Convergence of Gossip Algorithms for Consensus in Wireless Sensor Networks with Intermittent Links and Mobile Nodes

We study the convergence of pairwise gossip algorithms and broadcast gossip algorithms for consensus with intermittent links and mobile nodes. By nonnegative matrix theory and ergodicity coefficient theory, we prove gossip algorithms surely converge as long as the graph is partitionally weakly connected which, in comparison with existing analysis, is the weakest condition and can be satisfied for most networks. In addition we characterize the supremum for the mean squared error of convergence as a function associated with the initial states and the number of nodes. Furthermore, on the condition that the graph is partitionally strongly connected, the rate of convergence is proved to be exponential and governed by the second largest eigenvalue of expected coefficient matrix. For partitionally strongly connected digraphs, simulation results illustrate that gossip algorithms actually converge, and broadcast gossip algorithms can converge faster than pairwise gossip algorithms at the cost of larger error of convergence.