Motivated by detection of cascades of defaults in economy, we developed a detection framework for endogenous spreading based on causal motifs we define in this paper. We assume that vertex change of state can be triggered by endogenous or exogenous event, that underlying network is directed and that times when vertices changed their states are available. In addition to data of company defaults we use, we simulate cascades driven by different stochastic processes on different synthetic networks. We also extended an approximate master equation method to directed networks with temporal stamps in order to understand in which cases detection is possible. We show that some of the smallest motifs can robustly detect cascades.

中文翻译：

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应用于公司违约的有向网络中的因果主题和内生级联的存在

受检测经济中的级联违约的启发，我们开发了一个基于我们在本文中定义的因果主题的内生传播检测框架。我们假设状态的顶点变化可以由内生或外生事件触发，底层网络是有向的，并且顶点改变其状态的时间是可用的。除了我们使用的公司默认数据外，我们还模拟了由不同合成网络上的不同随机过程驱动的级联。我们还将近似主方程方法扩展到具有时间标记的有向网络，以了解在哪些情况下可以进行检测。我们表明，一些最小的图案可以稳健地检测级联。