Convergence rate on periodic gossiping

In a sensor network in which each sensor controls a real-valued state, the goal of a distributed averaging problem is to compute the global average in a decentralized way, which is the average of all sensors’ initial state values across the entire network. A T-periodic gossiping protocol can solve such a problem, which stipulates that each agent must gossip with each of its neighbors exactly once every T time unit. The convergence rate of a T-periodic gossiping protocol is determined by the magnitude of the second largest eigenvalue of the stochastic matrix associated with the gossip sequence occurring over one period. An interesting result is that when the allowable gossip graph is a tree, the convergence rate is independent of gossip orders within one period. This paper will prove this result by developing several properties of doubly stochastic matrices. The properties derived also can be used in analyzing convergence rate problems of other periodic gossip protocols.