Consensus over Random Graph Processes: Network Borel-Cantelli Lemmas for Almost Sure Convergence

TL;DR

This paper establishes conditions for almost sure consensus in distributed systems over random graph processes, using Borel-Cantelli lemmas, and extends results to a wide class of models including Erdős-Rényi and Markovian graphs.

Contribution

It introduces network Borel-Cantelli lemmas for almost sure convergence and generalizes independence concepts to $*$-mixing, broadening applicability to many random graph models.

Findings

Sharp threshold for consensus zero-one law

Convergence rates via $oldsymbol{ extepsilon}$-computation time bounds

Almost sure finite-time convergence conditions

Abstract

Distributed consensus computation over random graph processes is considered. The random graph process is defined as a sequence of random variables which take values from the set of all possible digraphs over the node set. At each time step, every node updates its state based on a Bernoulli trial, independent in time and among different nodes: either averaging among the neighbor set generated by the random graph, or sticking with its current state. Connectivity-independence and arc-independence are introduced to capture the fundamental influence of the random graphs on the consensus convergence. Necessary and/or sufficient conditions are presented on the success probabilities of the Bernoulli trials for the network to reach a global almost sure consensus, with some sharp threshold established revealing a consensus zero-one law. Convergence rates are established by lower and upper bounds of the $\epsilon$-computation time. We also generalize the concepts of connectivity/arc independence to their analogues from the $*$-mixing point of view, so that our results apply to a very wide class of graphical models, including the majority of random graph models in the literature, e.g., Erd\H{o}s-R\'{e}nyi, gossiping, and Markovian random graphs. We show that under $*$-mixing, our convergence analysis continues to hold and the corresponding almost sure consensus conditions are established. Finally, we further investigate almost sure finite-time convergence of random gossiping algorithms, and prove that the Bernoulli trials play a key role in ensuring finite-time convergence. These results add to the understanding of the interplay between random graphs, random computations, and convergence probability for distributed information processing.