Resilience in consensus dynamics via competitive interconnections

We show that competitive engagements within the agents of a network can result in resilience in consensus dynamics with respect to the presence of an adversary. We first show that interconnections with an adversary, with linear dynamics, can make the consensus dynamics diverge, or drive its evolution to a state different from the average. We then introduce a second network, interconnected with the original network via an engagement topology. This network has no information about the adversary and each agent in it has only access to partial information about the state of the other network. We introduce a dynamics on the coupled network which corresponds to a saddle-point dynamics of a certain zero-sum game and is distributed over each network, as well as the engagement topology. We show that, by appropriately choosing a design parameter corresponding to the competition between these two networks, the coupled dynamics can be made resilient with respect to the presence of the adversary. Our technical approach combines notions of graph theory and stable perturbations of nonsymmetric matrices. We demonstrate our results on an example of kinematic-based flocking in presence of an adversary.