Convergence Properties of the Asynchronous Maximum Model
J Larkin�- arXiv preprint arXiv:2406.01910, 2024 - arxiv.org
J Larkin
arXiv preprint arXiv:2406.01910, 2024•arxiv.orgLet $ 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�…
$\{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�…
Let be a connected directed graph on vertices. Assign values from the set to the vertices of 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 ) in and and these bounds are tight. We further characterise the convergence time in where is the vertex expansion of . This provides a better upper bound for a large class of graphs. Further, we show the number of rounds until convergence is in with high probability, where satisfies as . For a strongly connected directed graph the convergence time is shown to be in where is a parameter measuring directed cycle length and is a parameter measuring vertex expansion.
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