An inverse problem for fractional random walks on finite graphs

TL;DR

This paper investigates an inverse problem on finite graphs involving fractional random walks, aiming to recover graph properties and conductivities from limited observational data.

Contribution

It introduces a method to determine a gauge class of transition probabilities and recover graph structure and conductivities from partial fractional random walk data.

Findings

Transition probabilities determine a gauge class of the random walk.

Vertex count, edges, and conductivities can be recovered if transition matrix is known.

The data exhibits a new nonlocal property related to the transition matrices.

Abstract

We study an inverse problem on a finite connected graph G = (X, E), on whose vertices a conductivity {\gamma} is defined. Our data consists in a sequence of partial observations of a fractional random walk on G. The observations are partial in the sense that they are limited to a fixed, observable subset B of X, while the random walk is fractional in the sense that it allows long jumps with a probability P decreasing as a fractional power of the distance along the graph. The transition probability P also depends on {\gamma}. We show that this kind of random walk data allows for the determination of a gauge class to which the transition probability matrix P belongs, which we discuss. Moreover, we show that if the transition probability matrix P is itself known, then the amount of vertices |X|, the edge set E and the conductivity {\gamma} (up to a positive factor) can be recovered. We also show a characterization of the random walk data in terms of the corresponding transition matrices P , which highlights a new surprising nonlocal property. This work is motivated by the recent strong interest in the study of the fractional Calder\'on problem in the Riemannian setting.