Robustness analysis of network controllability

Lecture Notes in Computer Science, 2015

The notion of controllability, informally the ability to force a system into a desired state in a finite time or number of steps, is most closely associated with control systems such as those used to maintain power networks and other critical infrastructures, but has wider relevance in distributed systems. It is clearly highly desirable to understand under which conditions attackers may be able to disrupt legitimate control, or to force overriding controllability themselves. Following recent results by Liu et al., there has been considerable interest also in graphtheoretical interpretation of Kalman controllability originally introduced by Lin, structural controllability. This permits the identification of sets of driver nodes with the desired state-forcing property, but determining such nodes is a W [2]-hard problem. To extract these nodes and represent the control relation, here we apply the POWER DOMINATING SET problem and investigate the effects of targeted iterative multiple-vertex removal. We report the impact that different attack strategies with multiple edge and vertex removal will have, based on underlying noncomplete graphs, with an emphasis on power-law random graphs with different degree sequences.

Arxiv preprint arXiv: …, 2011

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Lui et al., Nature:473(7346), 167-173, 2011). While the integration of control theory and network analysis represents an important advance, we show that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set (PDS), is all that is needed for structural controllability, a result consistent with the well known fact that controllability (and its dual, observability) are generic properties of systems. We argue that more important than structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.

Physical review. E, 2017

When controlling a complex networked system it is not feasible to control the full network because many networks, including biological, technological, and social systems, are massive in size and complexity. But neither is it necessary to control the full network. In complex networks, the giant connected components provide the essential information about the entire system. How to control these giant connected components of a network remains an open question. We derive the mathematical expression of the degree distributions for four types of giant connected components and develop an analytic tool for studying the controllability of these giant connected components. We find that for both Erdős-Rényi (ER) networks and scale-free (SF) networks with p fraction of remaining nodes, the minimum driver node density to control the giant component first increases and then decreases as p increases from zero to one, showing a peak at a critical point p=p_{m}. We find that, for ER networks, the pe...

2020

Networks are all around us, telecommunication networks, road transportation networks, and the Internet are a few examples of networks that we encounter every day. The entities in a network are represented by nodes and the interconnections between them are represented by links. For example, in a telecommunication network, a node could be an end point for data transmission, a redistribution point or in physical terms, an entity that is capable of receiving, transmitting or redistributing information and a link could be a wired or a wireless connection between the nodes. It is of prime importance that the networks perform their functions properly. To ensure the effective operation of such networks, we need to control them by applying external inputs on some nodes which are known as driver nodes. We say that a network is controllable if it can be driven from any arbitrary state to any desired state in finite time under the control of driver nodes which are attached to the external input...

In this paper structural controllability of complex networks is analyzed. A new algorithm is proposed which constructs a structural control scheme for a given network by avoiding the absence of dilations and by guaranteeing the accessibility of all nodes. Such accessibility is solved via a wiring procedure; this procedure, based on determining the non-accessible regions of the network, has been improved in this new proposed algorithm. This way, the number of dedicated controllers is reduced with respect to the one provided by previous existing algorithms.

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.

Networks with a given degree distribution may be very resilient to one type of failure or attack but not to another. The goal of this work is to determine network design guidelines which maximize the robustness of networks to both random failure and intentional attack while keeping the cost of the network (which we take to be the average number of links per node) constant. We find optimal parameters for: (i) scale free networks having degree distributions with a single power-law regime, (ii) networks having degree distributions with two power-law regimes, and (iii) networks described by degree distributions containing two peaks. Of these various kinds of distributions we find that the optimal network design is one in which all but one of the nodes have the same degree, k1 (close to the average number of links per node), and one node is of very large degree, k2 ∼ N 2/3 , where N is the number of nodes in the network.

arXiv (Cornell University), 2008

With increasingly ambitious initiatives such as GENI and FIND that seek to design future internets, it becomes imperative to define the characteristics of robust topologies, and build future networks optimized for robustness. This paper investigates the characteristics of network topologies that maintain a high level of throughput in spite of multiple attacks. To this end, we select network topologies belonging to the main network models and some real world networks. We consider three types of attacks: removal of random nodes, high degree nodes, and high betweenness nodes. We use elasticity as our robustness measure and, through our analysis, illustrate that different topologies can have different degrees of robustness. In particular, elasticity can fall as low as 0.8% of the upper bound based on the attack employed. This result substantiates the need for optimized network topology design. Furthermore, we implement a tradeoff function that combines elasticity under the three attack strategies and considers the cost of the network. Our extensive simulations show that, for a given network density, regular and semi-regular topologies can have higher degrees of robustness than heterogeneous topologies, and that link redundancy is a sufficient but not necessary condition for robustness.

Vulnerability of a real-world complex network against unwanted attacks and random link failures is an issue of immense concern. A small attack or failure of the network, has the potential to trigger a global cascading breakdown, thereby raising questions with regard to the possible strategies to combat such a mishap. Many works have been published lately, that deals mainly with the revival of a complex network after an attack or failure. In this paper, we propose to build the network architecture in an efficient manner, so that the network can withstand attacks or link failures up to some certain pre-specified limit. We introduce a novel approach to enhance the robustness of a network from the prevention point of view, that is prior to an attack or failure. Simulation results reveal that with a slight increase in the number of driver nodes, from that obtained using the existing maximum matching algorithm, enhances the stability of the network up to a large number of link failures. We also observe that, the sparse and inhomogeneous networks are difficult to control and are less robust, compared to dense and homogeneous networks.

2015 54th IEEE Conference on Decision and Control (CDC), 2015

This paper studies the controllability degree of complex networks as a function of the network diameter and weights. We quantify the controllability degree of a network with the worst-case control energy to drive the network to an arbitrary state. We show that certain networks, including acyclic networks, are difficult to control whenever their diameter is a sublinear function of the network size, as the control energy grows exponentially with the network cardinality when the number of control nodes remains constant. Conversely, we show that certain anisotropic networks where the diameter depends linearly on the network cardinality are easy to control, as the control energy is bounded independently of the network cardinality and number of control nodes. We conjecture that the network diameter is a key topological property determining the controllability degree of a network.

We are concerned with an appropriate mathematical measure of resilience in the face of targeted node attacks for arbitrary degree networks, and subsequently comparing the resilience of different scale-free network models with the proposed measure. We strongly motivate our resilience measure termed vertex attack tolerance (VAT), which is denoted mathematically as τ (G) = minS⊂V

Machine Learning and Intelligent Communications, 2019

Complex network is a network structure composed of a large number of nodes and complex relationships between these nodes. Using complex network can model many systems in real life. The individual in the system corresponds to the node in the network and the relationship between these individuals corresponds to the edge in the network. The controllability of complex networks is to study how to enable the network to arrive at the desired state from any initial state by external input signals. The external input signals transmit to the whole network through some nodes in the network, and these nodes are called driver node. For the study of controllability of complex network, it is mainly to judge whether the network is controllable or not and how to select the appropriate driver nodes at present. If a network has a high controllability, the network will be easy to control. However, complex networks are vulnerable and will cause declining of controllability. Therefore, we propose in this paper a link prediction-based method to make the network more robust to different modes of attacking. Through experiments we have validated the effectiveness of the proposed method.

• This paper proposes a quantitative method for determining the robustness. • This method can deal with both node and edge attacks and assure the similar results.

RAIRO - Operations Research

The purpose of this work is four-fold: (1) We propose a new measure of network resilience in the face of targeted node attacks, vertex attack tolerance, represented mathematically as τ (G) = minS⊂V |S| |V -S-Cmax(V -S)|+1 , and prove that for d-regular graphs τ (G) = Θ(Φ(G)) where Φ(G) denotes conductance, yielding spectral bounds as corollaries. (2) We systematically compare τ (G) to known resilience notions, including integrity, tenacity, and toughness, and evidence the dominant applicability of τ for arbitrary degree graphs. (3) We explore the computability of τ , first by establishing the hardness of approximating unsmoothened vertex attack tolerance τ (G) = minS⊂V |S| |V -S-Cmax(V -S)| under various plausible computational complexity assumptions, and then by presenting empirical results on the performance of a betweenness centrality based heuristic algorithm applied not only to τ but several other hard resilience measures as well. (4) Applying our algorithm, we find that the random scale-free network model is more resilient than the Barabasi-Albert preferential attachment model, with respect to all resilience measures considered.

International Journal of Internet Technology and Secured Transactions, 2010

With increasingly ambitious initiatives such as GENI and FIND that seek to design future internets, it becomes imperative to define the characteristics of robust topologies, and build future networks optimized for robustness. This paper investigates the characteristics of network topologies that maintain a high level of throughput in spite of multiple attacks. To this end, we select network topologies belonging to the main network models and some real world networks. We consider three types of attacks: removal of random nodes, high degree nodes, and high betweenness nodes. We use elasticity as our robustness measure and, through our analysis, illustrate that different topologies can have different degrees of robustness. In particular, elasticity can fall as low as 0.8% of the upper bound based on the attack employed. This result substantiates the need for optimized network topology design. Furthermore, we implement a tradeoff function that combines elasticity under the three attack strategies and considers the cost of the network. Our extensive simulations show that, for a given network density, regular and semi-regular topologies can have higher degrees of robustness than heterogeneous topologies, and that link redundancy is a sufficient but not necessary condition for robustness.