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A Realistic Model for Failure Propagation in Interdependent Cyber-Physical Systems
Abstract
Modern cyber-physical systems are becoming increasingly interdependent. Such interdependencies create new vulnerabilities and make these systems more susceptible to failures. In particular, failures can easily spread across these systems, 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, called 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 use HINT to train machine learning methods based on novel features for predicting the effects of the cascading failures. Additionally, by using feature selection, we identify the most important features that characterize critical nodes. We compare HINT with two previously proposed models both on synthetic and real network topologies. Experimental results show that existing models oversimplify the failure evolution and network functionality requirements. In addition, the machine learning approaches accurately forecast the effects of the failure propagation in the considered scenarios. Finally, we show that by strengthening few critical nodes identified by the proposed features, we can greatly improve the network robustness.
... • Dependencies between different kinds of infrastructures are diverse and complex [26,29]. CFs have been extensively studied in single networks, such as electric networks [19,34]. ...
... • Extensive experiments demonstrate that comparing the state-of-the-art baselines, 3 can achieve a 31.94% improvement in terms of AUC, 18.03% in terms of precision, 29.17% in terms of recall, and 22.73% in terms of F1-score. For CF volume, it can achieve a 28.52% improvement in RMSE. ...
... Parandehgheibi et al predicted CF on an electric network-communication network and proposed a two-stage control strategy to mitigate cascading faults. Sturaro et al focused on the interdependence between grid-communication networks and proposed the HINT model for CF prediction [29]. ...
Cascading failures (CF) entail component breakdowns spreading through infrastructure networks, causing system-wide collapse. Predicting CFs is of great importance for infrastructure stability and urban function. Despite extensive research on CFs in single networks such as electricity and road networks, interdependencies among diverse infrastructures remain overlooked, and capturing intra-infrastructure CF dynamics amid complex evolutions poses challenges. To address these gaps, we introduce the \textbf{I}ntegrated \textbf{I}nterdependent \textbf{I}nfrastructure CF model (), designed to capture CF dynamics both within and across infrastructures. employs a dual GAE with global pooling for intra-infrastructure dynamics and a heterogeneous graph for inter-infrastructure interactions. An initial node enhancement pre-training strategy mitigates GCN-induced over-smoothing. Experiments demonstrate achieves a 31.94\% in terms of AUC, 18.03\% in terms of Precision, 29.17\% in terms of Recall, 22.73\% in terms of F1-score boost in predicting infrastructure failures, and a 28.52\% reduction in terms of RMSE for cascade volume forecasts compared to leading models. It accurately pinpoints phase transitions in interconnected and singular networks, rectifying biases in models tailored for singular networks. Access the code at https://github.com/tsinghua-fib-lab/Icube.
... Sturaro et al. [23] proposed a heterogeneous interdependent networks (HINT) approach for generating a dataset to train a ML model. The authors took into account the heterogeneity of the smart grid. ...
... We present a comparison of the RTAP model with the AWV [29], HINT [23], and GBFS [28] models. Our dataset comprises 12,000 samples, including 9,000 normal and 3,000 attacked instances. ...
... These failures affect generators' voltages and powers, as shown in Figure 6. Table 4 shows the attack detection accuracy and testing time results of the RTAP model compared with the AWV [29], HINT [23], and GBFS [28] models. We use the XG-Boost algorithm as the core of the RTAP model for comparison. ...
One main challenge of smart grid systems is the cascading failures caused by cyber-attacks, which can affect the power and communication networks. Many testbeds have been proposed to model the impact of cyber-attacks on these two networks; however, many lack a realistic propagation model or simulate real-time behavior. In this paper, we develop a novel real-time testbed that models both the power and communication networks to analyze cyber-attacks’ impacts on smart grid systems. Our proposed testbed can model various cyber-attacks on both networks and analyze the propagation of failure within the system. To create a realistic model of smart grid systems, we utilize real-time simulators and implement a failure propagation model. Using this testbed, we propose a prediction model to detect and predict failures after cyber-attacks have impacted the system. This model can detect cyber-attacks in the early stages of failure propagation and predict the state of each power and communication component following the propagation. We prove this model is realistic using the failure propagation factor and validate its effectiveness by employing an IEEE 14-bus test case, showcasing its high accuracy in detecting various types of attacks.
... They did not study how failures propagate through the communication network and how the failure of cyber components affects the power system. This causes these models to assume unrealistic failure propagation conditions, making them unable to identify all failed components [9], [10]. ...
... Some work focused on forecasting and predicting the condition of a system following failure propagation [10], [19]. For example, Pi et al. [19] developed a model that used Bayes networks to forecast the state of a smart grid system after failure propagation. ...
... Sturaro et al. [10] introduced a heterogeneous model-based approach (HINT) to generate a dataset for training ML models. This work accounted for the heterogeneity of the smart grid system, utilizing polynomial regression, decision trees, and neural networks to predict the number of failed nodes following failure propagation. ...
Cascading failures resulting from cyberattacks are one of the main concerns in smart grid systems. The use of machine learning (ML) algorithms has become more relevant in identifying and forecasting such cascading failures. In this article, we develop a real-time early stage mechanism (RESP) to predict cascading failures due to cyberattacks in smart grid systems using supervised ML algorithms. We use a realistic methodology to create a dataset to train the algorithms and predict the state of all components of the system after failure propagation. We utilize the extreme gradient boosting (XGBoost) algorithm and consider the features of both the power and communication networks to improve the failure prediction accuracy. We use the real-time digital simulator (RTDS) to simulate the power system and make the system more applicable. We evaluate the mechanism's effectiveness using the IEEE 14-bus system, which results in the XGBoost algorithm achieving a 96.25% prediction accuracy rate in random attacks. We show that RESP can accurately predict the state of a power system in the early stages of failure propagation using real-time data. Furthermore, we show that RESP can identify the initial failure locations, which can aid in further protection plans and decisions.
... Considering the complexity of these phenomenon and challenges in controlling them, a large body of work has been formed in understanding cascading failures and mitigating their effects in power grids. Particularly, cascading failures in power grids have been studies using power physics-based approaches [11]- [13], simulation-based techniques [14]- [16], probabilistic models [17]- [19], and graph-based modeling and analyses [20], [21]. Despite all the studies and developed techniques, due to the large size and geographical scale, complex and at-distance underlying interactions among the components, and new attributes and dynamics of modern power grids (for instance, deployment of stochastic renewable resources), cascading failures have remained, although not very common, but a complex and costly threat to these systems. ...
... Researchers have investigated the general form of the distribution of cascading failures and have observed its heavy-tailed power-law nature [12], [19], [99]- [103]. Similarly, the study of interdependent power and communication systems has also contributed to the understanding of cascade size distributions in interdependent systems [20], [104]. Characterizing the cascade size distribution unique to each state of the system and the initial disturbances is one of the focuses in this category of problems. ...
... The majority of the work in characterizing the probability distribution of cascade sizes primarily relies on data-driven and statistical methods, while the direct application of ML to this problem remains limited. From the available works, the work in [20], applies linear and polynomial regression, decision tree, and deep neural network to predict the total number of components that failed after the initial faults while considering the topological features such as degree and betweenness among the nodes. In [101], the interaction among the lines of the grid is learned by the expectation minimization algorithm to characterize the probability of small, medium, and large cascade sizes for the number of line failures. ...
Cascading failures pose a significant threat to power grids and have garnered considerable research interest in the power system domain. The inherent uncertainty and severe impact associated with cascading failures have raised concerns, prompting the development of various techniques to study these complex phenomena. In recent years, advancements in monitoring technologies and the availability of large volumes of data from power systems, coupled with the emergence of intelligent algorithms, have made machine learning (ML) techniques increasingly attractive for addressing cascading failure problems. This survey provides a comprehensive overview of ML-based techniques for analyzing cascading failures in power systems. The survey categorizes these techniques based on the evolutionary phases of the cascade process in power systems, as well as studies focusing on cascade resiliency before the occurrence of cascades and problems related to cascades after their termination. By organizing and presenting these works into relevant categories, this survey aims to offer insights and a systematic understanding the role of ML in mitigating cascading failures in power systems.
... In the CPS domain, an approach was designed to evaluate the consequences of attack propagation, without including the steps of anomaly detection [12]. Also, a power-grid specific failure propagation model named Heterogeneous Interdependent NeTwork (HINT), was developed to analyze three functionally separated subsystems [13]: generation, transmission and distribution. Since these subsystems are different from the main subsystems of a naval CPS -navigation, propulsion, safety, power, and utilities -the HINT approach cannot be adapted in a simple manner either. ...
... This effect was combined with changes of and in ì that represent an abnormal decrease in fuel and oil tanks levels. Table V highlights the features of the proposed method for anomaly propagation analysis, compared to two other previously described approaches [13], [12]. ...
... Further work will consist on studying higher complexity CPS to evaluate the interest of weighted dependencies between variables, completing the quality assessment with an enlarged set of quality dimensions, completed by the knowledge and wisdom concepts of the quality model. HINT model [13] Evaluation of anomaly propagation consequences [12] ...
As any other infrastructure relying on cyber-physical systems (CPS), naval CPS are highly interconnected and collect considerable data streams, on which depend multiple command and navigation decisions. Being a data-driven decision system requiring optimized supervisory control on a permanent basis, it is critical to examine the CPS vulnerability to anomalies and their propagation. This paper presents an approach to detect CPS anomalies and estimate their propagation applying a quality assessed graph, which represents the CPS physical and digital subsystems, combined with system variables dependencies and a set of data and information quality measures vectors. Following the identification of variables dependencies and high-risk nodes in the CPS, data and information quality measures reveal how system variables are modified when an anomaly is detected, also indicating its propagation path. Taking as reference the normal state of a naval propulsion management system, four anomalies in the form of cyber-attacks-port scan, programmable logical controller stop, and man in the middle to change the motor speed and operation of a tank valve-were produced. Three anomalies were properly detected and their propagation path identified. These results suggest the feasibility of anomaly detection and estimation of propagation estimation in CPS, applying data and information quality analysis to a system graph.
... Additional results on application of the proposed methods are presented in [17]. In [18], authors use a graph-theoretical presentation of a power grid system that is coupled with a communication network. To model the propagation of failures in a cyber-physical power system, supervised machine learning techniques including polynomial and decision trees regression models are trained with synthetic failure data. ...
... This work complements the study of Sturaro et al. in [18] by utilizing a domain-specific cyber-physical simulation tool to provide the resolution required for capturing behaviors of discrete-time cyber components in sensing, decision making, and control roles. The superiority of our method is that our simulation tool models the real-world physics of the underlying physical system, i.e., the interconnected electric delivery system and corresponding power components. ...
Interdependence is an intrinsic feature of cyber–physical systems. Cyber and physical components are tightly integrated with each other, and hence, a trivial impairment in a part of the system may affect several components, leading to a sequence of failures that collapses the entire system. In this paper, we seek to identify the interdependencies among the components of a cyber–physical system using correlation metrics as well as a heuristic causation analysis method. We also demonstrate applicability of neural networks for prediction of imminent failures given the current system state. The proposed prediction tool can help system operators to perform timely preventive actions and mitigate the consequences of accidental failures and malicious attacks. As a case study, we have analyzed two smart grid test cases based on IEEE power bus systems, namely, IEEE–14 and IEEE–57.
... Specifically, every computing equipment obtains real-time status information of each physical equipment from the data acquisition equipment connected with it and realizes the coordinated control and dynamic optimization of the power network through the two-way data interaction of the communication network. Accordingly, power network provides the cyber layer with the electricity needed for normal operation (Wang et al., 2016a;Huang et al., 2017;Tu et al., 2019;Sturaro et al., 2020). Such interdependencies within the CPPS is conducive to the expansion of the network scale and intelligent management of the power network. ...
... Accordingly, the regional control for large CPPSs can be simulated. Sturaro et al. (2020) considered two types of communication nodes, including control centers and relays. The types of the power node are considered when selecting the power source for the communication nodes in their model. ...
The Cyber-Physical Power System (CPPS) is one of the most critical infrastructure systems in a country because a stable and secure power supply is a key foundation for national and social development. In recent years, resilience has become a major topic in preventing and mitigating the risks caused by large-scale blackouts of CPPSs. Accordingly, the concept and significance of CPPS resilience are at first explained from the engineering perspective in this study. Then, a review of representative quantitative assessment measures of CPPS resilience applied in the existing literature is provided. On the basis of these assessment measures, the optimization methods of CPPS resilience are reviewed from three perspectives, which are mainly focused on the current research, namely, optimizing the recovery sequence of components, identifying and protecting critical nodes, and enhancing the coupling patterns between physical and cyber networks. The recent advances in modeling methods for cascading failures within the CPPS, which is the theoretical foundation for the resilience assessment and optimization research of CPPSs, are also presented. Lastly, the challenges and future research directions for resilience optimizing of CPPSs are discussed.
... However, because of the changing nature of power grids, it is necessary to model and assess the resilience of smart grids from the cyber-physical perspective. Several studies have modelled CPPSs to study the effects of and defence against cyberattacks in smart city applications [11], the effects of cascading failures [12], and system reliability [13]. The multiagent nature of CPPSs increases the risk of cyberattacks in open network environments [14]. ...
As power grids develop, their structure becomes more complex, multi‐dimensional, and digitalised—hence, it is referred to as cyber‐physical infrastructure. The sensitivity of grids to extreme cyber and physical events is becoming a hot research topic due to the increasing rate of such events and their catastrophic consequences. To produce accurate and comprehensive measures, modelling and assessing the resilience of power systems must include both the physical and cyber domains. However, resilience quantification models that include both domains have not received sufficient attention. A novel resilience model and quantification framework are proposed. The model is based on a resilience trapezoid that depicts the different phases of the cyber and physical domains during severe natural or anthropogenic events. A resilience index is also proposed to measure the resilience levels of local nodes and entire systems, including various factors that contribute to the modelled degradation states. Severe weather conditions were modelled to examine the impact of this category of events on the proposed resilience model.
... Based on stochastic currents, reference [20] adopts power node degree, power node betweenness and aggregation coefficients to identify vulnerable nodes of the grid, and use entropy and ideal solutions to obtain grid critical nodes. In [21], a heterogeneous interdependent network model is proposed to study the evolution of cascading faults for the coupling relationship between grid and communication networks, while fault evolution feature selection is used to obtain the critical nodes. Reference [22] proposes seven centrality metrics, which are combined with the correlation coefficient of integrated entropy and inter-attribute correlation of Spearman. ...
The rapid development of communication services in the power distribution network poses challenges for existing wireless communications, and the deployment of a fiber optic network is costly and difficult. The emerging 5G technology has been piloted in power distribution networks, though the cost-effectiveness of its large-scale deployment remains unclear. This paper proposes an economic evaluation method for 5G planning in power distribution networks, considering the coupling relationship between power distribution and communication networks and the identification of important network nodes. First, the objective function to solve the planned number of 5G base stations is established. This is solved using the adaptive particle swarm algorithm and K-means algorithm. Second, the coupling relationship between the distribution and communication networks is discussed and quantified. The node importance of the coupling network is analyzed to identify the important nodes, and micro base stations or optical fibers are added to improve the reliability of the distribution network at the communication level. Finally, an economic evaluation index of 5G planning of the distribution network is established. The paper compares the economic solutions of 5G and 4G communications in city and town scenarios using the IEEE 123-node network as an example, and concludes that the economics of 5G are better than those of 4G.
... In order to solve these problems, the authors propose the so-called HINT model, another failure propagation formulation for interdependent power and communication networks. The research work reported in [11] expands upon this model, again focusing on interconnected and interdependent networks. The heterogeneity of the different networks is considered in failure propagation, unlike in many generic models where all nodes are considered to be the same. ...
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.
... However, while the fusion of power systems with communication technology has brought a wealth of benefits, it has also introduced potential threats to power systems [4][5][6]. For example, changes in the network topology of the information system have the propensity to cause delays or even obstructions in the transmission of information, thus disrupting real-time monitoring of the power system [7]. ...
In cyber–physical power systems (CPPSs), system collapse can occur as a result of a failure in a particular component. In this paper, an approach is presented to build the load-capacity model of CPPSs using the concept of electrical betweenness and information entropy, which takes into account real-time node loads and the allocation of power and information flows within CPPSs. By introducing an innovative load redistribution strategy and comparing it with conventional load distribution strategies, the superior effectiveness of the proposed strategy in minimizing system failures and averting system collapses has been demonstrated. The controllability of the system after cascading failures under different coupling strategies and capacity parameters is investigated through the analysis of different information network topologies and network parameters. It was observed that CPPSs constructed using small-world networks, which couple high-degree nodes from the information network to high-betweenness nodes from the power grid, exhibit improved resilience. Furthermore, increasing the capacity parameter of the power network yields more favorable results compared to increasing the capacity parameter of the information network. In addition, our research results are validated using the IEEE 39-node system and the Chinese 132-node system.
... For example, a power system can be regarded as a composition of a power network and a Supervisory Control And Data Acquisition (SCADA) network. Later studies that extended or generalized this model include [3], [8], [9]. ...
In interdependent systems, such as electric power systems, entities or components mutually depend on each other. Due to these interdependencies, a small number of initial failures can propagate throughout the system, resulting in catastrophic system failures. This paper addresses the problem of finding the set of entities whose failures will have the worst effects on the system. To this end, a two-phase algorithm is developed. In the first phase, the tight bound on failure propagation steps is computed using a Boolean Satisfiablility (SAT) solver. In the second phase, the problem is formulated as an Integer Linear Programming (ILP) problem using the obtained step bound and solved with an ILP solver. Experimental results show that the algorithm scales to large problem instances and outperforms a single-phase algorithm that uses a loose step bound.
... It makes the research stay in the theoretic stage rather than put it into practice. In [14], the future control center is designed. It intends to build and operate perfect communication networks for optimizing the information networks. ...
The deep integration of power grids and communication networks is the basis for realizing the complete observability and controllability of power grids. The communication node or link is always built according to the physical nodes. This step is alternatively known as “designing with the same power tower”. However, the communication networks do not form a “one-to-one correspondence” relationship with the power physical network. The existing theory cannot be applied to guide the practical power grid planning. In this paper, a local evolution model of a communication network based on the physical power grid topology is proposed in terms of reconnection probabilities. Firstly, the construction and upgrading of information nodes and links are modeled by the reconnection probabilities. Then, the power flow entropy is employed to identify whether the power cyber-physical system (CPS) is at the self-organized state, indicating the high probability of cascading failures. In addition, on the basis of the cascading failure propagation model of the partially dependent power CPS, operation reliabilities of the power CPS are compared with different reconnection probabilities using the cumulative probability of load loss as the reliable index. In the end, a practical provincial power grid is analyzed as an example. It is shown that the ability of the power CPS to resist cascading failures can be improved by the local growth evolution model of the communication networks. The ability is greater when the probability of reconnection is p = 0.06. By updating or constructing new links, the change in power flow entropy can be effectively reduced.
... Related studies about vulnerability evaluation of CPPS are primarily based on two kinds of modeling, cosimulation [9][10][11][12] and complex networks [13][14][15][16][17][18]. e advantages of cosimulation lie in clear physical meaning and accurate calculation results. ...
With the deep coupling between the cyber side and the physical side of power systems, the failure of any link of both sides may lead to power outages, so it is necessary to analyze their vulnerability and vulnerable links for targeted improvement of systems. By dynamically attacking the coupled network nodes, this paper proposes a multilevel model and node-to-edge cyber-physical power system and the corresponding indexes system to analyze the vulnerability of the coupled power grid and its key components. The results showed that in the order of the indexes proposed in this paper, attacking surviving power nodes and cyber nodes results in a network crash rate of 25.0% and 66.7% faster than that in the order of “betweenness” and that attacking surviving cyber nodes results in a network crash rate of 89.4% faster than that in the order of “degree.” In terms of attacking power nodes, the index proposed in this paper has the same rate as “degree.” Therefore, the proposed model can better describe the vulnerability of the power grid to withstand attacks.
... Instead, the inner control loops are robust against the cyber-attacks as it works with a tracking objective for each state. The inner control loop is full against cyber-attacks only when the outer control loop is unattacked [155]. Via employing (5), the control inputs can either be manipulated in the communication links or the controller through an external entity. ...
This paper presents an inclusive review of the cyber-physical (CP) attacks, vulnerabilities, mitigation approaches on the power electronics and the security challenges for the smart grid applications. With the rapid evolution of the physical systems in the power electronics applications for interfacing renewable energy sources that incorporate with cyber frameworks, the cyber threats have a critical impact on the smart grid performance. Due to the existence of electronic devices in the smart grid applications, which are interconnected through communication networks, these networks may be subjected to severe cyber-attacks by hackers. If this occurs, the digital controllers can be physically isolated from the control loop. Therefore, the cyber-physical systems (CPSs) in the power electronic systems employed in the smart grid need special treatment and security. In this paper, an overview of the power electronics systems security on the networked smart grid from the CP perception, as well as then emphases on prominent CP attack patterns with substantial influence on the power electronics components operation along with analogous defense solutions. Furthermore, appraisal of the CPS threats attacks mitigation approaches, and encounters along the smart grid applications are discussed. Finally, the paper concludes with upcoming trends and challenges in CP security in the smart grid applications.
Inter-network degree correlation (IDC) has been widely proven to be important to the performance of cyber-physical systems (CPSs) facing cascading failures by comparing the limited extreme cases, such as maximally-positive, maximally-negative, and completely random interdependent networks. In this paper, we develop a novel system model for providing a more realistic and detailed analysis on how IDC observed in an actual CPS relates to cascades occurring on it. Unlike existing models that often assume homogeneous network operating characteristics, and only focus on symmetric interdependency between coupled networks while ignoring asymmetric interdependency, our proposed model captures heterogeneous flow characteristics of a physical network and a cyber network composing a specific CPS and considers two types of their symmetric and asymmetric interdependency to simulate cascading failure processes in practical situations. Furthermore, not limited to the extreme IDCs, CPSs with tunable IDC are generated and investigated systematically by introducing null models. Experimental results reveal that the relationship between IDC and cascading robustness of a CPS is dependent on its interdependency mechanism. In the case of symmetric interdependency, increasing IDC can significantly make the system more robust; while in the case of asymmetric one, the robustness is nearly unchanged. The study highlights the necessity of considering realistic interdependency mechanisms when designing inter-network structure optimization schemes against cascading failures in CPSs.
The cyber-physical power system (CPPS) stands as one of a nation's most critical infrastructures, as an unwavering and secure power supply forms the cornerstone of national and societal development. Recently, the concept of resilience has become a trending topic in preventing and mitigating the risks caused by large-scale CPPS blackouts. This article introduces the concept of resilience in complex engineering systems, detailing quantitative assessment metrics used to evaluate CPPS resilience. These metrics underpin the capacity of CPPSs to recover from disruptions. The article reviews current research focused on enhancing CPPS resilience, covering three main aspects: 1) optimizing component recovery sequences, 2) identifying critical nodes, and 3) optimizing physical-cyber coupling patterns. Recent advances in modeling methodologies for CPPS resilience against cascading failures are also discussed. Finally, the article delves into the challenges and future direction for CPPSs resilience enhancement and modeling approaches.
In this paper, we study the dynamic of cascading failure iterating between physical power grid and communication network, and further propose the attack strategy to locate the critical nodes which can seriously damage the cyber-physical power systems. First, the coupling connection between physical power grid and communication network is modeled, along with the power flow of physical power grid and the data flow of communication network. Then, the zone-based coupling model is proposed to detail the failure cascading in systems and its impact on coupled networks. Third, based on electrical and structural characteristic, the criticality of nodes is defined to devise attack strategy to locate the vulnerability of systems. The simulation has analyzed the impact of failure nodes on coupled networks and verified the effectiveness of proposed attack strategy.
With the continuous development of information technology, a spontaneous interdependent network has formed within the air traffic control network. Due to the internal interdependence, any small, failed node may trigger a cascade failure of the entire system. The purpose of this study is to investigate the resilience of air traffic control networks. Based on air traffic management regulations, a new cascading failure model for air traffic control networks is proposed, which is based on the theory of interdependent networks. The model establishes a dual-layer dependency relationship between the control coordination network and the air route facility network, including control dependency and service dependency. Through experiments, targeted measures are proposed to improve the safety and reliability of air traffic control. This model introduces parameters such as control cost and node control capability, and reflects the resilience of the air traffic control network, based on the final number of failed nodes after the steady-state of the cascade failure, the network’s cascade failure rate, and the system’s load failure threshold. Simulation results show that enhancing the control capability and increasing the number of control positions can improve the control cost of the air traffic control network. The higher the control cost, the better the resilience of the air traffic control network. Improving the control capability of control nodes has a greater impact on the resilience of the air traffic control network, compared to increasing the number of control nodes. The degree attack on route nodes has a greater impact on the cascade failure of the air traffic control network, compared to random attacks and facility node degree attacks. The cascade failure model proposed in this paper provides a new method for guiding the air traffic control network to resist cascade failure attacks and enhance its resilience.
We consider the cascading failure process in interdependent power-communication networks, where the power grid provides the required energy for the communication nodes, and the communication network facilitates the monitoring and controlling the power networks. The proposed system model considers the flow dynamics in both networks and the failure rollover to study the cascade process in the system and capture the possible beneficial and adverse effects of interdependency between the networks. We suggest weak and strong interdependencies models that determine how and to what extent the loss of controllability after failures impacts the power network and a congestion-aware load balancing scheme that exploits the system state to decrease the density of severe cascades. The results of the cascading failure processes on data from two power networks are provided and discussed in terms of the average unserved load in the power network and the number of failed nodes in the communication layer in different scenarios. We find that increasing the coupling is beneficial in most cases; however, considering the robustness of each network and the nature of the interdependencies between the two networks, over-coupling can decrease the system's robustness against failure cascading in certain scenarios.
In this paper, the problem of attack detection for actuator attacks and power balance on cyber physical power systems (CPPSs) is investigated. An economic optimization strategy based on observer data is proposed to enable optimal power balance of the attacked system. System states and cyber attacks are estimated by a sliding mode observer satisfying the H∞ performance index. Then, the system power is redistributed according to the optimal economic dispatch. Finally, several simulation cases are presented to illustrate the effectiveness of attack detection and economic optimization strategy.
This article proposes an approach based on the stochastic process and approximate dynamic behavior for modeling cascading failures of the cyberphysical power system (CPPS) considering component multistate failures. Our approach focuses on modeling the coupling effects of interdependencies of the integrated power and communication networks, including control and power supply dependencies as well as the effects of the degradation of one network performance on the other. Nine failure states of each component and the performance/capacity degradation in different failure states are incorporated into the modeling approach. The state transitions between nine failure states are modeled as a discrete time Markov process whose transition probabilities are time-varying. In addition, the article proposes a new robustness measure for multistate CPPS, which integrates the information of the local and global topology, the damage state of components, as well as the performance degradation of the CPPS. The system that couples the IEEE 118-bus model with a small-world communication network is used as a testbed to demonstrate the feasibility and effectiveness of the proposed modeling approach, and the comparison with existing robustness measures shows the superiority of the proposed measure.
Cyber-Physical System (CPS) is an integration of physical components like actuators, sensors and various types of equipment with the Internet possessing computational ability for efficient communication. A Heterogeneous Independent Network (HINT) is a realistic model that is used for the analysis of inter-dependability between the power grid and communications network. In the traditional Deep [Formula: see text]-Learning method, action needs to be stored in the [Formula: see text] table for the prediction. In real case studies, many state and action values affect the performance of the model. Existing Deep [Formula: see text]-Network (DQN) model generates all possible actions for the [Formula: see text]-values and this involves the generation of excessive information that causes the model to overfit. In this research, the Neural Network is applied to estimate the state–action in the DQN and to store the particular state–action value instead of storing all the state–action values as followed in the traditional method. The HINT model provides realistic failure propagation in the network and its state–action value overfits the existing DQN method due to the presence of more information. The proposed DQN with reinforcement learning stores selected state–action values in the [Formula: see text] tables and eliminates irrelevant information that helps to increase the accuracy and reduce the computational time. The DQN with reinforcement learning is applied to adaptively learn the system to select the optimal action in a continuous interaction with a stochastic environment. The proposed DQN model involves the application of reward function to store state–action value with higher probability based on prediction and eliminates other state–action values. Features such as intra-degree, inter-betweenness, substation-betweenness, relay-betweenness and feature vector are extracted and given as input to the DQN to characterize the critical nodes. The proposed DQN method is evaluated on the HINT network and synthetic network to analyze its efficiency in fault detection. The result shows that the HINT network has a lower prediction error compared to the existing Deep Neural Network (DNN) method. The proposed DQN and LSTM models have accuracies of 98% and 93% in fault prediction, respectively.
The extensive deployment of information and communication technologies significantly changes the characteristics of power grids. In this paper, we propose an integrated modeling framework for studying cascading failure and assessing the robustness of cyber-coupled power grids. By taking the perspective of cyber–physical systems, this framework depicts the electrical characteristics of the physical network, the realistic monitoring, control, and protection functions provided by the coupled cyber network, and integration with decentralized functions. It also includes a flow chart that generates a sequence of failure events in cyber-coupled power grids for simulating cascading failure. Based on the framework, a series of specific models can be constructed by incorporating concrete considerations. We demonstrate the robustness assessment of cyber-coupled power grids by one specific case study based on the modeling framework with appropriate assumptions made. Simulation results on four power test cases show that the cyber network can help effectively mitigate cascading failure and thus enhance the robustness of the grid. Moreover, the faults from the cyber layer can intensify the failure cascade and lead to a catastrophic power outage.
The smart grid is a typical cyber physical system (CPS). Communication networks and power networks have the structural coupling and functional coupling characteristics, which may induce cyber-physical coupling failures. To reveal the vulnerability characteristics of CPS, this paper proposes the CPS interdependence model and analyzes the vulnerability of CPS. Firstly, considering structural coupling of the power and communication system, structural characteristics and networking constraints of fiber optic communication networks, the communication network topology generation model is proposed. Then, based on the generated communication network, the CPS interdependence model is built, which describes the structural and functional coupling between power and communication network. Finally, based on the fault chain theory, the CPS cascading failure analysis model is proposed to assess the vulnerability. In the case study, IEEE 118-bus system is taken as an example to generate communication network, and the validity of the communication network topology model is verified by comparing structural features with actual networks. The influence of the proportion of composite lines (optical fiber composite overhead ground wire, OPGW), and information failure on vulnerability of CPS are analyzed.
The failure propagation of a networked system is profoundly affected by the operation and dynamic behavior of its constituent systems. In this paper, we propose a novel network-based model to investigate the cascading failure of cyber-coupled power systems. Unlike existing models that assume oversimplified network functions, operational characteristics and failure evolution, our proposed model considers the practical differences between a communication network and a power network in terms of network structure, physical operation and dynamic behavior. A practical trip mechanism is derived according to the instantaneous data transmission overload and the stochastic power flow overload. The effects of communication network connection strength and coupling patterns on the robustness of interdependent systems are also studied. Our study shows that the thermal memory effect of power lines aggravates the failure propagation, and that the robustness of the interdependent system is strongly related to the coupled communication network’s tolerance. Furthermore, coupling patterns based on network similarity have better inhibition effects on cascading failure under random attacks, and node destructiveness is a suitable coupling similarity index that can be used to improve the robustness of the system. However, the strategy is inapplicable when the system is under intentional attacks.
L'émergence de la 4ème révolution industrielle a fortement démocratisé l'utilisation de systèmes cyber-physiques (Cyber Physical System (CPS)) dans le secteur maritime. À bord, ils se caractérisent par divers composants interdépendants qui assurent le contrôle d'opérations physiques critiques à partir de commandes numériques. Une anomalie visant l'un de ces systèmes peut profiter de cette caractéristique pour se propager dans l'ensemble du navire, et engendrer des conséquences irréversibles. Premièrement, nous avons considéré la problématique des dépendances à l'échelle globale du navire pour caractériser l'importance de celles associées aux CPS. Ensuite, un modèle innovant de graphe 3-couches (numérique, physique, et variables système) a été formulé pour fournir une analyse structurelle du CPS. Diverses métriques de détection d'anomalies, basées sur l'analyse de la qualité, y sont intégrées pour amorcer les processus d'évaluation de la propagation dans un CPS maritime. Cette solution a été éprouvée sur deux CPS maritimes responsables de fonctions critiques: la propulsion et la distribution de l'eau. De nombreuses perspectives émergent de ces travaux pour fournir une réponse globale à la problématique d'évaluation de la propagation d'anomalies dans un navire.
With the deepening deployment of information and communication technologies (ICT), the cyber network is playing an increasingly important role in determining the performance of a power system. In this paper, we assess the robustness of cyber-physical power systems and examine various impact factors on the system’s robustness. A model that integrates the operating characteristics of the physical network and the wide-area protection functions provided by the cyber network is proposed. Based on the model, cascading failure propagation processes in a cyber-physical power system triggered by initial failures are simulated. Two statistical metrics, the power outage risk and the cumulative power outage size distribution, are then used to quantify the robustness by processing numerous cascading failure simulation results. We conduct case studies on IEEE 57 Bus, IEEE 118 Bus, IEEE 145 Bus, and UIUC 300 Bus with the proposed method. Simulation results demonstrate the necessity to consider cyber coupling, the importance of developing advanced wide-area protection algorithms, and the threat posed by cyberattacks compromising the robustness of cyber-coupled power systems.
This paper deals with the problem of managing the risks of complex systems under targeted attacks. It is usually solved by using Defender–Attacker models or similar ones. However, such models do not consider the influence of the defending system structure on the expected attack outcome. Our goal was to study how the structure of an abstract system affects its integral risk. To achieve this, we considered a situation where the Defender knows the structure of the expected attack and can arrange the elements to achieve a minimum of integral risk. In this paper, we consider a particular case of a simple chain attack structure. We generalized the concept of a local risk function to account for structural effects and found an ordering criterion that ensures the optimal placement of the defending system’s elements inside a given simple chain structure. The obtained result is the first step to formulate the principles of optimally placing system elements within an arbitrarily complex network. Knowledge of these principles, in turn, will allow solving the problems of optimal allocation of resources to minimize the risks of a complex system, considering its structure.
Modern infrastructures are always spatially embedded and coupled with auxiliary information systems to form heterogeneous cyber-physical systems (CPSs). In this paper, considering the spatial information and overlap of components, a heterogeneous cyber-physical system model with weak dependency is proposed, in which the two networks overlap geographically and have heterogeneous models. Physical edges are divided into controllable or uncontrollable according to the connectivity between their dependent cyber nodes and the control center, corresponding to global and local load transfer strategies, respectively. Combined with the coordinated DoS (deny of service) attack on cyber nodes, the system vulnerability under physical edge-removal and edge-overload scenarios is studied. It can be concluded that although the two networks are weakly dependent without direct failure-induced dependency, a fragmented cyber system can still aggravate the failure of the physical system. In the edge-removal scenario, deliberately attacking high-load edges produces better results, while imposing the extra load on low-load edges performs better in the edge-overload scenario. In addition, blocking specific cyber nodes that make the attacked edges or their adjacent edges become uncontrollable can effectively improve the attack effect. This model might yield insight into modeling and protecting spatial cyber-physical systems.
It is found larger inter-network assortativity can mitigate(aggravate) cascading failures under random(targeted) attacks in cyber–physical systems(CPSs) according to the limited extreme cases, such as the max–min, max–max, and complete random degree-based interdependent networks. However, it is unclear how the assortative interdependence influences the robustness of real infrastructure systems, especially the cyber–physical power systems(CPPSs). We develop a new model called POD based on an optimized load shedding policy to simulate the Power-loss failures, Out-of-control failures and Data-blocking failures in cascade process. By extending the RAndom Interacting Network(RAIN) model, CPPSs with different ‘one-to-one’ assortative interdependence can be generated and studied systematically. The simulation results show that the relationship between assortative interdependence and the robustness of systems is much more complicated than expected. Fragmentation and compatibility are introduced to explain these results, and we find slighter fragmentation and better compatibility make the CPPSs more robust against cascading failures. The interplay between fragmentation and compatibility plays an important role in estimating the influence of assortative interdependence on the CPPS robustness.
The deployment of relays between Internet of Things (IoT) end devices and gateways can improve link quality. In cellular-based IoT, relays have the potential to reduce base station overload. The energy expended in single-hop long-range communication can be reduced if relays listen to transmissions of end devices and forward these observations to gateways. However, incorporating relays into IoT networks faces some challenges. IoT end devices are designed primarily for uplink communication of small-sized observations toward the network; hence, opportunistically using end devices as relays needs a redesign of both the medium access control (MAC) layer protocol of such end devices and possible addition of new communication interfaces. Additionally, the wake-up time of IoT end devices needs to be synchronized with that of the relays. For cellular-based IoT, the possibility of using infrastructure relays exists, and noncellular IoT networks can leverage the presence of mobile devices for relaying, for example, in remote healthcare. However, the latter presents problems of incentivizing relay participation and managing the mobility of relays. Furthermore, although relays can increase the lifetime of IoT networks, deploying relays implies the need for additional batteries to power them. This can erode the energy efficiency gain that relays offer. Therefore, designing relay-assisted IoT networks that provide acceptable trade-offs is key, and this goes beyond adding an extra transmit RF chain to a relay-enabled IoT end device. There has been increasing research interest in IoT relaying, as demonstrated in the available literature. Works that consider these issues are surveyed in this paper to provide insight into the state of the art, provide design insights for network designers and motivate future research directions.
The increasing penetration of wind power may have a profound effect on the cascading dynamics of power delivery systems. In this study, we consider the impacts of packet traffic congestion, power overloading and network interaction on the failure evolution, and investigate the impacts of wind uncertainty and penetration level on the vulnerability of the cyber–physical power system to cascading failure. The proposed model takes into account the modularity of the coupled system, and incorporates both uncertainty analysis and stochastic approach into the line trip mechanism. The discrete packet traffic model is adopted to describe the influence of communication delay on the cascading failure of the power grid. Moreover, we investigate the effect of community structure on the robustness of the modular coupled system. Results indicate that increased wind uncertainty can lead to the occurrence of cascading events and higher wind power penetration increases the vulnerability of the coupled system to cascading failure. In addition, the robustness of the modular coupled system can be enhanced by dissolving the community structure of the communication network, and the timely and efficient data transmission in the communication network can make the cyber-coupled power system more robust.
Protecting the critical nodes of a Cyber-Physical Power System (CPPS) is an effective strategy for mitigating the risk of incurring large-scale blackouts. A Gene Importance based Evolutionary Algorithm (GIEA) is proposed to identify a set of critical k nodes by maximizing the total load loss received by end-users. GIEA adopts an importance-based evolutionary strategy to improve the algorithm’s convergence and accuracy, in which the initial node importance metrics are assessed based on dynamic power flows and topology information. Both performance contribution (PC) and coupling failure impact (CFI) are considered in our importance evaluation framework. The impacts of different types of communication nodes on power networks are integrated into the proposed cascading failure model and CFI assessment. Based on the coupling and interdependence information, the strong coupling node pairs are identified to reduce the dimension of the decision vector to improve the computational efficiency of GIEA. The effectiveness and superiority of the proposed methods are illustrated through an example of a coupling CPPS consisting of the IEEE 30-bus model and a communication network with the small-world structure.
A cyber physical power system model based on the weighted tensor product is proposed in this paper to describe the interlayer and intralayer relationships of entities in both systems. The different functional attributes of nodes and the directivity of the energy flow in the system are respectively described as the heterogeneity of nodes and the directionality of edges. During failure propagation, the dynamic topology method is applied to convert the protection strategies and load redistribution measures into system connectivity rebuilding. The simulation results of fault propagation in the cyber physical power system under different initial failure ratios show that, compared with the undirected model, the directional interactions in the proposed approach can significantly improve the system survival rate during cascading failure. In addition, the physical system shows a lower vulnerability due to the protection provided by spare edges and power sources.
This article investigates optimal recovery strategy of components for maximizing the resilience of the cyber–physical power system (CPPS), where a component represents a unique node or branch, such as generating stations and communication transmission lines. The proposed optimization model is built as a multimode resource-constrained project scheduling problem to incorporate system resilience, cascading failures of the CPPS, the diversity of recovery resources, execution modes of recovery activities, precedence of damaged components, as well as the availability, cost, and time of recovery resources. The failure propagation mechanisms are characterized by a cascading failure model, which is further embedded in the optimization model to quantify the system real-time performance during the recovery process, and determine whether repaired components can be reconnected to the system. The system resilience is quantified using a proposed time-dependent annual composite resilience metric based on a compound Poisson process. The proposed optimization model is solved using a modified simulated annealing algorithm. The system that couples the IEEE 30-bus model and a small-world communication network is used as a testbed to demonstrate the feasibility and effectiveness of the proposed modeling approach. Comparisons with existing optimization models in the literature show the superiority of the proposed model.
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.
When multiple networks are interconnected because of mutual service interdependence, propagation of phenomena across the networks is likely to occur. Depending on the type of networks and phenomenon, the propagation may be a desired effect, such as the spread of information or consensus in a social network, or an unwanted one, such as the propagation of a virus or a cascade of failures in a communication or service network. In this paper, we propose a general analytic model that captures multiple types of dependency and of interaction among nodes of interdependent networks, that may cause the propagation of phenomena. The above model is used to evaluate the effects of different diffusion models in a wide range of network topologies, including different models of random graphs and real networks. We propose a new centrality metric and compare it to more traditional approaches to assess the impact of individual network nodes in the propagation. We propose guidelines to design networks in which the diffusion is either a desired phenomenon or an unwanted one, and consequently must be fostered or prevented, respectively. We performed extensive simulations to extend our study to large networks and to show the benefits of the proposed design solutions.
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.
Classification and regression trees are machine‐learning methods for constructing prediction models from data. The models are obtained by recursively partitioning the data space and fitting a simple prediction model within each partition. As a result, the partitioning can be represented graphically as a decision tree. Classification trees are designed for dependent variables that take a finite number of unordered values, with prediction error measured in terms of misclassification cost. Regression trees are for dependent variables that take continuous or ordered discrete values, with prediction error typically measured by the squared difference between the observed and predicted values. This article gives an introduction to the subject by reviewing some widely available algorithms and comparing their capabilities, strengths, and weakness in two examples. © 2011 John Wiley & Sons, Inc. WIREs Data Mining Knowl Discov 2011 1 14‐23 DOI: 10.1002/widm.8
This article is categorized under: Technologies > Classification
Technologies > Machine Learning
Technologies > Prediction
Algorithmic Development > Statistics
To answer the questions of how information about the physical world is sensed, in what form is information remembered, and how does information retained in memory influence recognition and behavior, a theory is developed for a hypothetical nervous system called a perceptron. The theory serves as a bridge between biophysics and psychology. It is possible to predict learning curves from neurological variables and vice versa. The quantitative statistical approach is fruitful in the understanding of the organization of cognitive systems. 18 references.
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.
Systems as diverse as genetic networks or the world wide web are best
described as networks with complex topology. A common property of many large
networks is that the vertex connectivities follow a scale-free power-law
distribution. This feature is found to be a consequence of the two generic
mechanisms that networks expand continuously by the addition of new vertices,
and new vertices attach preferentially to already well connected sites. A model
based on these two ingredients reproduces the observed stationary scale-free
distributions, indicating that the development of large networks is governed by
robust self-organizing phenomena that go beyond the particulars of the
individual systems.
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
We propose a new method for estimation in linear models. The ‘lasso’ minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree‐based models are briefly described.
We study the influence of degree correlations or network mixing in interdependent security. We model the interdependence in security among agents using a dependence graph and employ a population game model to capture the interaction among many agents when they are strategic and have various security measures they can choose to defend themselves. The overall network security is measured by what we call the average risk exposure (ARE) from neighbors, which is proportional to the total (expected) number of attacks in the network. We first show that there exists a unique pure-strategy Nash equilibrium of a population game. Then, we prove that as the agents with larger degrees in the dependence graph see higher risks than those with smaller degrees, the overall network security deteriorates in that the ARE experienced by agents increases and there are more attacks in the network. Finally, using this finding, we demonstrate that the effects of network mixing on ARE depend on the (cost) effectiveness of security measures available to agents; if the security measures are not effective, increasing assortativity of dependence graph results in higher ARE. On the other hand, if the security measures are effective at fending off the damages and losses from attacks, increasing assortativity reduces the ARE experienced by agents.
We study cascading failures in a system comprising interdependent networks/systems, in which nodes rely on other nodes both in the same system and in other systems to perform their function. The (inter-)dependence among nodes is modeled using a dependence graph, where the degree vector of a node determines the number of other nodes it can potentially cause to fail in each system through aforementioned dependency. In particular, we examine the impact of the variability and dependence properties of node degrees on the probability of cascading failures. We show that larger variability in node degrees hampers widespread failures in the system, starting with random failures. Similarly, positive correlations in node degrees make it harder to set off an epidemic of failures, thereby rendering the system more robust against random failures.
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.
In recent years, deep neural networks (including recurrent ones) have won
numerous contests in pattern recognition and machine learning. This historical
survey compactly summarises relevant work, much of it from the previous
millennium. Shallow and deep learners are distinguished by the depth of their
credit assignment paths, which are chains of possibly learnable, causal links
between actions and effects. I review deep supervised learning (also
recapitulating the history of backpropagation), unsupervised learning,
reinforcement learning & evolutionary computation, and indirect search for
short programs encoding deep and large networks.
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.
We describe a new learning procedure, back-propagation, for networks of neurone-like units. The procedure repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector. As a result of the weight adjustments, internal 'hidden' units which are not part of the input or output come to represent important features of the task domain, and the regularities in the task are captured by the interactions of these units. The ability to create useful new features distinguishes back-propagation from earlier, simpler methods such as the perceptron-convergence procedure.
We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together. The elastic net is particularly useful when the number of predictors (p) is much bigger than the number of observations (n). By contrast, the lasso is not a very satisfactory variable selection method in the p≫n case. An algorithm called LARS-EN is proposed for computing elastic net regularization paths efficiently, much like algorithm LARS does for the lasso.
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.
DNA micro-arrays now permit scientists to screen thousands of genes simultaneously and determine whether those genes are active, hyperactive or silent in normal or cancerous tissue. Because these new micro-array devices generate bewildering amounts of raw data, new analytical methods must be developed to sort out whether cancer tissues have distinctive signatures of gene expression over normal tissues or other types of cancer tissues.
In this paper, we address the problem of selection of a small subset of genes from broad patterns of gene expression data, recorded on DNA micro-arrays. Using available training examples from cancer and normal patients, we build a classifier suitable for genetic diagnosis, as well as drug discovery. Previous attempts to address this problem select genes with correlation techniques. We propose a new method of gene selection utilizing Support Vector Machine methods based on Recursive Feature Elimination (RFE). We demonstrate experimentally that the genes selected by our techniques yield better classification performance and are biologically relevant to cancer.
In contrast with the baseline method, our method eliminates gene redundancy automatically and yields better and more compact gene subsets. In patients with leukemia our method discovered 2 genes that yield zero leave-one-out error, while 64 genes are necessary for the baseline method to get the best result (one leave-one-out error). In the colon cancer database, using only 4 genes our method is 98% accurate, while the baseline method is only 86% accurate.
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.
NIST framework and roadmap for smart grid interoperability standards, release 1.0
- locke