Cascading Failures in Finite-Size Random Geometric Networks
  The problem of cascading failures in cyber-physical systems is drawing much attention in lieu of different network models for a diverse range of applications. While many analytic results have been reported for the case of large networks, very few of them are readily applicable to finite-size networks. This paper studies cascading failures in finite-size geometric networks where the number of nodes is on the order of tens or hundreds as in many real-life networks. First, the impact of the tolerance parameter on network resiliency is investigated. We quantify the network reaction to initial disturbances of different sizes by measuring the damage imposed on the network. Lower and upper bounds on the number of failures are derived to characterize such damages. Such finite-size analysis reveals the decisiveness and criticality of taking action within the first few stages of failure propagation in preventing a cascade. By studying the trend of the bounds as the number of nodes increases, we observe a phase transition phenomenon in terms of the tolerance parameter. The critical value of the tolerance parameter, known as the threshold, is further derived. The findings of this paper, in particular, shed light on how to choose the tolerance parameter appropriately such that a cascade of failures could be avoided. 
Ali Eslami, Chuan Huang, Junshan Zhang, and Shuguang Cui
cs.SI