Causal Node Failures and Computation of Giant and Small Components in Networks

Node failures have a terrible effect on the connectivity of the network. In traditional models, the failures of nodes affect their neighbors and may further trigger the failures of their neighbors, and so on. However, it is also possible that node failures would indirectly cause the failure of nodes that are not adjacent to the failed one. In a power grid, generators share the load. Failure of one generator induces extra load on other generators in the network, which could further trigger their failures. We call such failures causal failures. In this work, we consider the impact of causal failures on the network's connectivity in terms of the number of connected components and their sizes. More specifically, we formally define causal failures in a given network and propose two problems that address the network's robustness and vulnerability, respectively. The first problem that corresponds to robustness aims to find the maximum number of causal failures while maintaining a connected component with a size of at least a given integer. More specifically, we are looking into the number of causal failures we can tolerate yet have most of the system connected with α being used to parametrize. The second problem deals with vulnerability, wherein we aim to find the minimum number of causal failures such that there are at least
k
connected components remaining. We are looking for the set of causal failures that will result in the network being disconnected into
k
or more components. We show that the decision versions of both problems are NP-complete and correspondingly provide integer linear programming. Given the hardness of both problems, we design polynomial-time heuristic algorithms to solve them approximately. Finally, we present an illustrative example to compare the performance of heuristics with that of integer linear programming and analyze the scalability of the proposed algorithms by performing experiments on another two networks.