Weak reachability and strong recurrence for stochastic directed graphs in terms of auxiliary functions

In this paper, directed graphs where the network topology is governed by a stochastic process are addressed. A one-to-one correspondence between these graphs and stochastic discrete-time systems, with set-valued transition map, is stated. Hence, by taking advantage of this equivalence, mathematical tools available for these discrete-time systems are used to characterize connectivity properties of stochastic digraphs. Namely, we characterize reachability in finite steps, reachability in infinite steps and recurrence relatively to a given set for a stochastic digraph in terms of auxiliary functions.