Fast and accurate determination of modularity and its effect size

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