A Graph-Theoretic Equilibrium Analysis of Attacker-Defender Game on Consensus Dynamics Under H₂ Performance Metric

We propose a game-theoretic framework for improving the resilience of the consensus algorithm, under the H₂ performance metric, in the presence of an attacker. In this game, an attacker selects a subset of nodes to inject attack signals to maximize the H₂ norm of the system from the attack signal to the output of the system. The defender improves the resilience of the system by adding self-feedback loops to certain nodes of the network to minimize the system's norm. We investigate the interplay between the equilibrium strategies of the game and the underlying connectivity graph, using the H₂ performance metric as the game pay-off. The existence of a Nash equilibrium is studied under undirected and directed networks. For the cases where the attacker-defender game does not admit a Nash equilibrium, the Stackelberg equilibrium of the game is studied with the defender as the game leader. We show that the effective center of the graph, a new network centrality measure, captures the optimal location of defense nodes in undirected networks. In directed networks, the optimal locations of defenders are those nodes with small in-degrees. The theoretical results are applied to the design of a resilient formation of vehicle platoons.