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Many complex networks display a mesoscopic structure with groups of nodessharing many links with the other nodes in their group and comparatively fewwith nodes of different groups. This feature is known as community structureand encodes precious information about the organization and the function of thenodes. Many algorithms have been proposed but it is not yet clear how theyshould be tested. Recently we have proposed a general class of undirected andunweighted benchmark graphs, with heterogenous distributions of node degree andcommunity size. An increasing attention has been recently devoted to developalgorithms able to consider the direction and the weight of the links, whichrequire suitable benchmark graphs for testing. In this paper we extend thebasic ideas behind our previous benchmark to generate directed and weightednetworks with built-in community structure. We also consider the possibilitythat nodes belong to more communities, a feature occurring in real systems,like, e. g., social networks. As a practical application, we show howmodularity optimization performs on our new benchmark.

Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities