Analyzing community structure in 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 community membership matrix mapping vertices to overlapping or non-overlapping communities, we present a new unsupervised method for analyzing and ranking these 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. The weighted matrix decomposition we obtain has a simple combinatorial interpretation. We show that the proposed optimization problem reduces to a Nonnegative Least Squares problem, and design a fast algorithm for computing the community weights. We assess this problem formulation on a variety of synthetic and real-world networks, in order to gain insight into its advantages and limitations.