Finding Subgraphs with Maximum Total Density and Limited Overlap

Abstract

Finding dense subgraphs in large graphs is a key primitive in a variety of real-world application domains, encompassing social network analytics, event detection, biology, and finance. In most such applications, one typically aims at finding several (possibly overlapping) dense subgraphs which might correspond to communities in social networks or interesting events. While a large amount of work is devoted to finding a single densest subgraph, perhaps surprisingly, the problem of finding several dense subgraphs with limited overlap has not been studied in a principled way, to the best of our knowledge. In this work we define and study a natural generalization of the densest subgraph problem, where the main goal is to find at most $k$ subgraphs with maximum total aggregate density, while satisfying an upper bound on the pairwise Jaccard coefficient between the sets of nodes of the subgraphs. After showing that such a problem is NP-Hard, we devise an efficient algorithm that comes with provable guarantees in some cases of interest, as well as, an efficient practical heuristic. Our extensive evaluation on large real-world graphs confirms the efficiency and effectiveness of our algorithms.