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Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays

We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d.\ topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links.