A potential theory on weighted graphs

TL;DR

This paper develops a discrete potential theory framework for weighted graphs, introducing operators analogous to continuum potential theory, and demonstrates its application in graph cloaking strategies.

Contribution

It introduces a novel discrete potential theory formalism for weighted graphs, including operators and calculus analogous to classical potential theory, with practical applications.

Findings

Established a discrete Calderón calculus for weighted graphs

Derived spectral properties of the Neumann-Poincaré operator on graphs

Proposed a graph cloaking method using the new potential theory

Abstract

We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators, the single and double layer potential operators, and the boundary layer operators. As in the continuum, these operators can represent exterior or interior harmonic functions with different boundary conditions. The formalism we introduce includes a discrete Calder\'on calculus and brings some well known results from potential theory to weighted graphs, e.g. on the spectrum of the Neumann-Poincar\'e operator. We illustrate the formalism with a cloaking strategy on weighted graphs which allows to hide an anomaly from the perspective of electrical measurements made away from the anomaly.