Open multi-agent systems: Gossiping with random arrivals and departures

We consider open multi-agent systems. Unlike the systems usually studied in the literature, here agents may join or leave while the process studied takes place. The system composition and size evolve thus with time. We focus here on systems where the interactions between agents lead to pairwise gossip averages, and where agents either arrive or are replaced at random times. These events prevent any convergence of the system. Instead, we describe the expected system behavior by showing that the evolution of scaled moments of the state can be characterized by a 2-dimensional (possibly time-varying) linear dynamical system. We apply this technique to two cases : (i) systems with fixed size where leaving agents are immediately replaced, and (ii) systems where new agents keep arriving without ever leaving, and whose size grows thus unbounded.