Normalized Cuts Are Approximately Inverse Exit Times

The Normalized Cut is a widely used measure of separation between clusters in a graph. In this paper we provide a novel probabilistic perspective on this measure. We show that for a partition of a graph into two weakly connected sets, V = A ⊎ B, in fact Ncut(V ) = 1/τA→B + 1/τB→A, where τA→B is the uni-directional characteristic exit time of a random walk from subset A to subset B. Using matrix perturbation theory, we show that this approximation is exact to first order in the connection strength between the two subsets A and B, and derive an explicit bound for the approximation error. Our result implies that for a Markov chain composed of a reversible component A that is weakly connected to an absorbing cluster B, the easy-to-compute Normalized Cut measure is a reliable proxy for the chain’s spectral gap.