Load Balancing in Hypergraphs

TL;DR

This paper extends the concept of load balancing from finite graphs to hypergraphs, introducing a notion of unimodularity, and analyzes the properties, limits, and convergence of balanced load distributions in hypergraph models.

Contribution

It introduces a new framework for balanced load allocation in hypergraphs, extending existing graph theories and providing variational formulas and convergence results.

Findings

Characterization of balanced load distributions in hypergraphs

Extension of unimodularity to hypergraphs

Convergence results for maximum load in hypergraph processes

Abstract

Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be moved from a vertex to another that is carrying less load. We analyze the properties of balanced allocations of load. We extend the concept of balancedness from finite hypergraphs to their local weak limits in the sense of Benjamini and Schramm (2001) and Aldous and Steele (2004). To do this, we define a notion of unimodularity for hypergraphs which could be considered an extension of unimodularity in graphs. We give a variational formula for the balanced load distribution and, in particular, we characterize it in the special case of unimodular hypergraph Galton Watson processes. Moreover, we prove the convergence of the maximum load under some conditions. Our work is an extension to hypergraphs of Anantharam and Salez (2016), which considered load balancing in graphs, and is aimed at more comprehensively resolving conjectures of Hajek (1990).