Hierarchies of Predominantly Connected Communities

Abstract

We consider communities whose vertices are predominantly connected, i.e., the vertices in each community are stronger connected to other community members of the same community than to vertices outside the community. Flake et al. introduced a hierarchical clustering algorithm that finds such predominantly connected communities of different coarseness depending on an input parameter. We present a simple and efficient method for constructing a clustering hierarchy according to Flake et al. that supersedes the necessity of choosing feasible parameter values and guarantees the completeness of the resulting hierarchy, i.e., the hierarchy contains all clusterings that can be constructed by the original algorithm for any parameter value. However, predominantly connected communities are not organized in a single hierarchy. Thus, we develop a framework that, after precomputing at most $2(n-1)$ maximum flows, admits a linear time construction of a clustering $\C(S)$ of predominantly connected communities that contains a given community $S$ and is maximum in the sense that any further clustering of predominantly connected communities that also contains $S$ is hierarchically nested in $\C(S)$. We further generalize this construction yielding a clustering with similar properties for $k$ given communities in $O(kn)$ time. This admits the analysis of a network's structure with respect to various communities in different hierarchies.