Finding minimum node separators: A Markov chain Monte Carlo method

Abstract

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.

Introduction

Today’s critical infrastructures are organized in the form of a network such as the communication network, the smart electrical power grid or the water distribution system. In these networks, small node failures can have a significant impact on connectivity and lead to graph separation [2], [3], where components become more reliable as their sizes increase. For example, in a power grid, there should be a sufficient amount of power generation to serve the power loads while not every bus (frequently represented as nodes in the graph modeling power grid) can have a generator. Protecting minimum separating nodes can keep the sizes of components higher than a given threshold, thereby increasing the chance of load being served. Thus, identifying the weak parts in practical networks is a major concern in general. Graph separation can also incur further cascading failures, because the amount of resource in the remaining components (e.g., the amount of generated power) becomes so reduced that the remaining components are overloaded and subsequently failed [4]. Thus, one of the most important goals of network attackers is to separate nodes such that the sizes of connected components become small. However, attackers have resource constraints as well, and they would like to inflict the greatest harm with limited cost. A defender wants to identify these weakest points in advance. In this work, we consider the problem of finding a minimum α-separator that partitions the graph into connected components of sizes smaller than αn, where n is the number of nodes in the graph. Finding a minimum α-separator is proven to be NP-hard for a general topology for $\alpha \ge \frac{2}{3}$ [5]. For topologies such as trees and cycles, the authors in [6] have developed polynomial-time algorithms, yet they require knowledge of the type of the graph topology in advance.

To tackle the minimum α-separator problem without prior knowledge of the graph topology, we apply a Markov chain Monte Carlo (MCMC) method. The basic idea is to solve this combinatorial optimization problem by constructing a random walk over a Markov chain, which traverses through feasible solutions, where the transitions lead the system to move to states with desired objectives (smaller α-separator in our case). In other words, the stationary distribution is higher for states with better objective values (i.e., closer to the minimum). The Markov chain Monte Carlo method has been applied in many NP-hard problems in various applications [7], [8], [9], [10], [11]. In our random walk algorithm, we additionally design a simple data structure to quickly identify the sizes of the components that vary under the random walk in each step. This allows us to further reduce the computational complexity in each step.

We then analytically characterize how long it takes to obtain the optimal solution by our algorithm. The standard metric often used in the literature is the mixing time, the convergence time until the Markov chain is close to its stationary distribution, and there are several existing works showing the polynomial convergence of the chain. For the independent set problem, [12], [13], [14] used a coupling technique to show polynomial convergence. However, conditions for guaranteeing polynomial convergence are limited. For example, in [12], the maximum node degree of a graph should be less than or equal to five, for polynomial convergence. Hence, in this paper we present an approximate analysis on the first passage time, using a hierarchical structure of the underlying Markov chain. We then provide sufficient conditions for the expected first passage time to be polynomial. As an example, we compute the expected first passage time in star-like topologies, which is O(n²), where n is the number of nodes. The conditions also include simple topologies that the sizes of minimum α-separators do not scale with the network size, e.g., line, circle, and balanced tree.

To evaluate the performance and support our analytical results, we run our random walk algorithm over various topologies including real topologies from US fiber networks of 20 Internet providers [15] and Italian power grid and Internet networks [16], [17]. In all topologies, our random walk algorithm converges to a solution within O(n³) steps, for a network with n nodes. We also compare our random walk algorithm with other heuristic algorithms including highest-degree-first and greedy algorithms, and illustrate the performance improvements we obtain. We underscore that the solution from our algorithm allows us to characterize the weakest points in the network that need to be strengthened. In real topologies, we find that attacking a dense area may not be an efficient way to partition a network graph which can lead to large cascading effects. Finally, we discuss some future directions to apply our general results to defense problems and practical networks such as power grids and water distribution systems in consideration of physical dynamics. Even though our graphical model is simple to model physical dynamics in practical systems, our work can infer initial weak points in these complex systems. As a future work, we will work on developing an algorithm to find weak points in networks with complex physical dynamics (e.g., correlated failures).