A fast parametric maximum flow algorithm and applications
SIAM Journal on Computing
Very simple methods for all pairs network flow analysis
SIAM Journal on Computing
The Stanford GraphBase: a platform for combinatorial computing
The Stanford GraphBase: a platform for combinatorial computing
IEEE Transactions on Pattern Analysis and Machine Intelligence
On clusterings-good, bad and spectral
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
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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 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 further develop a framework that, after precomputing at most 2(n−1) maximum flows, admits a linear time construction of a clustering Ω(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 Ω(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.