Reaching ℒp consensus in a network of multiagents with stochastically switching topologies

We study consensus in networks of multiagents with stochastically switching topologies, where the switching topologies are described as an adapted process, a rather general process including the independent and identically distributed (i.i.d.) process and the Markov process as special cases. First, motivated by some works done in the field of stochastic stability theory, we introduce a new concept of consensus, ¿L_p consensus¿ with p¿1. Then sufficient conditions for a network with stochastically switching topologies to reach L_p consensus are derived for both discrete-time and continuous-time cases. In the discrete-time case, we show that the existence of a spanning tree in the conditional expectation of the union of the graphes of the network topologies across each T-length time interval for some T > 0 is sufficient for L_p consensus of the network. In the continuous-time case, we also give a similar sufficient condition involving the existence of a spanning tree. As direct consequences of the main results we also give some corollaries for two important stochastic processes: the i.i.d. process and homogenous Markov process. Moreover, we compare our results with the results existing in literatures.