Multi-agent systems provide a framework for studying distributed decision-making problems as a number of agents make local decisions by interacting with each other over networks [1]-[3]. Due to the rise in the use of general purpose networks and wireless communication channels for such systems, cyber security has become a major critical issue [4]. One of the common security threats in networked systems is jamming attacks, where the adversary can simply transmit interference signals to interrupt communication among agents. The danger level of jamming attacks may further increase if the attacker is more aware of system parameters.

Jamming attacks on networked systems were previously analyzed through game-theoretic approaches [5]-[12]. The works [8]-[10] model the activity of jamming and transmitting signals as zero-sum games where the payoff structure of the players is balanced. In [7], [11]-[12], the authors consider a Stackelberg game approach, in which the players decide their actions sequentially by following a certain hierarchy.

Multi-agent consensus problems in the presence of such jamming attacks have been studied in [13]-[15]. Jamming attack models with energy constraints for the adversary were introduced in [14]-[19] in the context of networked control. However, in the abovementioned works, it is not clear whether the attacker attacks optimally or not. In addition, in those works there is also no defense mechanism, which is crucial in resilient systems, that is able to mitigate the attacks and restore the communication without waiting for the attacks to end. In this study, we model the interaction between an attacker and a defender in a two-player game setting in terms of links and action durations. The attacker is motivated to disrupt the communication by attacking individual links whereas the defender attempts to recover some of the attacked links whenever possible. Both players are constrained in terms of their available energy for the actions of attacks and recovery.

The detailed formulation is shown in Fig. 1 below, where the attacker attacks certain links on the graph representing communication topology for certain duration. In response to the attack, the defender attempts to recover some of the attacked links to restore the communication among agents. By attacking and recovering, the players spend some amount of energy. One game is defined as one cycle of attack-recovery interval. This attack-recovery sequence occurs repeatedly over time, and hence the game is played repeatedly over k.

Fig.1: Attack-recovery sequence.

The players attack and recover based on the connectivity measures of the graphs, where the attacker aims to disconnect the graph as much as possible and the defender wants the opposite. Here we use generalized edge connectivity as connectivity measure, which considers edge connectivity and number of connected components of the graphs.

To characterize the effect of the two-player game to the agents, we utilize consensus dynamics for the state of each agent. In Fig. 2, we can see that the optimal attacks in a multi-agent system slow the convergence speed down, where the recoveries from the defender speed the consensus up by resuming communication on some of the attacked edges. These optimal attacks and recoveries are affected by the energy of both players.

Fig.2: Agents' states with consensus dynamics. The red and green areas indicating the times when the multi-agent system is attacked and recovered, respectively.

We have analyzed the game between two players, the attacker and the defender, in terms of communications among agents in a multi-agent system. The strategies of the players are in form of numbers of links and durations of action intervals, with the optimal strategies depend on the available energy of the players. We have shown that the time for the agents to achieve consensus will be delayed due to attacks.