Distributed Integer Weight Balancing in the Presence of Time Delays in Directed Graphs

Abstract

A digraph with positive weights on its edges is weight-balanced if, for each node, the sum of the weights of the incoming edges is equal to the sum of the weights of the outgoing edges. Weight-balanced digraphs play an important role in a variety of cooperative control problems, including formation control, event-triggered coordination, distributed averaging, and optimization. Classical distributed algorithms for asymptotic average consensus typically assume timely and reliable exchange of information between neighboring components of a given multicomponent system. These assumptions are not necessarily valid in practice due to varying delays that might affect computations at different nodes and/or transmissions at different links. In this paper, we propose a distributed algorithm that solves the integer weight balancing problem in the presence of arbitrary (time-varying and inhomogeneous) time delays that might affect the transmission at a particular link at a particular time. The algorithm converges after a finite number of steps (that we explicitly bound) in the presence of bounded delays and converges in finite time (with probability one) in the case of unbounded delays (packet drops). Furthermore, we show that the resulting weight-balanced digraph is unique regardless of how time delays or packet drops manifest themselves. We also analyze the computational and communication complexity of the proposed algorithm, and provide examples to illustrate its operation and performance.