Average consensus and gossip algorithms in networks with stochastic asymmetric communications

We consider that a set of distributed agents desire to reach consensus on the average of their initial state values, while communicating with neighboring agents through a shared medium. This communication medium allows only one agent to transmit unidirectionally at a given time, which is true, e.g., in wireless networks. We address scenarios where the choice of agents that transmit and receive messages at each transmission time follows a stochastic characterization, and we model the topology of allowable transmissions with asymmetric graphs. In particular, we consider: (i) randomized gossip algorithms in wireless networks, where each agent becomes active at randomly chosen times, transmitting its data to a single neighbor; (ii) broadcast wireless networks, where each agent transmits to all the other agents, and access to the network occurs with the same probability for every node. We propose a solution in terms of a linear distributed algorithm based on a state augmentation technique, and prove that this solution achieves average consensus in a stochastic sense, for the special cases (i) and (ii). Expressions for absolute time convergence rates at which average consensus is achieved are also given.