Shock Propagation in Cascade Systems: Analysis and Applications

In this paper, we study shock propagation in cascade systems that can be a model of outage of power systems and that of spreading a fad. While existing research works have dealt with a certain relationship between global cascade and system properties, such as degree and distribution of threshold values, and interconnection topology, this paper addresses new engineering problems of the cascade systems. Mainly, we would like to 1) identify subsystems contributing to a large cascade, 2) develop a strategy that prevents all initial failures within a designated region from cascading out of the region, and 3) develop a strategy that prevents all initial failures outside of the region from cascading into the region. In order to solve these challenging problems, we develop an invariant set analysis method for cascade models and derive some features inherent to the cascade systems. As the first feature, supposing that we know the resulting cascade region from the given initial failures in specific subsystems, we will show that the cascading of initial failures of any combination of the subsystems cannot exceed the region. The second feature is a graph theoretic distance condition, which guarantees that two cascading regions are never combined together. Based on the analytical results, we develop algorithms solving the aforementioned engineering problems, in which the complexity of the computation is reduced.