Predicting the cascading dynamics in complex networks via the bimodal failure size distribution

Cascading failure as a systematic risk occurs in a wide range of real-world networks. Cascade size distribution is a basic and crucial characteristic of systemic cascade behaviors. Recent research works have revealed that the distribution of cascade sizes is a bimodal form indicating the existence of either very small cascades or large ones. In this paper, we aim to understand the properties and formation characteristics of such bimodal distribution in complex networks and further predict the final cascade size. We first find that the bimodal distribution is ubiquitous under certain conditions in both synthetic and real networks. Moreover, the large cascades distributed in the right peak of bimodal distribution are resulted from either the failure of nodes with high load at the first step of the cascade or multiple rounds of cascades triggered by the initial failure. Accordingly, we propose a hybrid load metric (HLM), which combines the load of the initial broken node and the load of failed nodes triggered by the initial failure, to predict the final size of cascading failures. We validate the effectiveness of HLM by computing the accuracy of identifying the cascades belonging to the right and left peaks of the bimodal distribution. The results show that HLM is a better predictor than commonly used network centrality metrics in both synthetic and real-world networks. Finally, the influence of network structure on the optimal HLM is discussed.