Consensus in networks of multiagents with cooperation and competition via stochastically switching topologies

In this brief, we provide some theoretical analysis of the consensus for networks of agents via stochastically switching topologies. We consider both discrete-time case and continuous-time case. The main contribution of this brief is that the underlying graph topology is more general in both cases than those appeared in previous papers. The weight matrix of the coupling graph is not assumed to be nonnegative or Metzler. That is, in the model discussed here, the off-diagonal entries of the weight matrix of the coupling graph may be negative. This means that sometimes, the coupling may not benefit, but may prevent the consensus of the coupled agents. In the continuous-time case, the switching time intervals also take a more general form of random variables than those appeared in previous works. We focus our study on such networks and give sufficient conditions that ensure almost sure consensus in both discrete-time case and continuous-time case. As applications, we give several corollaries under more specific assumptions, i.e., the switching can be some independent and identically distributed (i.i.d.) random variable series or a Markov chain. Numerical examples are also provided in both discrete-time and continuous-time cases to demonstrate the validity of our theoretical results.