Modulus metrics on networks

TL;DR

This paper introduces a family of graph metrics based on the concept of p-modulus, generalizing existing metrics, and explores their properties, including the antisnowflaking exponent, with numerical evidence and explicit computations.

Contribution

It develops a new parametrized family of graph metrics using p-modulus, extending and unifying several well-known metrics, and investigates their properties and conjectures.

Findings

Numerical evidence supporting the antisnowflaking exponent conjecture

Explicit computations of the new metrics on selected graphs

Generalization of existing graph metrics through p-modulus

Abstract

The concept of $p$-modulus gives a way to measure the richness of a family of objects on a graph. In this paper, we investigate the families of connecting walks between two fixed nodes and show how to use $p$-modulus to form a parametrized family of graph metrics that generalize several well-known and widely-used metrics. We also investigate a characteristic of metrics called the "antisnowflaking exponent" and present some numerical findings supporting a conjecture about the new metrics. We end with explicit computations of the new metrics on some selected graphs.