Modulus Consensus over Networks with Antagonistic Interactions and   Switching Topologies
  In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. Following the work by Altafini [IEEE Trans. Automatic Control, 58 (2013), pp. 935--946], by an antagonistic interaction between a pair of nodes updating their scalar states we mean one node receives the opposite of the state of the other and naturally by an cooperative interaction we mean the former receives the true state of the latter. Here the pairwise communication can be either unidirectional or bidirectional and the overall network topology graph may change with time. The concept of modulus consensus is introduced to characterize the scenario that the moduli of the node states reach a consensus. It is proved that modulus consensus is achieved if the switching interaction graph is uniformly jointly strongly connected for unidirectional communications, or infinitely jointly connected for bidirectional communications. We construct a counterexample to underscore the 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 modulus consensus even under fixed topologies. 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 node dynamics. 
Ziyang Meng, Guodong Shi, Karl H. Johansson, Ming Cao, and Yiguang   Hong
math.OC