Design of Attack-Resilient Consensus Dynamics: A Game-Theoretic Approach

We propose a game-theoretic framework for improving the resilience of multi-agent consensus dynamics in the presence of a strategic attacker. In this game, the attacker selects a set of network nodes to inject the attack signals. The attacker's objective is to minimize the required energy for steering the consensus towards its desired direction. This energy is captured by the trace of controllability Gramian of the system when the input is the attack signal. The defender improves the resilience of dynamics by adding self-feedback loops to certain nodes of the system and its objective is to maximize the trace of controllability Gramian. The Stackelberg equilibrium of the game is studied with the defender as the game leader. When the underlying network topology is a tree and the defender can select only one node, we show that the optimal strategy of the defender is determined by a specific distance-based network centrality measure, called network's f-center. In addition, we show that the degree-based centralities solutions may lead to undesirable payoffs for the defender. At the end, we discuss the case of multiple attack and defense nodes on general graphs.