Practical Nearly-Linear-Time Approximation Algorithms for Hybrid and Overlapping Graph Clustering

In many graph-clustering applications, overwhelming empirical evidence suggests that communities and clusters are naturally overlapping, calling for novel overlapping graph-partitioning algorithms (OGP). In this work, we introduce a framework based on two novel clustering objectives, which naturally extend the well-studied notion of conductance to overlapping clusters and to clusters with hybrid vertex- and edge-boundary structure. Our main algorithmic contributions are nearly-linear-time algorithms O(log n)-approximation algorithms for both these objectives. To this end, we show that the cut-matching framework of Khandekar et al. (2014) can be extended to overlapping partitions and give novel cut-improvement primitives that perform a small number of s-t maximum flow computations over the instance graph to detect sparse overlapping partitions near an input partition. Crucially, we implement our approximation algorithm to produce both overlapping and hybrid partitions for large graphs, easily scaling to tens of millions of edges, and test our implementation on real-world datasets against other competitive baselines.