When Do Gossip Algorithms Converge in Finite Time?

In this paper, we study finite-time convergence 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 nodes, we prove that a fastest convergence can be reached in mn node updates via symmetric gossiping. On the other hand, for n = 2 + r nodes with 0 ≤ r < 2, it requires at least mn+ 2r node updates for achieving a finite-time convergence in cooperation with asymmetric interactions.