Distributed weight balancing under integer constraints in the presence of packet drops

Abstract

A digraph with positive integer weights on its (directed) edges is weight-balanced if, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. We develop a distributed iterative algorithm in which nodes are in charge of updating the weights on their outgoing edges based on certain values they maintain and update, using corresponding information available from their immediate in- and out-neighbors. We assume that communication between neighboring nodes is bidirectional, but unreliable in that it may result in possible packet drops, independently between different links and link directions. We show that, even when communication links drop packets occasionally (but not always) and the integer weights are constrained to be within an interval, captured by lower and upper limits, the proposed algorithm allows nodes to reach integer weight balancing after a finite number of iterations with probability one, as long as the necessary and sufficient circulation conditions on the lower and upper edge weight limits are satisfied. We also provide examples to illustrate the operation and performance of the proposed algorithm.