Joint Robustness of Time-Varying Networks and Its Applications to Resilient Consensus

The notion of network robustness reported in some existing literature well characterizes the graph-theoretic properties of networked agent systems (NASs) with a fixed communication graph for seeking resilient consensus. Yet, our systematic understanding of the graph properties for achieving resilient consensus of NASs with time-varying communication graphs remains limited. This study aims to investigate the resilient consensus problem of NASs with misbehaving agents and time-varying communication graphs under certain mean-subsequence-reduced algorithms, with particular attention to formulating and revealing the graph-theoretic properties of the time-varying interaction graphs responsible for resilient consensus. Specifically, to achieve resilient consensus, each agent will first discard some extremely large and small values received from its neighbors and then generate the control input based upon the remaining neighbors' values at every time instant. To characterize the graph-theoretic properties of NASs subject to time-varying communication graphs for realizing resilient consensus in spite of the influence of misbehaving individuals, a new graph-theoretic property of the NASs, namely the
joint robustness
, is introduced. This new property connects the network robustness of the fixed graph with the property of time-varying interaction graphs jointly containing a directed spanning tree. Several necessary and sufficient criteria for resilient consensus of NASs with first-order and second-order individual dynamics subject to time-varying communication graphs are, respectively, proven and analyzed in the framework of joint robustness. At last, the effectiveness of the analytical findings is validated by performing numerical simulations.