Interdependent Network Model Research Articles

Percolation of conditional dependency clusters based on edge-coupled interdependent networks

Considering the existence of multiple edge dependencies in realistic interdependent networks, we propose a model of edge-coupled interdependent networks with conditional dependency clusters (EINCDCs). In this model, the edges in network A depend on the edges in dependency clusters of size m in network B. If the failure rate of edges within the dependency clusters in network B exceeds the failure tolerance α, the corresponding edges in network A that depend on those clusters in network B will fail accordingly. By adopting the self-consistent probabilities approach, a theoretical analytical framework is established to quantitatively address this model. Specifically, we study the robustness of the system verified with numerical simulations in the effect of the cluster size and failure tolerance under random attacks on systems composed of two networks A and B constructed with Random Regular (RR), Erdös-Rényi (ER) and Scale Free (SF) models. Our results show that both networks A and B undergo a first-order or hybrid phase transition when the dependency cluster size does not exceed 2. However, when the cluster size of dependency clusters exceeds 2, a non-monotonic behavior is observed. In particular, when the failure tolerance is in the range from 0 to 0.5, the robustness of the system weakens with the growing in the number of dependency clusters of size 2. While, this tendency reverses when the failure tolerance is in the range from 0.5 to 1. Moreover, we observe that due to the asymmetric interdependency between the two networks, network B always undergoes first-order phase transition, whereas network A could exhibit different types of phase transitions, which depends on the size of dependency clusters. In addition, the failure tolerance may have opposite effects on the two networks with the growing of dependency cluster sizes. The conclusions of the study may provide useful implications and enrich the understanding in the robustness of edge-coupled interdependent networks.

Robustness of higher-order interdependent networks with reinforced nodes.

In reality, pairwise interactions are no longer sufficient to describe the higher-order interactions between nodes, such as brain networks, social networks, etc., which often contain groups of three or more nodes. Since the failure of one node in a high-order network can lead to the failure of all simplices in which it is located and quickly propagates to the whole system through the interdependencies between networks, multilayered high-order interdependent networks are challenged with high vulnerability risks. To increase the robustness of higher-order networks, in this paper, we proposed a theoretical model of a two-layer partial high-order interdependent network, where a proportion of reinforced nodes are introduced that can function and support their simplices and components, even losing connection with the giant component. We study the order parameter of the proposed model, including the giant component and functional components containing at least one reinforced node, via theoretical analysis and simulations. Rich phase transition phenomena can be observed by varying the density of 2-simplices and the proportion of the network's reinforced nodes. Increasing the density of 2-simplices makes a double transition appear in the network. The proportion of reinforced nodes can alter the type of second transition of the network from discontinuous to continuous or transition-free, which is verified on the double random simplicial complex, double scale-free simplicial complex, and real-world datasets, indicating that reinforced nodes can significantly enhance the robustness of the network and can prevent networks from abrupt collapse. Therefore, the proposed model provides insights for designing robust interdependent infrastructure networks.

Aviation armament system-of-systems modeling and identification method of vulnerable nodes based on interdependent network

Aiming at the problem that it is difficult to build model and identify the vulnerable equipment for aviation armament System-of-Systems (SoS) due to complex equipment interaction relationships and high confrontation, the interdependent network theory is introduced to solve it. Firstly, a two-layer heterogeneous interdependent network model for aviation armament SoS is proposed, which reflects the information interaction, functional dependency and inter-network dependence effectively. Secondly, using the attack cost to describe the confrontation process and taking the comprehensive impact on kill chains as the entry point, the node importance index and the attack cost measurement method are constructed. Thirdly, the identification of vulnerable nodes is transformed into the optimization problem of node combinatorial selection, and the vulnerable node identification method based on tabu search is proposed. Based on vulnerable nodes, a robustness enhancement strategy for aviation armament SoS network is presented. Finally, the above methods are used to an aerial confrontation SoS, and the results verify the rationality and effectiveness of the proposed methods.

Percolation behavior analysis on n-layer edge-coupled interdependent networks

The robustness of interdependent networks has always been studied for years and has attracted wide attention from researchers in different fields. Noting that not all networks are coupled by nodes, networks coupled by edge are also worth further studying. At present, most studies on the edge-coupled interdependent networks are built on double layers, however, researches on multi-layer edge-coupled interdependent networks have hardly been conducted. In this paper, an n-layer edge-coupled interdependent network model was proposed, and the cascading failure propagation mechanism caused by attacking a small number of edges was studied. Based on the self-consistent probability theory that can significantly simplify the mathematical analysis of such systems, the phase transition behavior was theoretically analyzed. The research results help to deepen the understanding of the robustness of multi-layer interdependent networks and provide some valuable advice for the protection and improvement of complex network systems.

Microscopic Intervention Yields Abrupt Transition in Interdependent Ferromagnetic Networks.

The study of interdependent networks has recently experienced a boost with the development of experimentally testable materials that physically realize their novel critical behaviors, calling for systematic studies that go beyond the percolation paradigm. Here we study the critical kinetics and phase transitions of a model of interdependent spatial ferromagnetic networks where dependency couplings between networks are realized by a thermal interaction having a tunable spatial range. We show how the critical phenomena and the phase diagram of this realistic model are highly affected by the range of thermal dissipation and how the latter influences the microscopic kinetics of the model. Furthermore, we show the existence of a new phase where localized microscopic interventions by heating or magnetic fields yield a macroscopic phase transition. Our results unveil rich phenomena and realistic protocols for controlling the macroscopic phases of interdependent materials by means of microscopic interventions.

Vulnerability Analysis of UAV Swarm Network with Emergency Tasks

With the rapid development of Unmanned Aerial Vehicle (UAV) technology, UAV swarms are used for emergency tasks in various scenarios such as area detection, fire rescue, logistics, and transportation. However, for complex scenarios, UAV swarms are prone to environmental interference that damages their equipment or disrupts their communication links, affecting the normal execution of tasks. In this paper, an information–communication interdependent network model is designed for the vulnerability analysis of UAV swarm networks with emergency tasks. Firstly, from the perspective of network functions of a UAV swarm, we introduce the theory of interdependent networks to abstract the relationship between the UAV swarm’s communication network and its information network, where the communication network represents its communication topology and the information network is related to the function of each UAV individual in the UAV swarm. Then, the vulnerability of the UAV swarm is analyzed according to the relationship between network construction costs and network connectivity under environmental interference. Finally, the effectiveness of the vulnerability analysis method is verified through simulation.

Analysis of seepage behaviour in a binary two-layer coupled network

To Existing research on interdependent networks defines network functions either entirely on nodes or entirely on edges, meaning interdependence between nodes and nodes, or interdependence between edges and edges. However, the reality is not characterized solely by interdependence between individual elements' functionalities, meaning it is not entirely a single-element coupled network. There are cases where both nodes and edges depend on each other. Taking into account this realistic scenario, the paper proposes a binary interdependent network model with coupling between nodes and edges (BINNEC), where both nodes and edges are interdependent. In this model, nodes in network A randomly depend on multiple edges in network B, forming edgedependent clusters. Additionally, a failure tolerance parameter, denoted as <i>μ</i>, is set for these edge-dependent clusters. When the failure rate of an edge-dependent cluster exceeds <i>μ</i>, it results in the failure of the nodes in network A that depend on it. Based on the self-balancing probability method, a theoretical analysis framework is established. Through computer simulation verification of BINNEC under three classical network structures, the model's phase transition behavior and critical thresholds in the face of random attacks are analyzed. The results reveal that BINNEC under three network structures is as fragile as a singleelement coupled network, exhibiting a first-order phase transition behavior. As the size of edge-dependent clusters <i>m</i> increases, network robustness enhances. Moreover, with a constant size of edge-dependent clusters, a larger tolerance for node failure <i>μ</i> leads to stronger network robustness. Finally, the research reveals that under the same conditions of <i>m</i> and <i>μ</i>, when the tolerance for node failure <i>μ</i> is insufficient to withstand the failure of a single edge, the degree distribution widens, and network robustness weakens. However, when the tolerance for node failure is sufficient to withstand the failure of at least one edge, network robustness actually strengthens as the degree distribution increases. These findings provide a theoretical basis for such binary coupled models and also offer guidance for the secure design of real-world networks.

Enhancement of cooperation induced by information-payoff evolution on two-layer complex networks

Human behavior will be influenced by information. Thus, we propose a model of interdependent scale-free networks to depict this situation. One layer represents the information network with cooperative and defective signals, and the other one indicates the game network with node-type and link-type players. Within this model, the nodes of the information network and the game network correspond one by one, but the types of signals and strategies, as well as the edges of the two networks, need not to be strictly consistent. Furthermore, signals in the information network are constantly updated in the unique way, which can influence dissemination of behavior in the game network. Herein, we investigate the impact of signal update probability and the proportion of link-type players on the evolution of cooperation. Lots of simulation results indicate that such simple setup can greatly facilitate the evolution of cooperation.

Tolerance Analysis of Cyber-Manufacturing Systems to Cascading Failures

In practical cyber-manufacturing systems (CMS), the node component is the forwarder of information and the provider of services. This dual role makes the whole system have the typical physical-services interaction characteristic, making CMS more vulnerable to cascading failures than general manufacturing systems. In this work, in order to reasonably characterize the cascading process of CMS, we first develop an interdependent network model for CMS from a physical-service networking perspective. On this basis, a realistic cascading failure model for CMS is designed with full consideration of the routing-oriented load distribution characteristics of the physical network and selective load distribution characteristics of the service network. Through extensive experiments, the soundness of the proposed model has been verified and some meaningful findings have been obtained: (1) attacks on the physical network are more likely to trigger cascading failures and may cause more damage; (2) interdependency failures are the main cause of performance degradation in the service network during cascading failures; and (3) isolation failures are the main cause of performance degradation in the physical network during cascading failures. The obtained results can certainly help users to design a more reliable CMS against cascading failures.

Percolation behavior analysis of weighted edge-coupled interdependent networks

The robustness of interdependent networks has received extensive attention from the inter-disciplinary fields. In reality, many interdependent systems have interdependence relationships not only between nodes but also between edges. Different from previous studies focusing on edge-coupled interdependent networks without consideration on coupling strength, in this paper, a weighted edge-coupled interdependent network model with coupling strength is proposed. Then, the cascading failure propagation mechanism caused by attacking a small percentage of edges is studied. Based on the self-consistent probability theory, theoretical framework is established to provide the mathematical analysis of robustness of weighted edge-coupled interdependent networks under edge attacks. The current results can help to understand the vulnerability of interdependent networks and provide useful suggestions for the protection and optimization of complex networked systems.

Percolation phase transition in the heterogeneous multi-coupled interdependent network

In previous studies, complex system models have often been set up in an oversimplified manner and the failure mechanism for nodes affected by each other is not correspond to reality. Accordingly, we develop a new model of heterogeneous multi-coupled interdependent networks and analyze the change in the giant component of the network after the attack in conjunction with the load redistribution mechanism. We first analyze the cascading failure process of the network analytically by recursive methods, estimate the critical phase transition threshold and the size of the giant component of the remaining network after network percolation, and find that the threshold of the WS sub-network is positively related to the average degree, and the BA sub-network is related to the power exponent λ of the distribution. When λ is less than or equal to 2, there is always a giant component in the network. After λ is greater than 3.4788⋯, there is no giant component. When λ is between 2 to 3.4788⋯, we found that the threshold of the sub-network is more influenced by the proportion of attacks (p1 and p2), and when p1 is larger, the sub-networks interact with each other, resulting in a smaller threshold for the BA sub-network, while the threshold of the WS sub-network is smaller in other cases. Finally, the cross-sectional comparison illustrates that as λ increases to 3.4788⋯, the robustness of the BA sub-network increases, contrary to the findings for single-layer networks. Overall, the heterogeneous multi-coupled interdependent network is more robust, and the giant component is greatly affected by the low robust sub-network, but its existence is related to the stock of nodes.

Analysis of Failure Propagation in Cyber-Physical Power Systems Based on an Epidemic Model

From the perspective of propagation dynamics in complex networks, failure propagation in cyber-physical power systems is analogous to the spread of diseases; subsequently, the cyber nodes and power nodes are regarded as individuals in each of their groups. In this study, a two-layer interdependent network model of the cyber-physical power system is proposed, where each subnetwork adopts the Susceptible-Infected-Susceptible (SIS) epidemic-spreading model. On this basis, we construct a failure cooperation propagation model of cyber-physical power systems. Furthermore, we introduce the node protection mechanism to ensure the normal operation of key nodes. The generated scale-free cyber network and IEEE118-bus power system are used for simulation to analyze the influence of the coupling effect between them on the final failure scale.

Modeling the Vulnerability and Resilience of Interdependent Transportation Networks under Multiple Disruptions

Because infrastructure systems are highly interconnected, it is crucial to analyze their vulnerability and resilience with the consideration of interdependencies. This paper constructed a bus–metro interdependent network model based on the passenger transfer relationship and used deep learning to identify the network topology attributes. The vulnerability process of the interdependent network to different disruptions under structural and functional perspective was studied. On this basis, this paper adopted a resilience assessment framework and mainly focused on modeling and resilience analysis of interdependent networks’ recovery processes. The optimal and instructive recovery strategy was determined, and it is shown that the increase of the coupling distance cannot alleviate the vulnerability of the interdependent network effectively; after the tolerance coefficient reaches the threshold, the effect on the vulnerability of the dependent network is weakened; and a betweenness-based strategy (BBS) works best in the preferential recovery of key nodes.

Control–Flight Conflict Interdependent Network Based Controllers’ Workload Prediction Evaluation Study

As the core link of the air traffic control system, the evaluation and management of controllers’ workload are of great significance for the rapid development of China’s civil aviation industry and aviation safety. To address the problems of strong subjectivity and lag in the current controller evaluation methods, we propose to construct a flight conflict network and control network based on inter-aircraft flight conflict, controller control relationship and control transfer relationship by using the interdependent network theory. Based on the urgency of inter-aircraft conflict and the control difficulty of controllers, we set the side rights and construct an interdependent network model. Based on the constructed network model, the controller workload is evaluated by selecting the interdependent network index characteristics. Finally, the experimental analysis is carried out by program simulation and control data of Takasaki Airport. The results demonstrate that the method is able to evaluate the controller’s workload. At the same time, it can quickly and accurately identify the key control nodes and provide assistance for controllers to allocate their efforts reasonably. Our proposed controller workload evaluation method enables quantitative analysis and predicts the future workload of controllers based on the potential conflict relationships between air posture and aircraft. The method has strong timeliness and objectivity. At the same time, the constructed interdependent network model can realize the identification of key aircraft in the control area and reflect the impact of key aircraft on the whole network, which can help controllers to perform better control work.

Security Situation Awareness Assessment of Heterogeneous Cyber-Physical Systems in Multiple Load Mode

As the Internet of Things technology develops rapidly, cyber-physical systems (CPSs) have provided critical technologies in the industrial field. A smart grid is a typical application of the CPS, which is formed by the coupling of the power network and communication network. In general, the two networks are heterogeneous. Considering the characteristics of the power network, the ordinary percolation network model is not suitable for the network carrying physical flow. Therefore, we propose an interdependent network model with a load. That is, nodes in the physical network have loaded. Then, we research the security of CPS under the multi-load mode. The rule for node failure is different from the percolation model. When a node’s load exceeds its capacity, the node fails. Through the theoretical analysis, we obtained the iterative equation of the percolation threshold and analyzed it through simulation experiments. We adopt both linear and nonlinear capacity models. It is found that appropriately increasing the average degree of ER network can improve its ability to resist attacks. In contrast, the power exponent of the SF network has less influence on the critical value of network percolation. In addition, we found that increasing the capacity parameter in the linear capacity model can improve the robustness of the network, where the variation of the ER-ER network is more evident than that of the SF-SF network. Increasing the capacity parameter in the nonlinear capacity model can significantly increase the network’s ability to resist attacks. However, due to the increase in economical cost, we can improve the network’s reliability by increasing the node capacity while controlling the cost.

Study on Cascading Failure and Elasticity of UAV Swarm Communication Network

As an important new force in information warfare, the UAV swarm has attracted more and more attention from militarists and scholars of various countries. At present, large-scale and low-cost UAV swarms have been gradually applied to reconnaissance operations. The invulnerability and elasticity of its communication network are the key factors that determine whether the swarm can successfully complete the reconnaissance mission. The research object of this study is the UAV swarm communication network, which performs reconnaissance missions. The swarm network is extracted as a three-layer interdependent network model. We apply scale-free network theory and the new connection probability equation to establish three subnetworks. Then, a three-layer interdependent network is constructed by using the interlayer connection rules of the nodes with the largest degree. The classical M-L model is extended to an interdependent network. An interlayer load redistribution rule is proposed to simulate the cascading failure process of swarm networks. Then, we attack network nodes randomly and observe the invulnerability and elasticity of this network. The simulation results show that the performance of the interdependent network is better than that of the single-layer network due to the effect of the interdependent links. By analyzing the experimental results, some feasible suggestions are put forward to improve the invulnerability and elasticity of reconnaissance UAV swarm communication network.

Study on Invulnerability of Material Emergency Transportation System Based on Three-Layer Interdependent Network

The material transportation capacity under emergency conditions is an important guarantee for the country to deal with war, epidemic outbreak, and other crisis situations. Under emergency conditions, some nodes of the material transportation system may fail to work normally, which may lead to the collapse of the whole system. Based on analyzing the characteristics of the material emergency transportation system, this article builds a three-layer interdependent network model and uses the improved M-L model to describe the failure propagation mechanism of node damage in the three-layer network. Then, the network model is attacked randomly, and the relationship between invulnerability of the three-layer network and the three main indexes of network flow, average degree, and probability of interdependence are studied. Afterwards, the propagation of cascading failure among three subnetworks in the interdependent network is analyzed and compared. This article provides a theoretical basis for building an efficient and robust material emergency transportation system.

Importance Assessment of Communication Equipment in Cyber-Physical Coupled Distribution Networks Based on Dynamic Node Failure Mechanism

Electrical power communication networks undertake complicated power services. They interact with physical power grids all the time, which will result in a high risk to power grids when their electric power communication network is attacked. Therefore, to improve the reliability of power systems, it is of great significance to identify the key links of communication networks. Existing research studies focus more on the risk of power communication services undertaken by communication links, which ignores the frequent interaction between power grids and communications networks. In this paper, from a multi-layer network perspective, an assessment method of communication equipment is proposed in view of the operation characteristics of nodes in different systems as well as the dynamic fault propagation between homogeneous communication nodes. Firstly, we build a model of cyber-physical power interdependent network. Then, for communication nodes, a dynamic model is adopted for cascading failure among the nodes based on the load-capacity model. Afterward, we define an index to evaluate the node importance considering characteristics of the operation of both communication networks and power grids. Finally, a local power grid is taken as an example for verification by comparing the effects of different indexes. The simulation results demonstrate the validity of the evaluation method proposed.

Multiple phase transitions in ER edge-coupled interdependent networks

Considering the real-world scenarios that there are interactions between edges in different networks and each network has different topological structure and size, we introduce a model of interdependent networks with arbitrary edge-coupling strength, in which q A and q B are used to represent the edge-coupling strength of network A and network B respectively. A mathematical framework using generating functions is developed based on self-consistent probabilities approach, which is verified by computer simulations. In particular, we carry out this mathematical framework on the Erdös–Rényi edge-coupled interdependent networks to calculate the values of phase transition thresholds and the critical coupling strengths which distinguish different types of transitions. Moreover, as contrast to the corresponding node-coupled interdependent networks, we find that for edge-coupled interdependent networks the critical coupling strengths are smaller, and the critical thresholds as well, which means the robustness of partially edge-coupled interdependent networks is better than that of partially node-coupled interdependent networks. Furthermore, we find that network A will have hybrid percolation behaviors as long as the coupling strength q A belongs to a certain range, and the range does not affected by average degree of network A. Our findings may fill the gap of understanding the robustness of edge-coupled interdependent networks with arbitrary coupling strength, and have significant meaning for network security design and optimization.

Weighted interdependent network disintegration strategy based on Q-learning

The problem of network disintegration is one of the core topics in the field of network science. Currently, most of the existing research is based on homogeneous and single-layer networks of nodes. However, the various components of complex systems in the real world are often interdependent, and the cost of attacking different units is diverse, causing the traditional network disintegration method to lack good applicability. This paper establishes a weighted interdependent network (WIN) model, and based on this, a WIN disintegration strategy based on Q-learning is proposed. First, the network nodes are divided into multiple node sets according to the dependencies between the nodes, and the state and action space of Q-learning are determined. Next, the disintegration cost constraints and Q-learning parameters are defined to perform iterative learning. Then, the optimal network disintegration strategy is calculated according to the iterative Q-table. The results show that when the cost sensitivity factor (p) is fixed, DSQ can maintain good results in disintegrating different types of networks under different cost constraints, while the baseline methods have difficulty guaranteeing the disintegration effect in the face of different types of networks. Furthermore, we perform a sensitivity analysis on the p value and find that the effect of most of the baseline methods worsens as the p value increases, while DSQ maintains a good effect.