Flow Metrics on Graphs

Abstract: Given a graph with non-negative edge weights, there are various ways to interpret the edge weights and induce a metric on the vertices of the graph. A few examples are shortest-path, when interpreting the weights as lengths; resistance distance, when thinking of the graph as an electrical network and the weights are resistances; and the inverse of minimum $st$-cut, when thinking of the weights as capacities. It is known that the 3 above-mentioned metrics can all be derived from flows, when formalizing them as convex optimization problems. This key observation led us to studying a family of metrics that are derived from flows, which we call flow metrics, that gives a natural interpolation between the above metrics using a parameter $p$. We make the first steps in studying the flow metrics, and mainly focus on two aspects: (a) understanding basic properties of the flow metrics, either as an optimization problem (e.g. finding relations between the flow problem and the dual potential problem) and as a metric function (e.g. understanding their structure and geometry); and (b) considering methods for reducing the size of graphs, either by removing vertices or edges while approximating the flow metrics, and thus attaining a smaller instance that can be used to accelerate running time of algorithms and reduce their storage requirements. Our main result is a lower bound for the number of edges required for a resistance sparsifier in the worst case. Furthermore, we present a method for reducing the number of edges in a graph while approximating the flow metrics, by utilizing a method of [Cohen and Peng, 2015] for reducing the size of matrices. In addition, we show that the flow metrics satisfy a stronger version of the triangle inequality, which gives some information about their structure and geometry.