Minimal subfamilies and the probabilistic interpretation for modulus on graphs

TL;DR

This paper introduces minimal subfamilies for the $p$-modulus on graphs, showing they have at most |E| elements and relate to a probability measure when p=2, providing a new perspective on graph richness.

Contribution

It develops the concept of minimal subfamilies using Lagrangian duality for $p$-modulus, revealing their size bounds and probabilistic interpretation at p=2.

Findings

Minimal subfamilies have at most |E| elements.

Elements carry importance weights related to the $p$-modulus.

When p=2, importance measures become probability distributions.

Abstract

The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies have at most $|E|$ elements and that these elements carry a weight related to their "importance" in relation to the corresponding $p$-modulus problem. When $p=2$, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.