Distributed balancing of a digraph with integer weights

We address the integer weight-balancing problem for a distributed system whose components (nodes) can exchange information via interconnection links (edges) that form an arbitrary, possibly directed, communication topology (digraph). A weighted digraph is balanced if, for each of its nodes, the sum of the weights of the edges outgoing from the node is equal to the sum of the weights of the edges incoming to the node. Weight-balanced digraphs play a key role in a number of applications, including distributed optimization, cooperative control, and distributed averaging problems. In this paper, we develop a distributed iterative algorithm, which can be used to reach weight balance, by assigning a positive integer weight on each edge, as long as the underlying communication topology forms a strongly connected digraph (or is a collection of strongly connected digraphs).