Cascading Edge Failures: A Dynamic Network Process

This paper studies a network process that can be used to model cascading failures in networks. The Dynamic Bond Percolation (DBP) process models, through stochastic local rules, the failure or recovery of an edge/link (i;j) in a network. The probability that a working link fails or a failed link recovers may be independent of the state of other links or may be dependent locally on the state of neighboring links as described by a cascade function f. In applications, this means that failures or recovery of links may have a regional preference, or, alternatively, relationships between neighbors in the network can lead to changes in the links between neighbors of neighbors. This paper shows that the dynamic evolution of P(A; t), the probability that the network is in some state A, describing the collective states of all the links at time t, converges to a stationary distribution. We use this distribution to study the emergence of global behaviors like consensus (i.e., catastrophic failure or full recovery of all the edges) or mixed (i.e., some failed and some working substructures). In particular, we show that, depending on the local dynamical rule, different network substructures, such as hub or triangle subgraphs, are more prone to failure.