Convergence Properties of the Asynchronous Maximum Model
John Larkin
TL;DR
This work analyzes the asynchronous maximum model on graphs, proving that convergence to a uniform value occurs on strongly connected digraphs and characterizing the convergence time across graph classes. It introduces a general iterative graph framework and the Markov chain of possibilities to formalize period and convergence, with period shown to be 1 in both undirected and strongly connected directed cases. For undirected graphs, convergence time is tightly bounded between $\Omega(n\log n)$ and $O(n^2)$, and can be improved to $O\left(\frac{n}{\phi}\log n\right)$ using vertex expansion $\phi$, including high-probability concentration results. For strongly connected directed graphs, the bound is $O\left(nb^2 + \frac{n}{\phi'}\log n\right)$ where $b$ is a directed-cycle orbit parameter and $\phi' = \min(\phi_{out}, \phi_{in})$, highlighting a two-phase convergence: formation of a strong cycle set and subsequent propagation to all vertices. The results give a unified, quantitative understanding of asynchronous max dynamics and show how graph structure governs convergence through expansion-like parameters and orbit-based cycle measures, offering a framework applicable to a broader class of iterative graph processes.
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 $Ω(n\log n)$ and $O(n^2)$ and these bounds are tight. We further characterise the convergence time in $O(\frac{n}φ\log n)$ where $φ$ 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 $O((\frac{n}φ\log n)g(n))$ 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 $O(nb^2 + \frac{n}{φ'}\log n)$ where $b$ is a parameter measuring directed cycle length and $φ'$ is a parameter measuring vertex expansion.