Load Balancing in Hypergraphs

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 (Electron J Probab 6(23):13, 2001) and Aldous and Steele (in: Probability on discrete structures. Springer, Berlin, pp 1–72, 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 (Ann Appl Probab 26(1):305–327, 2016), which considered load balancing in graphs, and is aimed at more comprehensively resolving conjectures of Hajek (IEEE Trans Inf Theory 36(6):1398–1414, 1990).