When Do Gossip Algorithms Converge in Finite Time?

In this paper, we show that there exists a gossip algorithm that converges in finite time if and only if the number of network nodes is a power of two. In addition, an "all-or-nothing" theorem is established on the finite-time convergence of distributed averaging algorithms: for a class of averaging algorithms defined by matrix sequences selected from a countable set of stochastic matrices, finite-time convergence either holds for all initial conditions, or fails for almost all initial conditions.