Convergence Properties of the Asynchronous Maximum Model

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.