A Graph-Theoretic Equilibrium Analysis of Attacker-Defender Game on Consensus Dynamics Under $\mathcal {H}_2$ Performance Metric

Abstract

In this paper, we propose a game-theoretic framework for improving the resilience of the consensus algorithm, under the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> performance metric, in the presence of a strategic attacker. In this game, an attacker selects a subset of nodes in the network to inject attack signals. Its objective is to maximize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> performance metric as the game pay-off. The equilibrium of the (zero-sum) attacker-defender game determines the optimal location of the defense nodes in the network. The existence of a Nash equilibrium for consensus dynamics is studied under undirected and directed network topologies. 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. Our results indicate that the equilibrium strategies of the game are characterized by graph-theoretic notions such as network centrality metrics. In particular, we show that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">effective center</i> of the graph, a new network centrality measure, captures the optimal location of defense nodes in undirected networks. In directed networks, however, 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.