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We present a fast spectral algorithm for community detection in complexnetworks. Our method searches for the partition with the maximum value of themodularity via the interplay of several refinement steps that include bothagglomeration and division. We validate the accuracy of the algorithm byapplying it to several real-world benchmark networks. On all these, ouralgorithm performs as well or better than any other known polynomial scheme.This allows us to extensively study the modularity distribution in ensembles ofErd\H{o}s-R\'enyi networks, producing theoretical predictions for means andvariances inclusive of finite-size corrections. Our work provides a way toaccurately estimate the effect size of modularity, providing a $z$-scoremeasure of it and enabling a more informative comparison of networks withdifferent numbers of nodes and links.

Fast and accurate determination of modularity and its effect size