Reaching Approximate Byzantine Consensus with Multi-hop Communication

We address the problem of reaching consensus in the presence of Byzantine faults. Fault-tolerant consensus algorithms typically assume knowledge of nonlocal information and multi-hop communication; however, this assumption is not suitable for large-scale static/dynamic networks. A handful of iterative algorithms have been proposed recently under the assumption that each node (faulty or fault-free) can only access local information, thus is only capable of sending messages via one-hop communication. In this paper, we unify these two streams of work by assuming that each node knows the topology of up to $l^{th}$ hop neighborhood and can send messages to other nodes via up to $l$-hop transmission, where $1\le l\le n-1$ and $n$ is the number of nodes. We prove a family of necessary and sufficient conditions for the existence of iterative algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the initial states of the fault-free nodes. The following convergence requirement is imposed: for any $\epsilon>0$, after a sufficiently large number of iterations, the states of the fault-free nodes are guaranteed to be within $\epsilon$ of each other.