Resilient Vector Consensus in Multi-Agent Networks Using Centerpoints

In this paper, we study the resilient vector consensus problem in multi-agent networks and improve resilience guarantees of existing algorithms. In resilient vector consensus, agents update their states, which are vectors in ℝ
^d
, by locally interacting with other agents some of which might be adversarial. The main objective is to ensure that normal (non-adversarial) agents converge at a common state that lies in the convex hull of their initial states. Currently, resilient vector consensus algorithms, such as approximate distributed robust convergence (ADRC) are based on the idea that to update states in each time step, every normal node needs to compute a point that lies in the convex hull of its normal neighbors' states. To compute such a point, the idea of Tverberg partition is typically used, which is computationally hard. Approximation algorithms for Tverberg partition negatively impact the resilience guarantees of consensus algorithm. To deal with this issue, we propose to use the idea of centerpoint, which is an extension of median in higher dimensions, instead of Tverberg partition. We show that the resilience of such algorithms to adversarial nodes is improved if we use the notion of centerpoint. Furthermore, using centerpoint provides a better characterization of the necessary and sufficient conditions guaranteeing resilient vector consensus. We analyze these conditions in two, three, and higher dimensions separately. We also numerically evaluate the performance of our approach.