Load Balancing in Hypergraphs

We study the load balancing problem in networks. A task is viewed as a hyperedge in a hypergraph having access to the resources represented by its endpoints. For simplicity, we consider the case where each hyperedge represents one unit of load which should be distributed among its endpoints. An allocation is a way of distributing this load. The overall cost of an allocation is evaluated as the separable sum of a convex function of the total load at each resource. It is known that an allocation is optimal if it is balanced in the sense that no demand desires to change its allocation. Further, it is known that the net load faced by each resource is the same in any balanced allocation. We analyze the properties of the empirical distribution of the loads faced by the resources in balanced allocations for a sequence of hypergraphs as their size goes to infinity. In the special case of unimodular hypergraph Galton-Watson processes, we characterize the asymptotic empirical load distribution at a typical resource via a fixed point equation. Moreover, we are able to characterize the asymptotics of the maximum load at a resource under some additional conditions.