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

TL;DR

This paper develops a duality framework for $p$-modulus on networks, providing new proofs and generalizations of graph metrics, and connecting probabilistic interpretations with blocking duality.

Contribution

It introduces a general framework for $p$-modulus duality on networks, leading to new proofs of metric properties and connections to probabilistic interpretations.

Findings

Effective resistance is a metric on graphs.

A family of $p$-dependent graph metrics interpolates between shortest-path, resistance, and mincut.

Monotonicity properties of modulus are established and generalized.

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.