Finite-time Convergent Gossiping

TL;DR

This paper investigates finite-time convergence in gossip algorithms, revealing conditions under which symmetric and asymmetric algorithms converge in finite time, and applies findings to quantum networks where finite-time convergence is impossible.

Contribution

It characterizes conditions for finite-time convergence in deterministic gossip algorithms, including structural requirements and optimal convergence times, and extends analysis to quantum network control.

Findings

Symmetric gossip converges in finite time iff nodes are a power of two.

Asymmetric gossip always achieves finite-time convergence regardless of node count.

Finite-time convergence imposes strong structural constraints on interaction graphs.

Abstract

Gossip algorithms are widely used in modern distributed systems, with applications ranging from sensor networks and peer-to-peer networks to mobile vehicle networks and social networks. A tremendous research effort has been devoted to analyzing and improving the asymptotic rate of convergence for gossip algorithms. In this work we study finite-time convergence of deterministic gossiping. 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 an asymmetric gossip algorithm with finite-time convergence, independent of the number of nodes. For $n=2^m$ nodes, we prove that a fastest convergence can be reached in $nm=n\log_2 n$ node updates via symmetric gossiping. On the other hand, under asymmetric gossip among $n=2^m+r$ nodes with $0\leq r<2^m$, it takes at least $mn+2r$ node updates for achieving finite-time convergence. It is also shown that the existence of finite-time convergent gossiping often imposes strong structural requirements on the underlying interaction graph. Finally, we apply our results to gossip algorithms in quantum networks, where the goal is to control the state of a quantum system via pairwise interactions. We show that finite-time convergence is never possible for such systems.