In this paper, we study the discrete-time consensus problem over networkswith antagonistic and cooperative interactions. Following the work by Altafini[IEEE Trans. Automatic Control, 58 (2013), pp. 935--946], by an antagonisticinteraction between a pair of nodes updating their scalar states we mean onenode receives the opposite of the state of the other and naturally by ancooperative interaction we mean the former receives the true state of thelatter. Here the pairwise communication can be either unidirectional orbidirectional and the overall network topology graph may change with time. Theconcept of modulus consensus is introduced to characterize the scenario thatthe moduli of the node states reach a consensus. It is proved that modulusconsensus is achieved if the switching interaction graph is uniformly jointlystrongly connected for unidirectional communications, or infinitely jointlyconnected for bidirectional communications. We construct a counterexample tounderscore the rather surprising fact that quasi-strong connectivity of theinteraction graph, i.e., the graph contains a directed spanning tree, is notsufficient to guarantee modulus consensus even under fixed topologies. Finally,simulation results using a discrete-time Kuramoto model are given to illustratethe convergence results showing that the proposed framework is applicable to aclass of networks with general nonlinear node dynamics.
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