Complexity of optimally defending and attacking a network

Networks are common; numerous complex systems that we encounter in our daily life are based on an intricate network. Examples include society, the Internet, communication infrastructures, spread of infectious diseases and protein interactions. Recently, strategic aspects of network analysis has been a focus of research in many fields. Within this area, identifying the most important nodes/edges is a fundamental problem. Applications include disrupting or blocking fake news, weakening a terrorist network, and restricting a contagion. The problem is relevant to various fields and sectors such as epidemiology, sociology, physics, security and logistics.We address the two contrasting problems of optimally attacking and defending a network. The aim is to identify the most critical nodes or edges, whose removal has the most significant impact on the performance of the network. We choose to quantify the network performance by Inverse Geodesic Length (IGL). It equals the sum of the inverse distances between every two vertices. Our choice is driven by two factors. One, IGL has been frequently studied in the relevant literature as a global measure of robustness of a network. Two, IGL remains effective irrespective of the input graph structure. Interestingly, despite its widespread use as a network performance measure, optimization problems with respect to IGL have not been examined in detail previously.From the perspective of an attacker, we consider two problems, where given a network G, an integer k and a target IGL T , the question is whether there exits a set of vertices (edges, respectively) of size k such that, upon their removal, the IGL of the network is at most T. From a defender’s point of view, we propose a defender-attacker game on a network. In this game the defender seeks to maximize the IGL of the network by committing to protect some number of vertices of the network, while knowing that the attacker can delete the remaining (unprotected) vertices to weaken the network. We conduct a comprehensive complexity analysis of these problems and provide several intractability results. We complement these results by providing exact parameterized algorithms to compute optimal strategies for both the attacker and defender