Sequences of well-distributed vertices on graphs and spectral bounds on optimal transport

TL;DR

This paper introduces a greedy algorithm for selecting well-distributed vertices on graphs, linking spectral bounds to Wasserstein distance, with potential applications in sampling and graph analysis.

Contribution

It presents a novel greedy algorithm inspired by potential theory for vertex selection on graphs, and establishes spectral bounds relating Wasserstein distance to graph properties.

Findings

Algorithm performs well on various graphs

Spectral bounds relate Wasserstein distance to graph spectra

Potential for improved sampling methods

Abstract

Given a graph $G=(V,E)$, suppose we are interested in selecting a sequence of vertices $(x_j)_{j=1}^n$ such that $\left\{x_1, \dots, x_k\right\}$ is `well-distributed' uniformly in $k$. We describe a greedy algorithm motivated by potential theory and corresponding developments in the continuous setting. The algorithm performs nicely on graphs and may be of use for sampling problems. We can interpret the algorithm as trying to greedily minimize a negative Sobolev norm; we explain why this is related to Wasserstein distance by establishing a purely spectral bound on the Wasserstein distance on graphs that mirrors R. Peyre's estimate in the continuous setting. We illustrate this with many examples and discuss several open problems.