Behaviors of networks with antagonistic interactions and switching topologies

In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. A cooperative interaction between two nodes takes place when one node receives the true state of the other while an antagonistic interaction happens when the former receives the opposite of the true state of the latter. We adopt a quite general model where the node communications can be either unidirectional or bidirectional, the network topology graph may vary over time, and the cooperative or antagonistic relations can be time-varying. It is proven that, the limits of all the node states exist, and the absolute values of the node states reach consensus if the switching interaction graph is uniformly jointly strongly connected for unidirectional topologies, or infinitely jointly connected for bidirectional topologies. These results are independent of the switching of the interaction relations. We construct a counterexample to indicate a rather surprising fact that quasi-strong connectivity of the interaction graph, i.e., the graph contains a directed spanning tree, is not sufficient to guarantee the consensus in absolute values even under fixed topologies. Based on these results, we also propose sufficient conditions for bipartite consensus to be achieved over the network with joint connectivity. Finally, simulation results using a discrete-time Kuramoto model are given to illustrate the convergence results showing that the proposed framework is applicable to a class of networks with general nonlinear dynamics.