On consensus protocols allowing joint-agent interactions

In this note we consider consensus protocols where an agent would not be influenced by any of his neighbours singularly taken, but might be sensitive to the simultaneous and coherent influence of two or more of them. This may resemble several common behaviours in social, economic and opinion networks (i.e. conformity, risk aversion, social inertia, herding). We derive novel graph-theoretical concepts to describe and analyze the ability of general networks with joint-agent interactions to converge towards consensus. Interestingly, and for the first time, we borrow to this end the language of Petri Nets as a convenient way to describe bipartite directed graphs, showing how the notion of siphon is helpful in characterizing the flow of information across the network and its ability to induce attainment of consensus among agents in the considered set-up.