Blocking duality for $p$-modulus on networks and applications

Abstract

This paper explores the implications of blocking duality---pioneered by Fulkerson et al.---in the context of $p$-modulus on networks. Fulkerson's blocking duality is an analogue on networks to the method of conjugate families of curves in the plane. The technique presented here leads to a general framework for studying families of objects on networks; each such family has a corresponding dual family whose $p$-modulus is essentially the reciprocal of the original family's. As an application, we give a modulus-based proof for the fact that effective resistance is a metric on graphs. This proof immediately generalizes to yield a family of graph metrics, depending on the parameter $p$, that continuously interpolates among the shortest-path metric, the effective resistance metric, and the mincut ultrametric. In a second application, we establish a connection between Fulkerson's blocking duality and the probabilistic interpretation of modulus. This connection, in turn, provides a straightforward proof of several monotonicity properties of modulus that generalize known monotonicity properties of effective resistance. Finally, we use this framework to expand on a result of Lov\'asz in the context of randomly weighted graphs.