网络级联过程统一框架中的系统性风险
Systemic Risk in a Unifying Framework for Cascading Processes on Networks
摘要：
我们介绍了一个一般的框架模型的级联和网络上的传染过程，以确定它们的共性和差异。特别是，模型的社会和金融级联，以及纤维束模型，选民模型，流行病传播的模型恢复为特殊情况。为了统一它们的描述，我们定义了一个节点的净脆弱性，这是它的脆弱性和确定其故障的阈值之间的差异。如果节点的净脆弱性增长到零以上，则节点会失效，而节点的失效会增加相邻节点的脆弱性，从而可能引发级联。在这个框架中，我们确定了三个类，这取决于一个节点的脆弱性的方式增加了一个邻居的失败。在微观层面上，我们用具体的例子说明了故障传播模式如何随着触发级联的节点而变化，这取决于它在网络中的位置和程度。在宏观层面上，系统性风险被测量为失效节点的最终分数，$X^\ast$，并且对于三个类别中的每一个，我们推导出一个递归方程来计算其值。$X^\ast$的相图作为初始条件的函数，因此允许预测系统性风险以及比较三种不同的模型类别。我们可以识别哪种模型类别导致系统性风险的一阶相变，即初始条件的微小变化可能导致全局失效的情况。最后，我们推广我们的框架，包括随机传染模型。这表明有可能进一步推广。
英文摘要：
We introduce a general framework for models of cascade and contagion processes on networks, to identify their commonalities and differences. In particular, models of social and financial cascades, as well as the fiber bundle model, the voter model, and models of epidemic spreading are recovered as special cases. To unify their description, we define the net fragility of a node, which is the difference between its fragility and the threshold that determines its failure. Nodes fail if their net fragility grows above zero and their failure increases the fragility of neighbouring nodes, thus possibly triggering a cascade. In this framework, we identify three classes depending on the way the fragility of a node is increased by the failure of a neighbour. At the microscopic level, we illustrate with specific examples how the failure spreading pattern varies with the node triggering the cascade, depending on its position in the network and its degree. At the macroscopic level, systemic risk is measured as the final fraction of failed nodes, $X^\ast$, and for each of the three classes we derive a recursive equation to compute its value. The phase diagram of $X^\ast$ as a function of the initial conditions, thus allows for a prediction of the systemic risk as well as a comparison of the three different model classes. We could identify which model class lead to a first-order phase transition in systemic risk, i.e. situations where small changes in the initial conditions may lead to a global failure. Eventually, we generalize our framework to encompass stochastic contagion models. This indicates the potential for further generalizations.
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