Optimal control of networks in the presence of attackers and defenders (original) (raw)

The Optimal Defense of Networks of Targets

Economic Inquiry, 2018

This paper examines a game-theoretic model of attack and defense of multiple networks of targets in which there exist intra-network strategic complementarities among targets. The defender's objective is to successfully defend all of the networks and the attacker's objective is to successfully attack at least one network of targets. Although there are multiple equilibria, we characterize correlation structures in the allocations of forces across targets that arise in all equilibria. For example, in all equilibria the attacker utilizes a stochastic 'guerrilla warfare' strategy in which a single random network is attacked.

Strategic Defense and Attack of Complex Networks

2007

This article shows how policy choices about defense and attack investments at the component level can be made for arbitrarily complex networks and systems. Components can be in series, parallel, interdependent, interlinked, independent, or combinations of these. Investments and utilities are determined for the defender and attacker dependent on their unit costs of investment and contest intensity for each component, and their evaluations of the value of system functionality.

Vulnerability and controllability of networks of networks

Chaos, Solitons & Fractals, 2015

Network science is a highly interdisciplinary field ranging from natural science to engineering technology and it has been applied to model complex systems and used to explain their behaviors. Most previous studies have been focus on isolated networks, but many real-world networks do in fact interact with and depend on other networks via dependency connectivities, forming "networks of networks" (NON). The interdependence between networks has been found to largely increase the vulnerability of interacting systems, when a node in one network fails, it usually causes dependent nodes in other networks to fail, which, in turn, may cause further damage on the first network and result in a cascade of failures with sometimes catastrophic consequences, e.g., electrical blackouts caused by the interdependence of power grids and communication networks. The vulnerability of a NON can be analyzed by percolation theory that can be used to predict the critical threshold where a NON collapses. We review here the analytic framework for analyzing the vulnerability of NON, which yields novel percolation laws for n-interdependent networks and also shows that percolation theory of a single network studied extensively in physics and mathematics in the last 50 years is a specific limited case of the more general case of n interacting networks. Understanding the mechanism behind the cascading failure in NON enables us finding methods to decrease the vulnerability of the natural systems and design of more robust infrastructure systems. By examining the vulnerability of NON under targeted attack and studying the real interdependent systems, we find two methods to decrease the systems vulnerability: (1) protect the high-degree nodes, and (2) increase the degree correlation between networks. Furthermore, the ultimate proof of our understanding of natural and technological systems is reflected in our ability to control them. We also review the recent studies and challenges on the controllability of networks and temporal networks.

The Optimal Defense of Networks of Targets The Optimal Defense of Networks of Targets 1

2010

This paper examines a game-theoretic model of attack and defense of multiple networks of targets in which there exist intra-network strategic complementarities among targets. The defender's objective is to successfully defend all of the networks and the attacker's objective is to successfully attack at least one network of targets. In this context, our results highlight the importance of modeling asymmetric attack and defense as a conflict between "fully" strategic actors with endogenous entry and force expenditure decisions as well as allowing for general correlation structures for force expenditures within and across the networks of targets. JEL Classification: C7, D74

Robustness of Network of Networks under Targeted Attack

The robustness of a network of networks (NON) under random attack has been studied recently [Gao et al., Phys. Rev. Lett. 107, 195701 (2011)]. Understanding how robust a NON is to targeted attacks is a major challenge when designing resilient infrastructures. We address here the question how the robustness of a NON is affected by targeted attack on high-or low-degree nodes. We introduce a targeted attack probability function that is dependent upon node degree and study the robustness of two types of NON under targeted attack: (i) a tree of n fully interdependent Erdős-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdős-Rényi networks. For any tree of n fully interdependent Erdős-Rényi networks and scale-free networks under targeted attack, we find that the network becomes significantly more vulnerable when nodes of higher degree have higher probability to fail. When the probability that a node will fail is proportional to its degree, for a NON composed of Erdős-Rényi networks we find analytical solutions for the mutual giant component P ∞ as a function of p, where 1 − p is the initial fraction of failed nodes in each network. We also find analytical solutions for the critical fraction p c , which causes the fragmentation of the n interdependent networks, and for the minimum average degreek min below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdős-Rényi networks under targeted attack, we find the critical coupling strength q c for different n. When q > q c , the attacked system undergoes an abrupt first order type transition. When q q c , the system displays a smooth second order percolation transition. We also evaluate how the central network becomes more vulnerable as the number of networks with the same coupling strength q increases. The limit of q = 0 represents no dependency, and the results are consistent with the classical percolation theory of a single network under targeted attack.

Attack Strategies on Complex Networks

Lecture Notes in Computer Science, 2006

In this work, we estimate the resilience of scale-free networks on a number of different attack methods. We study a number of different cases, where we assume that a small amount of knowledge on the network structure is available, or can be approximately estimated. We also present a class of real-life networks that prove to be very resilient on intentional attacks, or equivalently much more difficult to immunize completely than most model scale-free networks. network, where the probability that a node has a given number of links decays as a power-law P (k) ∼ k −γ , it has been shown that the critical percentage f c of removed nodes that results in network desintegration is very low (less than f c = 0.07) . It is, thus, a well-established fact, supported by analytic results and simulations on model and real-life networks, that a scale-free network is very vulnerable to intentional attacks (where f c is close to 0), although the same network is extremely robust under random node failures (where f c 1) .

Minimizing Expected Attacking Cost in Networks

Electronic Notes in Discrete Mathematics, 2010

A branch-and-bound algorithm is devised to determine the optimal attack strategy to disconnect a network where the objective is to minimize the expected attacking cost. The attacker cannot launch an attack if its cost is beyond his available budget or its probability of success falls below a threshold level. The proposed branch-andbound algorithm includes, among other features, a dynamic programming-based lower bound as well as a preprocessing algorithm which aims at identifying unattackable links and removing irrelevant ones. Extensive use of the min-cut algorithm is made to derive valid upper bounds and to perform feasibility tests. Preliminary numerical implementation shows potential to provide exact solutions for medium-sized networks within reasonable time.

Stability and Topology of Scale-Free Networks under Attack and Defense Strategies

Physical Review Letters, 2005

We study tolerance and topology of random scale-free networks under attack and defense strategies that depend on the degree k of the nodes. This situation occurs, for example, when the robustness of a node depends on its degree or in an intentional attack with insufficient knowledge of the network. We determine, for all strategies, the critical fraction p c of nodes that must be removed for disintegrating the network. We find that, for an intentional attack, little knowledge of the well-connected sites is sufficient to strongly reduce p c . At criticality, the topology of the network depends on the removal strategy, implying that different strategies may lead to different kinds of percolation transitions.

The Optimal Defense of Network Connectivity

Game Theory & Bargaining Theory eJournal, 2015

Maintaining the security of critical infrastructure networks is vital for a modern economy. This paper examines a game-theoretic model of attack and defense of a network in which the defender’s objective is to maintain network connectivity and the attacker’s objective is to destroy a set of nodes that disconnects the network. The conflict at each node is modeled as a contest in which the player that allocates the higher level of force wins the node. Although there are multiple mixed-strategy equilibria, we characterize correlation structures in the players’ multivariate joint distributions of force across nodes that arise in all equilibria. For example, in all equilibria the attacker utilizes a stochastic ‘guerrilla warfare’ strategy in which a single random [minimal] set of nodes that disconnects the network is attacked.

Survivable network design under optimal and heuristic interdiction scenarios

Journal of Global Optimization, 2006

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer's network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.