Detectability of hierarchical communities in networks.

We study the problem of recovering a planted hierarchy of partitions in a network. The detectability of a single planted partition has previously been analyzed in detail and a phase transition has been identified below which the partition cannot be detected. Here we show that, in the hierarchical setting, there exist additional phases in which the presence of multiple consistent partitions can either help or hinder detection. Accordingly, the detectability limit for nonhierarchical partitions typically provides insufficient information about the detectability of the complete hierarchical structure, as we highlight with several constructive examples.