Escape of resources in distributed clustering processes

TL;DR

This paper investigates resource flow in a distributed clustering process, demonstrating that under certain initial distributions, resources can escape to infinity despite previous results suggesting eventual stabilization.

Contribution

It proves the existence of translation-invariant initial distributions where resources escape to infinity, answering an open question in the field.

Findings

Resources can escape to infinity under specific initial distributions.

Previous results on stabilization do not hold universally.

The paper constructs explicit examples of such distributions.

Abstract

In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret \cite{coffman91}, each vertex of $\mathbb{Z}^d$ receives an initial amount of a resource, and, at each iteration, transfers all of its resource to the neighboring vertex which currently holds the maximum amount of resource. In \cite{hlrnss} it was shown that, if the distribution of the initial quantities of resource is invariant under lattice translations, then the flow of resource at each vertex eventually stops almost surely, thus solving a problem posed in \cite{berg91}. In this article we prove the existence of translation-invariant initial distributions for which resources nevertheless escape to infinity, in the sense that the the final amount of resource at a given vertex is strictly smaller in expectation than the initial amount. This answers a question posed in \cite{hlrnss}.