Fixation for Distributed Clustering Processes

TL;DR

This paper proves that in a discrete-time resource flow model on integer lattices, the process at each vertex terminates finitely, answering a longstanding question and extending the result to other graphs.

Contribution

It establishes finite termination of resource flow in a lattice model, solving a generalized problem from 1991 and employing the mass-transport principle.

Findings

Flow terminates finitely at each vertex almost surely

Results extend to other graph structures

Addresses a longstanding open problem

Abstract

We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs.