Normalized cuts are approximately inverse exit times

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 {multiset union} B, the multiway normalized cut is approximately MN cut ≈ 1/τ_A→B + 1/τ_B→A, where τ_A→B is the unidirectional 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 we derive an explicit bound for the approximation error. Our result implies that for a Markov chain composed of a reversible subset A that is weakly connected to an absorbing subset B, the easy-to-compute normalized cut measure is a reliable proxy for the chain's spectral gap.