Scalable detection of statistically significant communities and hierarchies, using message-passing for modularity

Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory "communities" in random graphs where none exist. We address this problem by using the modularity as a Hamiltonian at finite temperature, and using an efficient Belief Propagation algorithm to obtain the consensus of many partitions with high modularity, rather than looking for a single partition that maximizes it. We show analytically and numerically that the proposed algorithm works all the way down to the detectability transition in networks generated by the stochastic block model. It also performs well on real-world networks, revealing large communities in some networks where previous work has claimed no communities exist. Finally we show that by applying our algorithm recursively, subdividing communities until no statistically-significant subcommunities can be found, we can detect hierarchical structure in real-world networks more efficiently than previous methods.