Abstract Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d -dimensional torus and identify the threshold values.

中文翻译：

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自举渗透的阈值行为

摘要 考虑一个图 G 和一个初始随机配置，其中每个节点都是黑色的，概率为 p，否则为白色，独立地。在离散时间轮中，如果每个节点至少有 r 个黑色邻居，则每个节点变为黑色，否则为白色。我们证明，当底层图形是 ad 维环面时，这个基本过程表现出具有两个相变的阈值行为并识别阈值。