Interdependent Network Model Research Articles

Robustness and recovery mechanism under the interaction of dependent networks

With the development of society and the progress of science and technology, the process of urban infrastructure construction is accelerating, various infrastructure networks are constantly improving, and the links between different infrastructure networks are getting closer. Compared with a single network, this kind of interdependent network is more complex, and the research results of the existing single network are difficult to explain the nature and phenomenon of this kind of network. This article mainly introduces the research on the robustness and recovery mechanism of interdependent networks. From the perspective of a complex network, this paper combines the interdependence between the networks in the actual system and the node load and builds an interdependent network model. On the basis of the load capacity model, an interdependent network error model is established. And through matlab simulation experiments, the fault propagation characteristics of dependent networks under three conditions and the reliability attack methods of dependent networks are studied. The experimental results in this article show that dependent networks show exceptional vulnerability under deliberate attack functions, while dependent networks show good robustness under random attack modes. In addition, increasing the network node tolerance coefficient can improve the robustness of the interdependent network. When the tolerance is increased from 1 to 10, the robustness of the dependent network is increased by 18%.

Robustness assessment of Hetero-functional graph theory based model of interdependent urban utility networks

Increasing urban population imposes a substantial and growing burden on the supporting infrastructure, such as electricity, water, heating, natural gas, road transportation. This paper presents a Hetero-functional graph theory (HFGT) based modeling framework for these integrated infrastructures followed by an analysis of network robustness. The supporting infrastructures along with the infrastructure repair facilities are considered. In contrast to conventional graph representations, a weighted HFGT model is used to capture the system processes and mutual dependencies among resources. To assess robustness of interdependent networks, impacts of complete/partial and random/targeted attacks are quantified. Specifically, various attack scenarios are simulated and the vulnerability of the network is evaluated. Additionally, several robustness metrics are used to provide a comprehensive evaluation of system robustness. The proposed weighted HFGT modeling and robustness assessment approach is tested using a synthetic interdependent network, comprising of an electrical power system, a water network, a district heating network, a natural gas system and a road transportation network. Results demonstrate that system robustness can be enhanced via securing system information and mitigating attack strength.

Third party interventions mitigate conflicts on interdependent networks

Interventions from third parties, such as governmental agencies or organizations, play an important role in mitigating conflicts in modern human societies. The goal thereby is to pacify disputants, although this is subject to failure on account of self-interest from all involved. To study how relationships between disputants and third parties evolve, we propose an interdependent network model, where one layer is occupied by disputants and the other layer is occupied by third parties. Disputants play a prisoner’s dilemma game, where defection is the dominant strategy, whereas third parties play a snowdrift game, where cooperation and defection coexist more commonly. Moreover, third parties have the ability to mediate a conflict on the other layer by enforcing a snowdrift game onto disputants, for which they can receive a fee. We show that third party interventions improve the evolution of cooperation between disputants, and also, that the improvement of cooperation in turn promotes interventions. Nonetheless, non-intervention does not go extinct, which enables defectors to survive, thus creating a feedback loop between the two networks layers. The evolutionary dynamics is characterized by fascinating spatial pattern formation, which we explore by means of Monte Carlo simulations and via replicator equations.

Research on Robustness of Emergency Transportation System of Materials Based on the Three-layer Interdependent Network

The ability to transport materials under emergency conditions is one of the important capabilities of the country to respond to crises. In order to study the ability of the emergency material transportation system to resist external damage, this paper builds a three-layer interdependent network model based on capacity-load under the analysis of the basic characteristics of the system, and conducts random attacks on the network, studies the relationship between overall robustness and transportation flow of this three-layer network . In addition, the robustness of the three sub-networks is studied, which provides a theoretical basis for constructing a reasonable and robust emergency material transportation system.

Interdependent transport via percolation backbones in spatial networks

The functionality of nodes in a network is often described by the structural feature of belonging to the giant component. However, when dealing with problems like transport, a more appropriate functionality criterion is for a node to belong to the network’s backbone, where the flow of information and of other physical quantities (such as current) occurs. Here we study percolation in a model of interdependent resistor networks and show the effect of spatiality on their coupled functioning. We do this on a realistic model of spatial networks, featuring a Poisson distribution of link-lengths. We find that interdependent resistor networks are significantly more vulnerable than their percolation-based counterparts, featuring first-order phase transitions at link-lengths where the mutual giant component still emerges continuously. We explain this apparent contradiction by tracing the origin of the increased vulnerability of interdependent transport to the crucial role played by the dangling ends. Moreover, we interpret these differences by considering an heterogeneous k-core percolation process which enables to define a one-parameter family of functionality criteria whose constraints become more and more stringent. Our results highlight the importance that different definitions of nodes functionality have on the collective properties of coupled processes, and provide better understanding of the problem of interdependent transport in many real-world networks.

Memory-based prisoner's dilemma game with history optimal strategy learning promotes cooperation on interdependent networks

Cooperation is regarded as the architect of the evolution process. The research on the influence of memory mechanism on human cooperative behavior has been a hot issue in recent years. However, most previous researchers usually concentrated their efforts on the single-layer networks and the impact of memory length on cooperative behavior, but paid less attention to the interdependent networks and other factors affecting the decision-making. In this paper, we presented a two-layer interdependent Holme-Kim network model with scale-free, high clustering and interconnection characteristics and a history optimal strategy learning mechanism fully considering the historical strategy, payoffs information and memory length, and combined them to discuss the emergence and maintenance of cooperation behavior. Through modeling simulation, we studied the parameter r for network construction, memory length M and defection to temptation T. Game participants, equipped with the same memory length, played Prisoner's dilemma games with their neighbors located on home or adjacent layer (if any), and selected their following strategies for the next game round through the history optimal strategy learning mechanism. The results show that density of cooperation is inversely related to parameter r and positively related to memory length M. The newly proposed network model can improve network reciprocity and the history optimal strategy learning mechanism can find out the strategy that can bring the maximum revenue in a certain period of time. The smaller r is, the larger M is, the more conductive to cooperation.

Modelling and impact analysis of interdependent characteristics on cascading overload failure of syncretic railway networks.

To study the performance and mutual influence of a syncretic railway network (SRN) that comprises high-speed railway, regional railway, and urban rail transit under the condition of traffic overload during peak hours, we discuss the interdependent characteristics on cascading overload failure of SRNs under the cooperative organization from the perspective of an interdependent network. However, most existing research on cascading failure in interdependent network ignores the inconsistency between the physical structure and transportation organization of the subnetwork in an actual network, in addition to the restrictions on the load redistribution strategy of stations and sections in the load-capacity model of the interdependent network; especially, the influence of transfer behavior on the load redistribution inter subnetwork. In this study, we investigate the robustness of an interdependent SRN under overload and risk propagation. We propose a partially interdependent network model of a multimode rail transit, develop a novel cascading overload failure model with tunable parameters of load redistribution inter subnetwork, and analyze interdependent characteristics, cascade failure process, and robustness of an SRN under multiscene conditions, i.e., different attack and load distribution strategies, via simulations. A case study of an SRN in the metropolitan area of Chengdu, China is presented; the results indicate that, when the reserve coefficient of the metro subnetwork is 0.4 and the overload coefficient of the regional railway subnetwork is greater than 1.2, the station reserve capacity and overload capacity of the SRN is appropriately improved. When passenger load increases to a certain range, the reserve and overload capacities of stations in the regional railway subnetwork do not considerably contribute to robustness. Thus, a surplus load distribution strategy is recommended to improve robustness. The results of this paper have considerable significance for the planning, structural optimization, and operation safety of SRNs.

Effects of mutual traffic redistribution on robustness of interdependent networks to cascading failures under fluctuant load

Failure in real interdependent networks can be induced by temporal and fluctuant overload on nodes. Moreover, failure of nodes will lead to loss of flow to some extent due to the decline of system delivery ability. In this paper, we propose a cascading failure model of interdependent networks based on mutual traffic redistribution under fluctuant load. In the model, the flow loss that is correlated with network existing resource is considered by defining a traffic loss parameter. The interdependence between coupled layers is realized by mutual traffic redistribution. Different types of artificial interdependent networks and a Bus-Metro network example are investigated. The results show larger node tolerance and larger traffic loss parameter can lead to higher network robustness. Moreover, cascading time synchronization of different layers induced by mutual traffic redistribution is observed. Although the interdependence usually makes the entire systems fragile, the optimal mutual flow redistribution rules that are beneficial to the whole network robustness can be obtained by our model. The model and results in this paper can provide some references on robust traffic flow assignment and network topology design for real interdependent systems.

Optimal control of aging in complex networks

Many complex systems experience damage accumulation, which leads to aging, manifest as an increasing probability of system collapse with time. This naturally raises the question of how to maximize health and longevity in an aging system at minimal cost of maintenance and intervention. Here, we pose this question in the context of a simple interdependent network model of aging in complex systems and show that it exhibits cascading failures. We then use both optimal control theory and reinforcement learning alongside a combination of analysis and simulation to determine optimal maintenance protocols. These protocols may motivate the rational design of strategies for promoting longevity in aging complex systems with potential applications in therapeutic schedules and engineered system maintenance.

Control-aircraft state interdependent network model and characteristic analysis

Control-aircraft state interdependent network model and characteristic analysis

Cooperation on Interdependent Networks by Means of Migration and Stochastic Imitation.

Evolutionary game theory in the realm of network science appeals to a lot of research communities, as it constitutes a popular theoretical framework for studying the evolution of cooperation in social dilemmas. Recent research has shown that cooperation is markedly more resistant in interdependent networks, where traditional network reciprocity can be further enhanced due to various forms of interdependence between different network layers. However, the role of mobility in interdependent networks is yet to gain its well-deserved attention. Here we consider an interdependent network model, where individuals in each layer follow different evolutionary games, and where each player is considered as a mobile agent that can move locally inside its own layer to improve its fitness. Probabilistically, we also consider an imitation possibility from a neighbor on the other layer. We show that, by considering migration and stochastic imitation, further fascinating gateways to cooperation on interdependent networks can be observed. Notably, cooperation can be promoted on both layers, even if cooperation without interdependence would be improbable on one of the layers due to adverse conditions. Our results provide a rationale for engineering better social systems at the interface of networks and human decision making under testing dilemmas.

Analysis of Research Parameters for Cascading Failures of Interdependent Network Based on the Giant Component

With the deepening of the research on complex network, the cascading failures problem of the interdependent network is one of the hot research issues in the field. Setting reasonable parameters in the cascading failures analysis of interdependent network is of great significance for subsequent research. The mechanism of common models generation is analyzed, and its formula expression form is constructed. Starting from the single network cascading failures pattern, combined with the interdependent network theory research, a cascading failures model of interdependent network based on the giant component is established. The robustness analysis of interdependent networks with different network scale and different average degree is carried out for three common interdependent networks: BA-BA, WS-WS and ER-ER. The best condition for the robustness study of the interdependent network is obtained. That is, the sub-network node scale NS = NC ≥ 100 and the average degree k = 6. The conclusions obtained in this paper can provide a reference for the study of cascading failures in power, communication and other interdependent networks.

Interconnections between networks acting like an external field in a first-order percolation transition.

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we show how the effects of such interconnections can be described as an external field for interdependent networks experiencing a first-order percolation transition. We find that the critical exponents γ and δ, related to the external field, can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents (for first and second order) can even be found within a single model of interdependent networks, depending on the dependency coupling strength. Nevertheless, in both cases both sets satisfy Widom's identity, δ-1=γ/β, which further supports the validity of their definitions. Furthermore, we find that both Erdős-Rényi and scale-free networks have the same values of the exponents in the first-order regime, implying that these models are in the same universality class. In addition, we find that in k-core percolation the values of the critical exponents related to the field are the same as for interdependent networks, suggesting that these systems also belong to the same universality class.

Asymmetric interdependent networks with multiple-dependence relation.

In this paper, we study the robustness of interdependent networks with multiple-dependence (MD) relation which is defined that a node is interdependent on several nodes on another layer, and this node will fail if any of these dependent nodes are failed. We propose a two-layered asymmetric interdependent network (AIN) model to address this problem, where the asymmetric feature is that nodes in one layer may be dependent on more than one node in the other layer with MD relation, while nodes in the other layer are dependent on exactly one node in this layer. We show that in this model the layer where nodes are allowed to have MD relation exhibits different types of phase transitions (discontinuous and hybrid), while the other layer only presents discontinuous phase transition. A heuristic theory based on message-passing approach is developed to understand the structural feature of interdependent networks and an intuitive picture for the emergence of a tricritical point is provided. Moreover, we study the correlation between the intralayer degree and interlayer degree of the nodes and find that this correlation has prominent impact to the continuous phase transition but has feeble effect on the discontinuous phase transition. Furthermore, we extend the two-layered AIN model to general multilayered AIN, and the percolation behaviors and properties of relevant phase transitions are elaborated.

Optimization of robustness of interdependent network controllability by redundant design.

Controllability of complex networks has been a hot topic in recent years. Real networks regarded as interdependent networks are always coupled together by multiple networks. The cascading process of interdependent networks including interdependent failure and overload failure will destroy the robustness of controllability for the whole network. Therefore, the optimization of the robustness of interdependent network controllability is of great importance in the research area of complex networks. In this paper, based on the model of interdependent networks constructed first, we determine the cascading process under different proportions of node attacks. Then, the structural controllability of interdependent networks is measured by the minimum driver nodes. Furthermore, we propose a parameter which can be obtained by the structure and minimum driver set of interdependent networks under different proportions of node attacks and analyze the robustness for interdependent network controllability. Finally, we optimize the robustness of interdependent network controllability by redundant design including node backup and redundancy edge backup and improve the redundant design by proposing different strategies according to their cost. Comparative strategies of redundant design are conducted to find the best strategy. Results shows that node backup and redundancy edge backup can indeed decrease those nodes suffering from failure and improve the robustness of controllability. Considering the cost of redundant design, we should choose BBS (betweenness-based strategy) or DBS (degree based strategy) for node backup and HDF(high degree first) for redundancy edge backup. Above all, our proposed strategies are feasible and effective at improving the robustness of interdependent network controllability.

Alleviate cascade with fault pair in interdependent smart grid: a communication view

Smart grid has brought an innovation of traditional power. Smart grid, which couples communication network with power network, tends to cascade because of the heterogeneous coupling. Some research works have focused on external network features to mitigate the cascade, such as vulnerability and reliability. These studies, however, put less attention to an internal fault and fail to grasp cascading details. In this paper, we propose a pairwise scheme to investigate fault propagation in an interdependent network. The interdependent network model is built to characterize the fault cascade. Then, we leverage fault pair and analyze the necessary condition of the cascade. To alleviate cascading, we resort to find the minimized node set with constraints from a communication perspective. Further, the heuristic DCDR algorithm is proposed to hold the necessary redundant node set to relieve cascade. Extensive experiments not only illustrate the effectiveness of our approach on networks with different parameters, but also highlight some important observations which may give more insight to interdependent networks.

Emergence of robustness in networks of networks.

A model of interdependent networks of networks (NONs) was introduced recently [Proc. Natl. Acad. Sci. (USA) 114, 3849 (2017)PNASA60027-842410.1073/pnas.1620808114] in the context of brain activation to identify the neural collective influencers in the brain NON. Here we investigate the emergence of robustness in such a model, and we develop an approach to derive an exact expression for the random percolation transition in Erdös-Rényi NONs of this kind. Analytical calculations are in agreement with numerical simulations, and highlight the robustness of the NON against random node failures, which thus presents a new robust universality class of NONs. The key aspect of this robust NON model is that a node can be activated even if it does not belong to the giant mutually connected component, thus allowing the NON to be built from below the percolation threshold, which is not possible in previous models of interdependent networks. Interestingly, the phase diagram of the model unveils particular patterns of interconnectivity for which the NON is most vulnerable, thereby marking the boundary above which the robustness of the system improves with increasing dependency connections.

Modeling the interdependent network based on two-mode networks

Among heterogeneous networks, there exist obviously and closely interdependent linkages. Unlike existing research primarily focus on the theoretical research of physical interdependent network model. We propose a two-layer interdependent network model based on two-mode networks to explore the interdependent features in the reality. Specifically, we construct a two-layer interdependent loan network and develop several dependent features indices. The model is verified to enable us to capture the loan dependent features of listed companies based on loan behaviors and shared shareholders. Taking Chinese debit and credit market as case study, the main conclusions are: (1) only few listed companies shoulder the main capital transmission (20% listed companies occupy almost 70% dependent degree). (2) The control of these key listed companies will be more effective of avoiding the spreading of financial risks. (3) Identifying the companies with high betweenness centrality and controlling them could be helpful to monitor the financial risk spreading. (4) The capital transmission channel among Chinese financial listed companies and Chinese non-financial listed companies are relatively strong. However, under greater pressure of demand of capital transmission (70% edges failed), the transmission channel, which constructed by debit and credit behavior, will eventually collapse.

Research on Methods for Improving Robustness of Cascading Failures of Interdependent Networks

The issue about cascading failures of interdependent networks is a hot research field based on the current complex network basic theories. This paper mainly researches methods for improving robustness of interdependent networks. At first, a model of interdependent network cascading failures was established to carry out robustness analysis of three different interdependent networks including BA–BA, WS–WS and ER–ER. Then, methods for improving robustness of interdependent networks under random attacks and targeted attacks were researched respectively. (1)Under random attacks, based on the overall idea of network addition, the paper summarized four common methods for improving robustness of a simple complex network, including random addition, low-degree node addition, low-betweenness node addition and addition based on algebra connectivity. After that, according to inherent characteristics of interdependent networks, the paper put forward an interdependent network addition algorithm based on internal similarities. It is shown in cascading failure simulation calculation results of three interdependent coupling networks, that the method put forward in this paper can most effectively improve robustness of interdependent networks under random attacks. (2)Based on the overall idea of protecting key nodes under targeted attacks, the paper put forward a method for analyzing key nodes of interdependent networks based on natural connectivity. Then, under the BA–BA interdependent network, the paper analyzed this method in comparison with the commonly approved Max-Cas algorithm. It is shown in simulation calculation results that, the method put forward in this paper can more obviously improve robustness of an interdependent network under targeted attacks and can be applied flexibly.

Analysis of Cascading Failures of Interdependent Networks under Random Attacks

With continuous deepening of researches on complex network problems, the cascading failure problem of interdependent networks becomes one of the hot research topics in this field at present. This paper establishes a cascading failure model of interdependent networks based on a load-capacity model. And then we focus on the robustness problem of interdependent networks under random attacks based on four different network combinations. At last, an interdependent coupling network constituted of two networks with the node quantity of ten is taken as the example to verify conclusions obtained in this paper. Research results show that under random attacks, robustness of the asymmetric interdependent networks is similar with the previous research results, namely, robustness of the networks decreases with the increase of removal proportion and increases with the increase of tolerance coefficient. In addition, BA-BA interdependent networks and disassortative link networks perform better than other network combinations and coupling manners.