The Shortest-Path distance on graphons

TL;DR

This paper introduces a novel shortest-path distance analogue for graphons, extending Varadhan's formula, which links heat equations to geodesic distances, resulting in an integer-valued metric that generalizes traditional shortest-path measures.

Contribution

It proposes a new metric for graphons based on Varadhan's formula, connecting heat flow and geodesic distances, and relates it to communicability distance for a more comprehensive measure.

Findings

The new metric is integer-valued and aligns with shortest-path distance on finite graphs.

It establishes a link between Varadhan distance and communicability distance.

Provides a natural isometric embedding into a Hilbert space.

Abstract

We define an analogue of the shortest-path distance for graphons. The proposed method is rooted on the extension to graphons of Varadhan's formula, a result that links the solution of the heat equation on a Riemannian manifold to its geodesic distance. The resulting metric is integer-valued, and for step graphons obtained from finite graphs it is essentially equivalent to the usual shortest-path distance. We further draw a link between the Varadhan distance and the communicability distance, that contains information from all paths, not just shortest-paths, and thus provides a finer distance on graphons along with a natural isometric embedding into a Hilbert space.