The graph Merriman–Bence–Osher scheme produces, starting from an initial node subset, a sequence of node sets obtained by iteratively applying graph diffusion and thresholding to the characteristic (or indicator) function of the node subsets. One result in [14] gives sufficient conditions on the diffusion time to ensure that the set membership of a given node changes in one iteration of the scheme. In particular, these conditions only depend on local information at the node (information about neighbors and neighbors of neighbors of the node in question). In this paper we show that there does not exist any graph which satisfies these conditions. To make up for this negative result, this paper also presents positive results regarding the Merriman–Bence–Osher dynamics on star graphs and regular trees. In particular, we present sufficient (and in some cases necessary) results for the set membership of a given node to change in one iteration.

中文翻译：

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在星图和规则树上绘制MBO图。对DOI 10.1007 / s00032-014-0216-8进行了更正

图Merriman-Bence-Osher方案从初始节点子集开始生成一系列节点集，这些节点集是通过将图扩散和阈值迭代应用于节点子集的特征（或指标）函数而获得的。[14]中的一个结果给出了扩散时间的充分条件，以确保给定节点的集合成员资格在该方案的一次迭代中发生变化。特别地，这些条件仅取决于节点上的本地信息（有关所讨论节点的邻居和邻居的信息）。在本文中，我们表明不存在满足这些条件的图。为了弥补这一负面结果，本文还对星图和规则树上的Merriman-Bence-Osher动力学给出了积极的结果。特别是，