  This paper considers the resilient multi-dimensional consensus problem in
networked systems, where some of the agents might be malicious (or faulty). We
propose a multi-dimensional consensus algorithm, where at each time step each
healthy agent computes a "safe kernel" based on the information from its
neighbors, and modifies its own state towards a point inside the kernel.
Assuming that the number of malicious agents is locally (or globally) upper
bounded, sufficient conditions on the network topology are presented to
guarantee that the benign agents exponentially reach an agreement within the
convex hull of their initial states, regardless of the actions of the
misbehaving ones. It is also revealed that the graph connectivity and
robustness required to achieve the resilient consensus increases linearly with
respect to the dimension of the agents' state, indicating the existence of a
trade-off between the low communication cost and system security. Numerical
examples are provided in the end to validate the theoretical results.
