A measure of the local connectivity between graph vertices

Measuring the connectivity strength between a pair of vertices in a graph is one of the most vital concerns in numerous computational graph problems. In this paper we propose a local measure, together with an efficient algorithm that is scalable and parallelizable. In the heart of the algorithm is an iterative process that propagates and refines random values associated with each vertex. The connectivity measure hence defined, named algebraic distance, is the absolute difference between two values after a small number of iterations. This process is inspired by the bootstrap algebraic multigrid (BAMG), where a similar process is employed to expose connections between variables and to determine their role in convergence of a multigrid solver. We show convergence properties of the proposed measure, and provide a mutually reinforcing model to explain why the algebraic distances meaningfully measure the connectivity in a local sense. The practical effectiveness of the proposed measure is demonstrated on several computational (hyper)graph problems.