Inverse problems for discrete heat equations and random walks for a class of graphs

TL;DR

This paper investigates inverse problems on finite weighted graphs, establishing conditions under which the graph structure and random walk parameters can be uniquely recovered from partial observations of heat equations and first passage times.

Contribution

It introduces a novel equivalence between inverse heat problems and spectral problems on graphs, and proposes the Two-Points Condition for unique recovery of graph structure and transition matrices.

Findings

Unique recovery of graph structure under certain spectral conditions.

Recovery of transition matrices from first passage time distributions.

Introduction of the Two-Points Condition for random walk observations.

Abstract

We study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk.