We consider a dynamic network cascade process developed by Duncan Watts applied to a class of random networks, developed independently by Newman and Miller, which allows a specified amount of clustering (short loops). We adapt existing methods for locally tree-like networks to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a formula that has appeared in various forms in works by Hackett et al. and Miller which provides a threshold condition for certain extinction of the cascade. We find that clustering impedes cascade propagation for networks of low average degree by reducing the connectivity of the network, but for networks with high average degree, the presence of small cycles makes cascades more likely.

中文翻译：

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集群网络上的单种子级联

我们考虑将 Duncan Watts 开发的动态网络级联过程应用于由 Newman 和 Miller 独立开发的一类随机网络，它允许指定数量的聚类（短循环）。我们采用本地树状网络的现有方法来制定适当的两种分支过程来描述从单个活动节点开始的级联的传播，并获得一个不动点方程来隐式表达这种级联的灭绝概率。通过这样做，我们还恢复了在 Hackett 等人的作品中以各种形式出现的公式。和米勒，它为级联的某些灭绝提供了阈值条件。我们发现聚类通过降低网络的连通性来阻碍平均度数较低的网络的级联传播，