Finding minimum node separators: A Markov chain Monte Carlo method

In networked systems such as communication networks or power grids, graph separation from node failures can damage the overall operation severely. One of the most important goals of network attackers is thus to separate nodes so that the sizes of connected components become small. In this work, we consider the problem of finding a minimum α-separator, that partitions the graph into connected components of sizes at most αn, where n is the number of nodes. To solve the α-separator problem, we develop a random walk algorithm based on Metropolis chain. We characterize the conditions for the first passage time (to find an optimal solution) of our algorithm. We also find an optimal cooling schedule, under which the random walk converges to an optimal solution almost surely. Furthermore, we generalize our algorithm to non-uniform node weights. We show through extensive simulations that the first passage time is less than O(n3), thereby validating our analysis. The solution found by our algorithm allows us to identify the weakest points in the network that need to be strengthened. Simulations in real topologies show that attacking a dense area is often not an efficient solution for partitioning a network into small components.