Diffusion determines the recurrent graph

TL;DR

This paper investigates whether the structure of a weighted graph can be uniquely identified from the diffusion process it induces, revealing conditions under which the graph is determined up to isomorphism.

Contribution

It establishes that recurrent graphs are uniquely determined by their diffusion up to scaling, and explores conditions for non-recurrent graphs, extending diffusion theory to discrete structures.

Findings

Recurrent graphs are uniquely determined by diffusion up to scaling.

Order isomorphisms in diffusion are actually unitary maps (up to scaling).

Results extend diffusion analysis from Euclidean domains to discrete graphs.

Abstract

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass. These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.