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.

翻译：设 $G = (V,E)$ 为包含 $n$ 个顶点的连通有向图。将集合 $\{1,2,\dots,n\}$ 中的值分配给 $G$ 的顶点，并按照以下规则更新数值：均匀随机地选择一个顶点，将其值更新为其邻域内所有值的最大值。该顶点的值可能因此减小。此随机过程称为异步最大值模型。通过重复该过程，我们证明对于强连通有向图，最终所有顶点将具有相同值，此时称模型已实现 \textit{收敛}。在无向图情形下，证明期望收敛时间在 $n\to \infty$ 时渐近地处于 $\Omega(n\log n)$ 与 $O(n^2)$ 之间，且这些界是紧的。我们进一步刻画了 $O(\frac{n}{\phi}\log n)$ 范围内的收敛时间，其中 $\phi$ 为 $G$ 的顶点扩展度。该结果为一大类图提供了更优的上界。此外，我们证明以高概率收敛所需的轮数属于 $O((\frac{n}{\phi}\log n)g(n))$，其中 $g(n)$ 满足 $\frac{1}{g^2(n)} \to 0$（当 $n \to \infty$）。对于强连通有向图，证明其收敛时间属于 $O(nb^2 + \frac{n}{\phi'}\log n)$，其中 $b$ 为衡量有向环长度的参数，$\phi'$ 为衡量顶点扩展度的参数。