A low discrepancy sequence on graphs

TL;DR

This paper introduces a novel sampling scheme for graphs, inspired by Leja points, that provides low discrepancy estimates for approximating expectations on graph vertices, independent of graph size.

Contribution

It constructs a graph-based kernel ensuring the equilibrium distribution matches a fixed probability distribution, offering size-independent discrepancy bounds.

Findings

Provides a new low discrepancy sampling method for graph vertices.

Establishes size-independent error bounds for expectation approximation.

Connects potential theory concepts to graph sampling techniques.

Abstract

Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.