When Do Gossip Algorithms Converge in Finite Time?

In this paper, we study the finite-time convegrence of gossip algorithms. We show that there exists a symmetric gossip algorithm that converges in finite time if and only if the number of network nodes is a power of two, while there always exists a globally finite-time convergent gossip algorithm despite the number of nodes if asymmetric gossiping is allowed. For $n=2^m$ nodes, we prove that a fastest convergence can be reached in $mn$ node updates via symmetric gossiping. On the other hand, for $n=2^m+r$ nodes with $0\leq r<2^m$, it requires at least $mn+2r$ node updates for achiving a finite-time convergence when asymmetric gossiping is allowed.