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Towards a Realistic Model for Failure Propagation in Interdependent Networks
Abstract
Modern networks are becoming increasingly interdependent. As a prominent example, the smart grid is an electrical grid controlled through a communications network, which in turn is powered by the electrical grid. Such interdependencies create new vulnerabilities and make these networks more susceptible to failures. In particular, failures can easily spread across these networks due to their interdependencies, possibly causing cascade effects with a devastating impact on their functionalities.
In this paper we focus on the interdependence between the power grid and the communications network, and propose a novel realistic model, HINT (Heterogeneous Interdependent NeTworks), to study the evolution of cascading failures. Our model takes into account the heterogeneity of such networks as well as their complex interdependencies. We compare HINT with previously proposed models both on synthetic and real network topologies. Experimental results show that existing models oversimplify the failure evolution and network functionality requirements, resulting in severe underestimations of the cascading failures.
... In contrast, some following works identify vulnerable topologies in interdependent networks to avoid such fragile structures in the design phase by investigating the relation between node degree and failure impacts [9], or evaluating the importance of nodes exploiting the algebraic expression of dependencies [10]. Furthermore, other works propose design strategies in more realistic models to consider the impact of failures caused by a single component [11], integrated factors within and between layers [12], or the heterogeneity of nodes in each layer [13]. ...
... The works [9]- [13] focus on the design aspect of interdependent networks. The relation between the impact of failures and interdependencies is empirically demonstrated to decide appropriate dependency allocations in [9]. ...
... Combining multiple factors that make a node nonfunctional, their method adjusts the dependency of a node on the other nodes. The work in [13] also considers the influence within a singlelayer, supposing the heterogeneity of nodes. In this model, a network can have different types of nodes such as generating and relay nodes. ...
The interdependency between different network layers is commonly observed in Cyber Physical Systems and communication networks adopting the dissociation of logic and hardware implementation, such as Software Defined Networking and Network Function Virtualization. This paper formulates an optimization problem to improve the survivability of interdependent networks by restructuring the provisioning relations. A characteristic of the proposed algorithm is that the continuous availability of the entire system is guaranteed during the restructuring of dependencies by the preservation of certain structures in the original networks. Our simulation results demonstrate that the proposed restructuring algorithm can substantially enhance the survivability of interdependent networks, and provide insights into the ideal allocation of dependencies.
... Many works in modeling cascading failures study the effects of, and mitigation strategies for, a set of initiating element failures such as power lines or substations [5,7,8,9]. When J. Cunningham studying cascading failures, these works consider a maximum initiating failure size of anywhere from 10% to 35% of grid elements out of hundreds or thousands. ...
... When J. Cunningham studying cascading failures, these works consider a maximum initiating failure size of anywhere from 10% to 35% of grid elements out of hundreds or thousands. These targets are selected either randomly or via a deterministic heuristic [5,7]. However, an adversary will usually not behave randomly, but instead seek to cause maximum damage given their limited resources [10]. ...
... As communication and power networks have trended towards interdependence, there have been several works that have investigated modeling failure propagation between them. Sturaro et al. [7] introduce a heterogeneous model of cascading failure in interconnected communication and power grids that captures some of the complex and realistic interdependencies between the two network domains on networks as large as 1022 nodes. They show that under both random attacks and targeted attacks (based on a node degree heuristic) of up to 15% of nodes that simpler models tend to underestimate the cascading effect of failures in the coupled network. ...
Power grids that are interdependent with communication networks create more possible modes of failure (e.g., cyberattacks) as well as more complex propagation of failure through the coupled networks. To ensure robust defense of smart grids, it is important to model both attacker and defender as intelligent, a scenario that the framework of game theory provides methods to analyze. However, prior works in applying game theoretic models to smart grid security limit the problem space to a small number targets under threat due to the inability of state-of-the-art methods to scale to large networks. Our method scales to large networks by combining neural networks that use featurized action representations with an approximation of large combinatorial actions to generalize knowledge about the best targets to attack/defend across graphs of various topologies and sizes. Our model’s invariance to the size of the input graph allows us to transfer knowledge from games played on small graphs during training to large graphs during evaluation. Our experiments show that our method can learn Nash equilibrium strategies on small networks, and demonstrate low exploitability when generalized to large networks, especially compared to the common heuristics currently used to simulate attacks on large graphs.
... In contrast, some following works identify vulnera- ble topologies in interdependent networks to avoid such fragile structures in the design phase by investigating the relation between node degree and failure impacts [9], or evaluating the importance of nodes exploiting the algebraic expression of dependencies [10]. Furthermore, other works propose design strategies in more realistic models to consider the impact of failures caused by a single component [11], integrated factors within and between layers [12], or the heterogeneity of nodes in each layer [13]. ...
... The works [9]- [13] focus on the design aspect of interde- pendent networks. The relation between the impact of failures and interdependencies is empirically demonstrated to decide appropriate dependency allocations in [9]. ...
... Combining multiple factors that make a node nonfunctional, their method adjusts the dependency of a node on the other nodes. The work in [13] also considers the influence within a single- layer, supposing the heterogeneity of nodes. In this model, a network can have different types of nodes such as generating and relay nodes. ...
The interdependency between different network layers is commonly observed in Cyber Physical Systems and communication networks adopting the dissociation of logic and hardware implementation, such as Software Defined Networking and Network Function Virtualization. This paper formulates an optimization problem to improve the survivability of interdependent networks by restructuring the provisioning relations. A characteristic of the proposed algorithm is that the continuous availability of the entire system is guaranteed during the restructuring of dependencies by the preservation of certain structures in the original networks. Our simulation results demonstrate that the proposed restructuring algorithm can substantially enhance the survivability of interdependent networks, and provide insights into the ideal allocation of dependencies.
https://arxiv.org/abs/1903.01583
... In contrast, some following works try to identify vulnerable topologies in interdependent networks to avoid such fragile structures in the design phase [5], [6]. Furthermore, other works propose design strategies in more realistic models to consider the impact of failures caused by a single component [7], integrated factors within and between layers [8], or the heterogeneity of nodes in each layer [9]. This paper discusses a design problem for interdependent networks to improve their survivability, which is a measure of the robustness against a whole network failure, by modifying an existing network topology. ...
... The work in [8] considers dependency relations not only between layers but also within a single layer. In [9], the heterogeneity of nodes in each network is taken into account. Zhao et al. [7] formulate an optimization problem enhancing the system robustness using an ILP. ...
... The existing works on designing interdependent networks [5]- [9] assume that an entire network is designed and constructed at the same time, though it seems difficult to redesign all the systems simultaneously in practical infrastructure networks. Thus, our paper is aimed at propounding a method to redesign a portion of an existing network so that the network can be functional even during the redesign. ...
... One example of recent work is [4], which demonstrates the efficacy of using graphical models to improve the resiliency of critical infrastructures. Several additional examples of modeling failure propagation in graphs are [6][7][8][9][10]. However, most of these research works assume factors that may reduce problem complexity, which loses some of the realistic aspects of the system. ...
... However, most of these research works assume factors that may reduce problem complexity, which loses some of the realistic aspects of the system. The work in [6] compares its model with the uniform model reported in [7] and the small clusters model reported in [10]. The results indicate that the uniform model is too simplistic to apply to all systems due to the underlying assumptions it makes. ...
A wide range of critical infrastructures are connected via wide area networks as well as the Internet-of-Thing (IoT). Apart from natural disasters, these infrastructures, providing services such as electricity, water, gas, and Internet, are vulnerable to terrorist attacks. Clearly, damages to these infrastructures can have dire consequences on economics, health services, security and safety, and various business sectors. An infrastructure network can be represented as a directed graph in which nodes and edges denote operation entities and dependencies between entities, respectively. A knowledgeable attacker who plans to harm the system would aim to use the minimum amount of effort, cost, or resources to yield the maximum amount of damage. Their best strategy would be to attack the most critical nodes of the infrastructure. From the defender’s side, the strategy would be to minimize the potential damage by investing resources in bolstering the security of the critical nodes. Thus, in the struggle between the attacker and defender, it becomes important for both the attacker and defender to identify which nodes are most critically significant to the system. Identifying critical nodes is a complex optimization problem. In this paper, we first present the problem model and then propose a solution for computing the optimal cost attack while considering the failure propagation. The proposed model represents one or multiple interconnected infrastructures. While considering the attack cost of each node, the proposed method computes the optimal attack that a rational attacker would make. Our problem model simulates one of two goals: maximizing the damage for a given attack budget or minimizing the cost for a given amount of damage. Our technique obtains solutions to optimize the objective functions by utilizing integer-linear programming while observing the constraints for each of the specified goals. The paper reports an extensive set of experiments using various graphs. The results show the efficacy of our technique in terms of its ability to obtain solutions with fast turnaround times.
... A combination of power grid and computer network has been intensively studied as a typical instance of the interdependent networks. Each distribution node in the power grid must be supplied with sufficient electric power from the generation nodes in order to provide the computer network with the required electric power [17], [22]. Likewise, each edge node operating the power grid needs to receive a sufficient amount of control messages from the server nodes via the computer network. ...
... A graph theoretic metric for measuring the cascade effect on the interdependent networks and significance metrics to assess the impact of the individual nodes on the spreading of failures have been proposed [14]- [16]. A realistic model of the interdependent networks was proposed to study the evolution of the cascading failures [17]. The robustness of interdependent networks has been analyzed on a basis of the percolation theory [18]- [21]. ...
Cyber-physical systems comprise multiple interdependent networks supporting each other. This paper considers a combination of power grid and computer network as a typical instance of the interdependent networks. A node in an interdependent network fails if it cannot receive the required volume of support from the nodes in the other interdependent network. Meanwhile, each node supporting the other network is required to receive a sufficient volume of intra-network flow named “support flow”, which corresponds to the flow of electric power in the power grid and the flow of control messages in the computer network. Each sink node of the support flow must terminate a sufficient volume of support flow generated at source nodes in the same network in order to achieve the support of the other network. This means that the support flow is one of the essential elements to make the interdependent networks operable. The conditions related to the support flow increases the possibility of a small initial node failure invoking a cascading failure with a wide range. This paper proposes a robust design method for the interdependent networks to maximize the robustness against the cascading failures while assuring satisfaction of the constraints on the support flow. The method can derive the interdependency to minimize the impact of the cascading failures induced from a given set of possible initial node failures when the configuration of the respective interdependent networks is specified beforehand. The robust design problem is formulated using a mixed integer linear programming model and two kinds of metaheuristic methods are proposed to solve the problem efficiently even if a great number of initial node failures are specified. Several characteristics of the risk of the cascading failures are revealed from the results of the robust design. Simulation experiments demonstrate the effectiveness of the proposed metaheuristic methods.
... A combination of power grid and computer network has been intensively studied as a typical instance of the interdependent networks. Each distribution node in the power grid must be supplied with sufficient electric power from the generation nodes in order to provide the computer network with the required electric power [17], [22]. Likewise, each edge node operating the power grid needs to receive a sufficient amount of control messages from the server nodes via the computer network. ...
... A graph theoretic metric for measuring the cascade effect on the interdependent networks and significance metrics to assess the impact of the individual nodes on the spreading of failures have been proposed [14]- [16]. A realistic model of the interdependent networks was proposed to study the evolution of the cascading failures [17]. The robustness of interdependent networks has been analyzed on a basis of the percolation theory [18]- [21]. ...
p>Cyber-physical systems comprise multiple interdependent networks supporting each other. This paper considers a combination of power grid and computer network as a typical instance of the interdependent networks. A node in an interdependent network fails if it cannot receive the required volume of support from the nodes in the other interdependent network. Meanwhile, each node supporting the other network is required to receive a sufficient volume of intra-network flow named “support flow”, which corresponds to the flow of electric power in the power grid and the flow of control messages in the computer network. Each sink node of the support flow must terminate a sufficient volume of support flow generated at source nodes in the same network in order to achieve the support of the other network. This means that the support flow is one of the essential elements to make the interdependent networks operable. The conditions related to the support flow increases the possibility of a small initial node failure invoking a cascading failure with a wide range. This paper proposes a robust design method for the interdependent networks to maximize the robustness against the cascading failures while assuring satisfaction of the constraints on the support flow. The method can derive the interdependency to minimize the impact of the cascading failures induced from a given set of possible initial node failures when the configuration of the respective interdependent networks is specified beforehand. The robust design problem is formulated using a mixed integer linear programming model and two kinds of metaheuristic methods are proposed to solve the problem efficiently even if a great number of initial node failures are specified. Several characteristics of the risk of the cascading failures are revealed from the results of the robust design. Simulation experiments demonstrate the effectiveness of the proposed metaheuristic methods. </p
... In most of the interdependent networks, each node cannot demonstrate its ability to support the other interdependent networks when it is disconnected with particular nodes [17], [22]. For example, each distribution node in the power grid requires to be supplied with the electric power from the generation nodes to provide the computer network with the electric power. ...
... A graph theoretic metric for measuring the cascade effect on the interdependent networks and significance metrics to assess the impact of the individual nodes on the spreading of failures have been proposed [14]- [16]. A realistic model of the interdependent networks was proposed to study the evolution of the cascading failures [17]. The robustness of interdependent networks has been analyzed on a basis of the percolation theory [18]- [21]. ...
p>Cyber-physical systems comprise multiple interdependent networks supporting each other. A node in an interdependent network fails when it cannot receive a sufficient volume of support from normal nodes in the other interdependent networks. Meanwhile, each node loses its support capability if it cannot terminate intra-network flow named “support flow”, which corresponds to the flow of electric power in the power grid and the flow of control messages in the computer network. This means that each sink node of the support flow is regarded as being in a failure state when it is disconnected with all the normal source nodes of the support flow in the same network. This criterion of the sink node failure increases the possibility of an initial node failure invoking a cascading failure with a wide range. This paper proposes a robust design method for the interdependent networks that takes the support flow into account. The proposed method can derive the optimum interdependency to minimize the greatest impact of the cascading failures induced from a given set of the initial node failures. The robust design problem is formulated using a mixed integer linear programming model and two kinds of metaheuristic methods are proposed to solve the problem efficiently even when numerous initial node failures are specified. Several characteristics of the risk of the cascading failures are revealed from the results of the proposed robust design. Simulation experiments demonstrate the effectiveness of the proposed metaheuristic methods.</p
... In most of the interdependent networks, each node cannot demonstrate its ability to support the other interdependent networks when it is disconnected with particular nodes [17], [22]. For example, each distribution node in the power grid requires to be supplied with the electric power from the generation nodes to provide the computer network with the electric power. ...
... A graph theoretic metric for measuring the cascade effect on the interdependent networks and significance metrics to assess the impact of the individual nodes on the spreading of failures have been proposed [14]- [16]. A realistic model of the interdependent networks was proposed to study the evolution of the cascading failures [17]. The robustness of interdependent networks has been analyzed on a basis of the percolation theory [18]- [21]. ...
p>Cyber-physical systems comprise multiple interdependent networks supporting each other. A node in an interdependent network fails when it cannot receive a sufficient volume of support from normal nodes in the other interdependent networks. Meanwhile, each node loses its support capability if it cannot terminate intra-network flow named “support flow”, which corresponds to the flow of electric power in the power grid and the flow of control messages in the computer network. This means that each sink node of the support flow is regarded as being in a failure state when it is disconnected with all the normal source nodes of the support flow in the same network. This criterion of the sink node failure increases the possibility of an initial node failure invoking a cascading failure with a wide range. This paper proposes a robust design method for the interdependent networks that takes the support flow into account. The proposed method can derive the optimum interdependency to minimize the greatest impact of the cascading failures induced from a given set of the initial node failures. The robust design problem is formulated using a mixed integer linear programming model and two kinds of metaheuristic methods are proposed to solve the problem efficiently even when numerous initial node failures are specified. Several characteristics of the risk of the cascading failures are revealed from the results of the proposed robust design. Simulation experiments demonstrate the effectiveness of the proposed metaheuristic methods.</p
... The propagation of disturbances with a focus on the availability of components in CPESs is discussed in Lu et al. (2017) and Sturaro et al. (2016). These works model both power and ICT systems, focusing on how component failures in one domain propagate to the other, thereby impacting the combined CPES. ...
Modern power systems, referred to as cyber-physical energy systems (CPESs), are complex systems with strong interdependencies between power and information and communication technology (ICT) systems. CPESs also have dependencies between the essential grid services. For instance, coordinated voltage control depends on state estimation, which depends on measurement acquisition. Since the operation of CPESs is largely influenced by these grid services, assessing their performance is crucial for assessing the performance of a CPES. Most of these grid services are enabled by the ICT system, i.e., they rely to a high degree on ICT. Hence, properties such as availability, correctness and timeliness, which depend on the involved software, hardware and data of the ICT system, must be considered for assessing the performance of an ICT-enabled grid service. Disturbances and repairs in CPESs impact these properties, which can then propagate and affect the performance of a grid service as well as other dependent grid services. There is, therefore, a need to model the influence of the properties of software, hardware and data on ICT-enabled grid services for single services as well as across several services, resulting in a propagation of these parameters. Current literature lacks such a model, which can used not only to investigate but also to visualise the impact of these properties on the overall perfromance of a grid service as well as other dependent grid services. This paper proposes a meta model for assessing the performance of ICT-enabled grid services, which can be instantiated for different grid services considering their dependencies. A multi-dimensional operational state space, which serves as a visualisation of the performance of grid services in terms of their state trajectory, is also proposed in this paper. The contributions are then demonstrated by a case study with a state estimation service and the widely-used CIGRE medium voltage benchmark power grid augmented with an ICT system. Three scenarios with disturbances are presented to show the benefits of the contributions. Specifically, the performance of the state estimation service considering the disturbances is investigated using the meta model, and the change in performance is visualised as trajectories using the operational state space. These contributions enable new possibilities for planning and vulnerability analyses: property changes in parts of the ICT system can be simulated to investigate their consequences throughout the ICT-enabled grid services. A trajectory representing their performance can then be visualized in the state space based on which measures could be implemented to potentially improve the resilience of the service against the considered disturbances.
... To predict the CCF path, the CCF model [12] is used to simulate the propagation process of CCF in ADN. Each node in ADN is simulated with 1000 times for CCF. ...
Abstract—Cascading failures may lead to large scale outages,
which brings about significant economic losses and serious social
impacts. It is very important to predict cross-domain cascading
failures paths for identification of weak nodes, which contributes
to the control policies for preventing cascading failures and
blocking their propagation between cyber domain and physical
domain in cyber-physical active distribution networks. This paper
proposes an algorithm based on the Frequent-Patterns-Growth
(FP-Growth) to predict cascading failure paths, which predicts the
potential failure node set by analyzing a large number of
simulation datum and mining the hidden association relationship
among datum. To demonstrate the effectiveness of the proposed
cascading failure path prediction approach, an empirical study on
a cyber-physical active distribution network, named CEPRI-CPS
from Electric Power Research Institute of China, is performed,
and the result shows the robustness of cyber-physical active
distribution networks can be improved with prediction approach
in this paper.
Keywords—cascading failure path prediction, association
rules, cyber–physical active distribution networks, FP-Growth
algorithm
... De plus, le couplage de ressources différentes rend les réseaux interdépendants très sensibles aux diverses perturbations qui peuvent survenir, c'est-à-dire que ce couplage augmente leur vulnérabilité aux aléas [Buldyrev et al. 2010 ;Havlin et al. 2010]. Par exemple, la panne d'électricité qui a frappé l'Italie le 28 septembre 2003 était due à deux réseaux : le réseau électrique (utilisé pour le contrôle de surveillance) et le réseau Internet (utilisé pour le système d'acquisition des données [Sturaro et al. 2016 ;Rosato et al. 2008]. La nature des systèmes et leur mode de fonctionnement génèrent donc des dégradations de certains composants plus sensibles, pouvant créer un défaut de fonctionnement. ...
... With the use of wind power generation growing rapidly, lightning damage must receive more attention [3,4]. A lightning strike is a tremendously powerful phenomenon that can produce overvoltages in various components of a wind turbine [5,6], which can easily spread across a network of turbines [7]. ...
Wind generators are exposed to numerous destructive forces such as lightning and are therefore vulnerable to these phenomena. To evaluate the transient behavior of a wind power plant during direct and indirect strikes, modeling of all relevant components is required. Among the protective and control components of wind turbines, the grounding system is the most important element for protection against lightning strikes. This paper examines the impact of nonlinear soil ionization behavior and frequency dependency on a wind turbine in order to model a sufficient protection scheme to reduce overvoltage and make the system tolerable against transitions. The high frequency models of other equipment such as transformers, horizontal conductors, vertical rods, surge arresters and underground cables must also be taken into account to design the grounding system. Our Proposed Modified Grounding Scheme (PMGS) is to reduce the maximum transient overvoltages. We simulate the model in a restructured version of the Electromagnetic Transient Program (EMTP-RV) software to examine the effectiveness of the system. We then apply the simulated results to pair of turbines that are interconnected with a frequency-dependent cable. We carry out the simulation for direct and indirect lightning strikes. The results indicate that the MGS can lead to considerably more than a 50% reduction in transient voltages for lightning and thus leads to more reliable networks.
... Cascading failures in interdependent networks have been studied in several works [11,12,13,14,15,16]. The existing works on interdependent networks can be broadly classified into three categories: 1) those which study the interaction through percolation theory [14,15,16,17], 2) works which try to identify most vulnerable nodes and design failure resilient networks [11,18,19,20,21], 3) and the works which try to find the root cause of failures [22,23]. ...
Vulnerability due to inter-connectivity of multiple networks has been observed in many complex networks. Previous works mainly focused on robust network design and on recovery strategies after sporadic or massive failures in the case of complete knowledge of failure location. We focus on cascading failures involving the power grid and its communication network with consequent imprecision in damage assessment. We tackle the problem of mitigating the ongoing cascading failure and providing a recovery strategy. We propose a failure mitigation strategy in two steps: 1) Once a cascading failure is detected, we limit further propagation by redistributing the generator and load's power. 2) We formulate a recovery plan to maximize the total amount of power delivered to the demand loads during the recovery intervention. Our approach to cope with insufficient knowledge of damage locations is based on the use of a new algorithm to determine consistent failure sets (CFS). We show that, given knowledge of the system state before the disruption, the CFS algorithm can find all consistent sets of unknown failures in polynomial time provided that, each connected component of the disrupted graph has at least one line whose failure status is known to the controller.
In real life, complex networks may encounter uncertain attack events at any time, which may lead to large-scale interruption of various services provided by the network. In order to effectively prevent the occurrence of cascading failures in complex networks, this paper studies the impact of edge weight and capacity allocation strategies on complex network attack resistance. By analyzing the dynamic evolution mechanism of edge cascading fault propagation, a complex network edge cascading fault load model with parameters is constructed. Using the failure normalized F index to analyze the network crash threshold under different attack methods, an improved edge remaining capacity allocation strategy is proposed and a simulated annealing algorithm is used to solve the problem. The focus of this paper is to improve the overall resilience of complex networks against cascading failures, by using the strategy proposed in this paper, we observe the change of network invulnerability after attacks on real express networks and scale-free networks. The experimental results show that comparing the five capacity allocation strategies, the improved side residual capacity allocation strategy greatly improves the network crash threshold of the two networks under different attack methods, and enhances the dynamic resistance of the network.KeywordsCascading FailureCapacity Allocation StrategyInvulnerabilityBA NetworkExpress Network
Cyber-human-physical systems are deeply intertwined systems of physical components, cyber components, and human activities. Cyber components provide communication and control to the physical systems. In addition, human activities can affect or be affected by the behaviors of cyber and physical components. Modern critical infrastructures are a perfect example of such systems. Specifically, physical infrastructures (e.g., power grid, transportation) rely heavily on cyber infrastructures [e.g., supervisory control and data acquisition (SCADA) system] for control and monitoring. Also, humans interact with critical infrastructures in a complex way such that the resilience of critical infrastructures can be heavily affected by human activities. Indeed, critical infrastructures can suffer as a result of both large-scale events such as natural disasters and all kinds of other types of events such as cyber attacks and human operational errors. Therefore, the modeling and analysis of such complex systems are desirable. Such complex systems and their behavior can be modeled appropriately by networks for the purpose of assessing their key attributes, such as resilience. This paper provides an in-depth review of the network modeling and analysis approaches for cyber-human-physical systems. The focus of this paper is critical infrastructures, while the reviewed approaches can also be applied in other application domains.
This position paper highlights security and privacy issues in smart environments based on cyber-physical systems. It also summarizes some of our recent research activities and projects in this area.
Network science has attracted much attention in recent years due to its interdisciplinary applications. We witnessed the revolution of network science in 1998 and 1999 started with small-world and scale-free networks having now thousands of high-profile publications, and it seems that since 2010 studies of ‘network of networks’ (NON), sometimes called multilayer networks or multiplex, have attracted more and more attention. The analytic framework for NON yields a novel percolation law for n interdependent networks that shows that percolation theory of single networks studied extensively in physics and mathematics in the last 50 years is a specific limit of the rich and very different general case of n coupled networks. Since then, properties and dynamics of interdependent and interconnected networks have been studied extensively, and scientists are finding many interesting results and discovering many surprising phenomena. Because most natural and engineered systems are composed of multiple subsystems and layers of connectivity, it is important to consider these features in order to improve our understanding of such complex systems. Now the study of NON has become one of the important directions in network science. In this paper, we review recent studies on the new emerging area—NON. Due to the fast growth of this field, there are many definitions of different types of NON, such as interdependent networks, interconnected networks, multilayered networks, multiplex networks and many others. There exist many datasets that can be represented as NON, such as network of different transportation networks including flight networks, railway networks and road networks, network of ecological networks including species interacting networks and food webs, network of biological networks including gene regulation network, metabolic network and protein-protein interacting network, network of social networks and so on. Among them, many interdependent networks including critical infrastructures are embedded in space, introducing spatial constraints. Thus, we also review the progress on study of spatially embedded networks. As a result of spatial constraints, such interdependent networks exhibit extreme vulnerabilities compared with their non-embedded counterparts. Such studies help us to understand, realize and hopefully mitigate the increasing risk in NON.
From transportation networks to complex infrastructures, and to social and
communication networks, a large variety of systems can be described in terms of
multiplexes formed by a set of nodes interacting through different networks
(layers). Multiplexes may display an increased fragility with respect to the
single layers that constitute them. However, so far the overlap of the links in
different layers has been mostly neglected, despite the fact that it is an
ubiquitous phenomenon in most multiplexes. Here we show that the overlap among
layers can improve the robustness of interdependent multiplex systems and
change the critical behavior of the percolation phase transition in a complex
way.
We study both analytically and numerically the robustness of n interdependent networks with partial support-dependence relationship, which reflects real-world networks more realistically. For a starlike network of n Erdős-Rényi (ER) networks, we find that the system undergoes from second-order to first-order phase transition as coupling strength q increases. Moreover, we notice that the region of the first-order transition becomes larger, while the region of the second-order transition becomes smaller as the number of networks n increases. However, for a starlike network of n scale-free (SF) networks, the system undergoes from second-order through hybrid-order to first-order phase transition as q increases. Furthermore, we also observe that the region of the first-order transition remains constant and appears only for q = 1, however, the region of hybrid-order transition gradually becomes larger and the region of the second-order transition becomes smaller as n increases. For a looplike network of n ER networks, we find the giant component p∞ to be independent of the number of networks. Additionally, when the average degree of networks increases, the region of the first-order transition becomes smaller and the region of the second-order transition becomes larger. For the case of n ER networks with partial support-dependence relationship, as average supported degree , n coupled networks become independent and only second-order transition is observed, which is similar to q = 0.
As a typical emerging application of cyber physical system, smart power grid is composed of interdependent power grid and communication/control networks. The latter one contains relay nodes for communication and operation centers to control power grid. Failure in one network might cause failures in the other. Moreover, these failures may occur recursively between the two networks, leading to cascading failures. We propose a k-to-n interdependency model for smart grid. Each relay node and operation center is supported by only one power station, while each power station is monitored and controlled by k operation centers. Each operation center controls n power stations. We show that the system controlling cost is proportional to k. By calculating the fraction of functioning parts (survival ratio) using percolation theory and generating functions, we reveal the nonlinear relation between controlling cost and system robustness, and use graphic solution prove that a threshold exists for the proportion of faulty nodes, beyond which the system collapses. The extensive simulations validate our analysis, determine the percentage of survivals and the critical values for different system parameters. The mathematical and experimental results show that smart grid with higher controlling cost has a sharper transition, and thus is more robust. This is the first paper focusing on improving smart power grid robustness by changing monitoring strategies, from an interdependent complex networks perspective.
Real data show that interdependent networks usually involve inter-similarity.
Intersimilarity means that a pair of interdependent nodes have neighbors in
both networks that are also interdependent (Parshani et al \cite{PAR10B}). For
example, the coupled world wide port network and the global airport network are
intersimilar since many pairs of linked nodes (neighboring cities), by direct
flights and direct shipping lines exist in both networks. Nodes in both
networks in the same city are regarded as interdependent. If two neighboring
nodes in one network depend on neighboring nodes in the another we call these
links common links. The fraction of common links in the system is a measure of
intersimilarity. Previous simulation results suggest that intersimilarity has
considerable effect on reducing the cascading failures, however, a theoretical
understanding on this effect on the cascading process is currently missing.
Here, we map the cascading process with inter-similarity to a percolation of
networks composed of components of common links and non common links. This
transforms the percolation of inter-similar system to a regular percolation on
a series of subnetworks, which can be solved analytically. We apply our
analysis to the case where the network of common links is an
Erd\H{o}s-R\'{e}nyi (ER) network with the average degree K, and the two
networks of non-common links are also ER networks. We show for a fully coupled
pair of ER networks, that for any , although the cascade is reduced
with increasing K, the phase transition is still discontinuous. Our analysis
can be generalized to any kind of interdependent random networks system.
Complex networks appear in almost every aspect of science and technology. Although most results in the field have been obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of ‘networks of networks’ formed by interdependent random networks.
The robustness of a network of networks (NON) under random attack has been studied recently. 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 Erdos-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdos-Rényi networks. For any tree of n fully interdependent Erdos-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 Erdos-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 pc, which causes the fragmentation of the n interdependent networks, and for the minimum average degree k̄min below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdos-Rényi networks under targeted attack, we find the critical coupling strength qc for different n. When q>qc, the attacked system undergoes an abrupt first order type transition. When q≤qc, 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.
NetworkX is a Python language package for exploration and analysis of networks and network algorithms. The core package provides data structures for representing many types of networks, or graphs, including simple graphs, directed graphs, and graphs with parallel edges and self loops. The nodes in NetworkX graphs can be any (hashable) Python object and edges can contain arbitrary data; this flexibility mades NetworkX ideal for representing networks found in many different scientific fields. In addition to the basic data structures many graph algorithms are implemented for calculating network properties and structure measures: shortest paths, betweenness centrality, clustering, and degree distribution and many more. NetworkX can read and write various graph formats for eash exchange with existing data, and provides generators for many classic graphs and popular graph models, such as the Erdoes-Renyi, Small World, and Barabasi-Albert models, are included. The ease-of-use and flexibility of the Python programming language together with connection to the SciPy tools make NetworkX a powerful tool for scientific computations. We discuss some of our recent work studying synchronization of coupled oscillators to demonstrate how NetworkX enables research in the field of computational networks.
Many real-world networks interact with and depend upon other networks. We develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks, each composed of the same number of nodes N. The dependency links connecting nodes from different networks establish a unique one-to-one correspondence between the nodes of one network and the nodes of the other network. We study the dynamics of the cascades of failures in such a network of networks (NON) caused by a random initial attack on one of the networks, after which a fraction p of its nodes survives. We find for the fully interdependent loopless NON that the final state of the NON does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the NON, which generalizes both the giant component of regular percolation of a single network (n=1) and the recently studied case of the mutual giant component of two interdependent networks (n=2). We also find that the mutual giant component does not depend on the topology of the NON and express it in terms of generating functions of the degree distributions of the network. Our results show that, for any n⩾2 there exists a critical p=pc>0 below which the mutual giant component abruptly collapses from a finite nonzero value for p⩾pc to zero for p<pc, as in a first-order phase transition. This behavior holds even for scale-free networks where pc=0 for n=1. We show that, if at least one of the networks in the NON has isolated or singly connected nodes, the NON completely disintegrates for sufficiently large n even if p=1. In contrast, in the absence of such nodes, the NON survives for any n for sufficiently large p. We illustrate this behavior by comparing two exactly solvable examples of NONs composed of Erdős-Rényi (ER) and random regular (RR) networks. We find that the robustness of n coupled RR networks of degree k is dramatically higher compared to the n-coupled ER networks of the same average degree k̅ =k. While for ER NONs there exists a critical minimum average degree k̅ =k̅ min∼lnn below which the system collapses, for RR NONs kmin=2 for any n (i.e., for any k>2, a RR NON is stable for any n with pc<1). This results arises from the critical role played by singly connected nodes which exist in an ER NON and enhance the cascading failures, but do not exist in a RR NON.
Robustness of two coupled networks system has been studied only for
dependency coupling (S. Buldyrev et. al., Nature, 2010) and only for
connectivity coupling (E. A. Leicht and R. M. D'Souza, arxiv:09070894). Here we
study, using a percolation approach, a more realistic coupled networks system
where both interdependent and interconnected links exist. We find a rich and
unusual phase transition phenomena including hybrid transition of mixed first
and second order i.e., discontinuities like a first order transition of the
giant component followed by a continuous decrease to zero like a second order
transition. Moreover, we find unusual discontinuous changes from second order
to first order transition as a function of the dependency coupling between the
two networks.
We investigate the consequence of failures, occurring on the electrical grid, on a telecommunication network. We have focused on the Italian electrical transmission network and the backbone of the internet network for research (GARR). Electrical network has been simulated using the DC power flow method; data traffic on GARR by a model of the TCP/IP basic features. The status of GARR nodes has been related to the power level of the (geographically) neighbouring electrical nodes (if the power level of a node is lower than a threshold, all communication nodes depending on it are switched off). The electrical network has been perturbed by lines removal: the consequent re-dispatching reduces the power level in all nodes. This reduces the number of active GARR nodes and, thus, its Quality of Service (QoS). Averaging over many configurations of perturbed electrical network, we have correlated the degradation of the electrical network with that of the communication network. Results point to a sizeable amplification of the effects of faults on the electrical network on the communication network, also in the case of a moderate coupling between the two networks.
We consider percolation on interdependent locally treelike networks, recently
introduced by Buldyrev et al., Nature 464, 1025 (2010), and demonstrate that
the problem can be simplified conceptually by deleting all references to
cascades of failures. Such cascades do exist, but their explicit treatment just
complicates the theory -- which is a straightforward extension of the usual
epidemic spreading theory on a single network. Our method has the added
benefits that it is directly formulated in terms of an order parameter and its
modular structure can be easily extended to other problems, e.g. to any number
of interdependent networks, or to networks with dependency links.
When an initial failure of nodes occurs in interdependent networks, a cascade of failure between the networks occurs. Earlier studies focused on random initial failures. Here we study the robustness of interdependent networks under targeted attack on high or low degree nodes. We introduce a general technique which maps the targeted-attack problem in interdependent networks to the random-attack problem in a transformed pair of interdependent networks. We find that when the highly connected nodes are protected and have lower probability to fail, in contrast to single scale-free (SF) networks where the percolation threshold pc = 0, coupled SF networks are significantly more vulnerable with pc significantly larger than zero. The result implies that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes that have been found useful to significantly improve robustness of single networks.
Complex systems are very often organized under the form of networks where
nodes and edges are embedded in space. Transportation and mobility networks,
Internet, mobile phone networks, power grids, social and contact networks,
neural networks, are all examples where space is relevant and where topology
alone does not contain all the information. Characterizing and understanding
the structure and the evolution of spatial networks is thus crucial for many
different fields ranging from urbanism to epidemiology. An important
consequence of space on networks is that there is a cost associated to the
length of edges which in turn has dramatic effects on the topological structure
of these networks. We will expose thoroughly the current state of our
understanding of how the spatial constraints affect the structure and
properties of these networks. We will review the most recent empirical
observations and the most important models of spatial networks. We will also
discuss various processes which take place on these spatial networks, such as
phase transitions, random walks, synchronization, navigation, resilience, and
disease spread.
Many complex systems, such as communication networks, display a surprising degree of robustness: while key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. In this paper we demonstrate that error tolerance is not shared by all redundant systems, but it is displayed only by a class of inhomogeneously wired networks, called scale-free networks. We find that scale-free networks, describing a number of systems, such as the World Wide Web, Internet, social networks or a cell, display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected by even unrealistically high failure rates. However, error tolerance comes at a high price: these networks are extremely vulnerable to attacks, i.e. to the selection and removal of a few nodes that play the most important role in assuring the network's connectivity. Comment: 14 pages, 4 figures, Latex
Under the Energy Independence and Security Act (EISA) of 2007, the National Institute of Standards and Technology (NIST) is assigned “primary responsibility to coordinate development of a framework that includes protocols and model standards for information management to achieve interoperability of Smart Grid devices and systems…” [EISA Title XIII, Section 1305]. There is an urgent need to establish these standards. Recognizing the urgency, NIST developed a three-phase plan to accelerate the identification of standards while establishing a robust framework for the longer-term evolution of the standards and establishment of testing and certification procedures. This report is the output of Phase 1. It describes a high-level reference model for the Smart Grid, identifies nearly 80 existing standards that can be used now to support Smart Grid development, identifies 14 high priority gaps, plus cyber security, for which new or revised standards are needed, documents action plans with aggressive timelines by which designated Standards Development Organizations are tasked to fill these gaps, and describes the strategy being pursued to establish standards for ensuring cyber security of the Smart Grid.
In cyber physical system (CPS), computational resources and physical resources are strongly correlated and mutually dependent. Cascading failures occur between coupled networks, cause the system more fragile than single network. Besides widely used metric giant component, we study small cluster (small component) in interdependent networks after cascading failures occur. We first introduce an overview on how small clusters distribute in various single networks. Then we propose a percolation theory based mathematical method to study how small clusters be affected by the interdependence between two coupled networks. We prove that the upper bounds exist for both the fraction and the number of operating small clusters. Without loss of generality, we apply both synthetic network and real network data in simulation to study small clusters under different interdependence models and network topologies. The extensive simulations highlight our findings: except the giant component, considerable proportion of small clusters exists, with the remaining part fragmenting to very tiny pieces or even massive isolated single vertex; no matter how the two networks are tightly coupled, an upper bound exists for the size of small clusters. We also discover that the interdependent small-world networks generally have the highest fractions of operating small clusters. Three attack strategies are compared: Inter Degree Priority Attack, Intra Degree Priority Attack and Random Attack. We observe that the fraction of functioning small clusters keeps stable and is independent from the attack strategies.
We study a system composed of two partially interdependent networks; when nodes in one network fail, they cause dependent nodes in the other network to also fail. In this paper, the percolation of partially interdependent networks under targeted attack is analyzed. We apply a general technique that maps a targeted-attack problem in interdependent networks to a random-attack problem in a transformed pair of interdependent networks. We illustrate our analytical solutions for two examples: (i) the probability for each node to fail is proportional to its degree, and (ii) each node has the same probability to fail in the initial time. We find the following: (i) For any targeted-attack problem, for the case of weak coupling, the system shows a second order phase transition, and for the strong coupling, the system shows a first order phase transition. (ii) For any coupling strength, when the high degree nodes have higher probability to fail, the system becomes more vulnerable. (iii) There exists a critical coupling strength, and when the coupling strength is greater than the critical coupling strength, the system shows a first order transition; otherwise, the system shows a second order transition.
In order to design an efficient communication scheme and examine the efficiency of any networked control architecture in smart grid applications, we need to characterize statistically its information source, namely the power grid itself. Investigating the statistical properties of power grids has the immediate benefit of providing a natural simulation platform, producing a large number of power grid test cases with realistic topologies, with scalable network size, and with realistic electrical parameter settings. The second benefit is that one can start analyzing the performance of decentralized control algorithms over information networks whose topology matches that of the underlying power network and use network scientific approaches to determine analytically if these architectures would scale well. With these motivations, in this paper we study both the topological and electrical characteristics of power grid networks based on a number of synthetic and real-world power systems. The most interesting discoveries include: the power grid is sparsely connected with obvious small-world properties; its nodal degree distribution can be well fitted by a mixture distribution coming from the sum of a truncated geometric random variable and an irregular discrete random variable; the power grid has very distinctive graph spectral density and its algebraic connectivity scales as a power function of the network size; the line impedance has a heavy-tailed distribution, which can be captured quite accurately by a clipped double Pareto lognormal distribution. Based on the discoveries mentioned above, we propose an algorithm that generates random topology power grids featuring the same topology and electrical characteristics found from the real data.
Modern network-like systems are usually coupled in such a way that
failures in one network can affect the entire system. In
infrastructures, biology, sociology, and economy, systems are
interconnected and events taking place in one system can propagate to
any other coupled system. Recent studies on such coupled systems show
that the coupling increases their vulnerability to random failure.
Properties for interdependent networks differ significantly from those
of single-network systems. In this article, these results are reviewed
and the main properties discussed.
Networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links. Their percolation transition is usually of a first order compared to the second-order transition found in classical networks. We analytically analyze the effect of different distributions of dependencies links on the robustness of networks. For a random Erdös-Rényi (ER) network with average degree k that is divided into dependency clusters of size s, the fraction of nodes that belong to the giant component P(∞) is given by P(∞)=p(s-1)[1-exp(-kpP(∞))](s), where 1-p is the initial fraction of removed nodes. Our general result coincides with the known Erdös-Rényi equation for random networks for s=1. For networks with Poissonian distribution of dependency links we find that P(∞) is given by P(∞)=f(k,p)(P(∞))e(([s]-1)[pf(k,p)(P(∞))-1]), where f(k,p)(P(∞))≡1-exp(-kpP(∞)) and [s] is the mean value of the size of dependency clusters. For networks with Gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks.
We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes N{A} and N{B}, (ii) degree distributions of connectivity links P{A}(k) and P{B}(k), (iii) degree distributions of support links P̃{A}(k) and P̃{B}(k), and (iv) random attack removes (1-R{A})N{A} and (1-R{B})N{B} nodes form the networks A and B, respectively. We find the fractions of nodes μ{∞}{A} and μ{∞}{B} which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erdős-Rényi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees ã and b̃ in networks A and B, respectively, μ{∞}{A}=R{A}[1-exp(-ãμ{∞}{B})][1-exp(-aμ{∞}{A})] and μ{∞}{B}=R{B}[1-exp(-b̃μ{∞}{A})][1-exp(-bμ{∞}{B})]. In the limit of ã→∞ and b̃→∞, both networks become independent, and our model becomes equivalent to a random attack on a single Erdős-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
We study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes. We assume that both networks (A and B) have the same number of nodes N connected by the bidirectional dependency links establishing a one-to-one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links; i.e., their degrees coincide. This implies that both networks have the same degree distribution P(k). We call such networks correspondently coupled networks (CCNs). We assume that the nodes in each network are randomly connected. We define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional. We assume that initially a 1-p fraction of nodes are randomly removed because of an attack or failure and find analytically, for an arbitrary P(k), the fraction of nodes μ(p) that belong to the mutual giant component. We find that the system undergoes a percolation transition at a certain fraction p=p(c), which is always smaller than p(c) for randomly coupled networks with the same P(k). We also find that the system undergoes a first-order transition at p(c)>0 if P(k) has a finite second moment. For the case of scale-free networks with 2<λ≤3, the transition becomes a second-order transition. Moreover, if λ<3, we find p(c)=0, as in percolation of a single network. For λ=3 we find an exact analytical expression for p(c)>0. Finally, we find that the robustness of CCN increases with the broadness of their degree distribution.
Current network models assume one type of links to define the relations between the network entities. However, many real networks can only be correctly described using two different types of relations. Connectivity links that enable the nodes to function cooperatively as a network and dependency links that bind the failure of one network element to the failure of other network elements. Here we present an analytical framework for studying the robustness of networks that include both connectivity and dependency links. We show that a synergy exists between the failure of connectivity and dependency links that leads to an iterative process of cascading failures that has a devastating effect on the network stability. We present exact analytical results for the dramatic change in the network behavior when introducing dependency links. For a high density of dependency links, the network disintegrates in a form of a first-order phase transition, whereas for a low density of dependency links, the network disintegrates in a second-order transition. Moreover, opposed to networks containing only connectivity links where a broader degree distribution results in a more robust network, when both types of links are present a broad degree distribution leads to higher vulnerability.
Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.
Minnesota high voltage transmission line and substation data
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Minnesota Dept. of Commerce, "Minnesota high voltage transmission
line and substation data," 2015.
Aurora fiber-optic networks
- Minnesota Fiber-Optic Network
Minnesota Fiber-Optic Network, "Aurora fiber-optic networks," 2015.
NIST framework and roadmap for smart grid interoperability standards, release 1.0
- locke
G. Locke and P. D. Gallagher, "NIST framework and roadmap for
smart grid interoperability standards, release 1.0," National Institute of
Standards and Technology, 2010.
Blackout in the United States and Canada
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U.S.-Canada Power System Outage Task Force, "Final Report on the
August 14, 2003 Blackout in the United States and Canada," 2004.
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J. Gao, D. Li, and S. Havlin, "From a single network to a network of
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