Triangle-aware Spectral Sparsifiers and Community Detection

Abstract

Triangle-aware graph partitioning has proven to be a successful approach to finding communities in real-world data [8, 40, 51, 54]. But how can we explain its empirical success? Triangle-aware graph partitioning methods rely on the count of triangles an edge is contained in, in contrast to the well-established measure of effective resistance [12] that requires global information about the graph. In this work we advance the understanding of triangle-based graph partitioning in two ways. First, we introduce a novel triangle-aware sparsification scheme. Our scheme provably produces a spectral sparsifier with high probability [46, 47] on graphs that exhibit strong triadic closure, a hallmark property of real-world networks. Importantly, our sampling scheme is amenable to distributed computing, since it relies simply on computing node degrees, and edge triangle counts. Finally, we compare our methods to the Spielman-Srivastava sparsification algorithm [46] on a wide variety of real-world graphs, and we verify the applicability of our proposed sparsification scheme on real-world networks. Secondly, we develop a data-driven approach towards understanding properties of real-world communities with respect to effective resistances, and triangle counts. Our empirical approach is mainly based on the notion of ground-truth communities in datasets made available originally by Yang and Leskovec [53]. We perform a study of triangle-aware measures, and effective resistances on edges within, and across communities, and we discover certain interesting empirical findings. For example, we observe that the Jaccard similarity of an edge used by Satuluri [40], and the closely related Tectonic similarity measure introduced by Tsourakakis et al. [51] provide consistently good signals of whether an edge is contained within a community or not.