Ranking Communities in Weighted Networks

Real-world networks exhibit significant community structure. Communities are sometimes explicitly known or defined (e.g., virtual groups that one joins in an online social network, departments in an organization), but are often determined using a community detection or a clustering algorithm. Given a weighted network, with edge weights denoting interaction strengths between vertices, and a mapping of vertices to overlapping or non-overlapping communities, we present a new unsupervised method for ranking communities, such that the computed nonnegative community weights seek to explain the edge weights. Our method is based on a new factorization of the weighted adjacency matrix. Unlike Nonnegative Matrix Factorization, our decomposition has a simple combinatorial interpretation. We show that the proposed ranking problem reduces to a Nonnegative Least Squares problem, and design a fast algorithm for computing the ranking. We assess this ranking problem formulation on a variety of synthetic and real-world networks, in order to gain insight into its advantages and limitations.