Equilibria in infinite random graphs

A load balancing problem is formulated for infinite networks or graphs. There are overlapping sets of locations, each set having an associated possibly random amount of load to be distributed. The total load at a location is the sum of the contributions due to the sets that contain it. Equilibrium is said to hold if the load corresponding to any one set cannot be reassigned to improve the balance of total loads. The set of possible equilibria, or balanced load vectors, is examined. The balanced load vector is shown to be unique for Euclidean lattice networks, in which the sets correspond to pairs of neighboring nodes in a rectangular lattice in finite dimensions. A method for computing the load distribution is explored for tree networks. An FKG type inequality is proved. The concept of load percolation is introduced and is shown to be associated with infinite sets of locations with identical load.