Consensus formation in networks with neighbor-dependent synergy and observer effect

In this paper, we study the consensus formation over a directed hypergraph, which is an important generalization of standard graph structure by allowing possible neighbor-dependent synergy. The proposed model is situated in the social dynamics providing key features including social observer effect and bounded confidence. Under the minimal siphon condition of a directed hypergraph (Petri net), we show that global consensus can be reached with the final consensus value residing in the common comfortable range if it is non-empty. To achieved this, we establish an equivalent condition for the commensurate graph of a finite state machine to be strongly connected. Convergence analysis is performed based on the proposed nonlinear dynamic system model and Petri net method. The consensus result holds for any non-negative confidence bound, which distinguishes from traditional bounded confidence opinion models as we measure the difference among neighbors rather than the gap between neighbors and the ego. Numerical studies are conducted to unravel some insights in relation to the influence of observers, hypergraph architecture, and confidence bounds on opinion evolution. The results and methodologies presented here facilitate research of social consensus and also offer a way to make sense of synergy in networked complex systems.