Detecting and generating overlapping nested communities

Nestedness has been observed in a variety of networks but has been primarily viewed in the context of bipartite networks. Numerous metrics quantify nestedness and some clustering methods identify fully nested parts of graphs, but all with similar limitations. Clustering approaches also fail to uncover the overlap between fully nested subgraphs, as they assign vertices to a single group only. In this paper, we look at the nestedness of a network through an auxiliary graph, in which a directed edge represents a nested relationship between the two corresponding vertices of the network. We present an algorithm that recovers this so-called community graph, and finds the overlapping fully nested subgraphs of a network. We also introduce an algorithm for generating graphs with such nested structure given by a community graph. This algorithm can be used to test a community detection algorithm of this kind and potentially to evaluate different metrics of nestedness as well. Finally, we evaluate our nested community detection algorithm on a large variety of networks, including biparite and non-bipartite ones too. We derive a new metric from the community graph to quantify the nestedness of both bipartite and non-bipartite networks.