Convergence Properties of the Asynchronous Maximum Model

TL;DR

This paper analyzes the convergence behavior of the asynchronous maximum model on directed graphs, establishing bounds on the expected convergence time and relating it to graph properties like expansion and cycle length.

Contribution

It provides the first rigorous bounds on convergence time for the asynchronous maximum model, linking it to graph expansion and cycle parameters.

Findings

Expected convergence time in undirected graphs is rac{n}{\

Bounds on convergence time depend on vertex expansion rac{n}{\

Abstract

Let $G = (V,E)$ be a connected directed graph on $n$ vertices. Assign values from the set $\{1,2,\dots,n\}$ to the vertices of $G$ and update the values according to the following rule: uniformly at random choose a vertex and update its value to the maximum of the values in its neighbourhood. The value at this vertex can potentially decrease. This random process is called the asynchronous maximum model. Repeating this process we show that for a strongly connected directed graph eventually all vertices have the same value and the model is said to have \textit{converged}. In the undirected case the expected convergence time is shown to be asymptotically (as $n\to \infty$) in $\Omega(n\log n)$ and $O(n^2)$ and these bounds are tight. We further characterise the convergence time in $O(\frac{n}{\phi}\log n)$ where $\phi$ is the vertex expansion of $G$. This provides a better upper bound for a large class of graphs. Further, we show the number of rounds until convergence is in $O((\frac{n}{\phi}\log n)g(n))$ with high probability, where $g(n)$ satisfies $\frac{1}{g^2(n)} \to 0$ as $n \to \infty$. For a strongly connected directed graph the convergence time is shown to be in $O(nb^2 + \frac{n}{\phi'}\log n)$ where $b$ is a parameter measuring directed cycle length and $\phi'$ is a parameter measuring vertex expansion.