avg_edge_overlap.tex b/doc/latex/latex/structure/metrics/avg_edge_overlap.tex new file mode 100644 index 0000000..4df2d51 --- /dev/null +++ b/doc/latex/latex/structure/metrics/avg_edge_overlap.tex @@ -0,0 +1,42 @@ +\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} |