Robustness of a partially interdependent network formed of clustered networks

Robustness of Interdependent Networks with Weak Dependency Based on Bond Percolation

Real-world systems interact with one another via dependency connectivities. Dependency connectivities make systems less robust because failures may spread iteratively among systems via dependency links. Most previous studies have assumed that two nodes connected by a dependency link are strongly dependent on each other; that is, if one node fails, its dependent partner would also immediately fail. However, in many real scenarios, nodes from different networks may be weakly dependent, and links may fail instead of nodes. How interdependent networks with weak dependency react to link failures remains unknown. In this paper, we build a model of fully interdependent networks with weak dependency and define a parameter α in order to describe the node-coupling strength. If a node fails, its dependent partner has a probability of failing of 1−α. Then, we develop an analytical tool for analyzing the robustness of interdependent networks with weak dependency under link failures, with which we can accurately predict the system robustness when 1−p fractions of links are randomly removed. We find that as the node coupling strength increases, interdependent networks show a discontinuous phase transition when α<αc and a continuous phase transition when α>αc. Compared to site percolation with nodes being attacked, the crossover points αc are larger in the bond percolation with links being attacked. This finding can give us some suggestions for designing and protecting systems in which link failures can happen.

Group percolation in interdependent networks with reinforcement network layer

In many real-world interdependent network systems, nodes often work together to form groups, which can enhance robustness to resist risks. However, previous group percolation models are always of a first-order phase transition, regardless of the group size distribution. This motivates us to investigate a generalized model for group percolation in interdependent networks with a reinforcement network layer to eliminate collapse. Some backup devices that are equipped for a density ρ of reinforced nodes constitute the reinforcement network layer. For each group, we assume that at least one node of the group can function in one network and a node in another network depends on the group to function. We find that increasing the density ρ of reinforcement nodes and the size S of the dependency group can significantly enhance the robustness of interdependent networks. Importantly, we find the existence of a hybrid phase transition behavior and propose a method for calculating the shift point of percolation types. The most interesting finding is the exact universal solution to the minimal density _ρ of reinforced nodes (or the minimum group size _S) to prevent abrupt collapse for Erdős-Rényi, scale-free, and regular random interdependent networks. Furthermore, we present the validity of the analytic solutions for a triple point _ρ (or _S ), the corresponding phase transition point _p , and second-order phase transition points _p in interdependent networks. These findings might yield a broad perspective for designing more resilient interdependent infrastructure networks.

Study of Cascading Failure in Multisubnet Composite Complex Networks

Current research on the cascading failure of coupling networks is mostly based on hierarchical network models and is limited to a single relationship. In reality, many relationships exist in a network system, and these relationships collectively affect the process and scale of the network cascading failure. In this paper, a composite network is constructed based on the multisubnet composite complex network model, and its cascading failure is proposed combined with multiple relationships. The effect of intranetwork relationships and coupling relationships on network robustness under different influencing factors is studied. It is shown that cascading failure in composite networks is different from coupling networks, and increasing the strength of the coupling relationship can significantly improve the robustness of the network.

Robustness improvement for cyber physical system based on an optimization model of interdependent constraints

With the rapid development of information technology, traditional infrastructure networks have evolved into cyber physical systems (CPSs). However, this evolution has brought along with it cyber failures, in addition to physical failures, which can affect the safe and stable operation of the whole system. In light of this, in this paper, we propose an interdependence-constrained optimization model to improve the robustness of the cyber physical system. The proposed model includes not only the realistic physical law but also the interdependence between the physical network and the cyber network. However, this model is highly nonlinear and cannot be solved directly. Therefore, we transform the model into a bi-level mixed integer linear programming problem, which can be easily and effectively solved in polynomial time. We conduct the simulation based on standard Institute of Electrical and Electronics Engineers test cases and study the impact of the disaster level and coupling strength on the robustness of the whole system. The simulation results show that our proposed model can effectively improve the robustness of the cyber physical system. Moreover, we compare the performance of the power supply in different CPSs, which have different network structures of the cyber network. Our work can provide useful instructions for system operators to improve the robustness of CPSs after extreme events happen in them.

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.

Asymmetry in interdependence makes a multilayer system more robust against cascading failures

Multilayer networked systems are ubiquitous in nature and engineering, and the robustness of these systems against failures is of great interest. A main line of theoretical pursuit has been percolation-induced cascading failures, where interdependence between network layers is conveniently and tacitly assumed to be symmetric. In the real world, interdependent interactions are generally asymmetric. To uncover and quantify the impact of asymmetry in interdependence on network robustness, we focus on percolation dynamics in double-layer systems and implement the following failure mechanism: Once a node in a network layer fails, the damage it can cause depends not only on its position in the layer but also on the position of its counterpart neighbor in the other layer. We find that the characteristics of the percolation transition depend on the degree of asymmetry, where the striking phenomenon of a switch in the nature of the phase transition from first to second order arises. We derive a theory to calculate the percolation transition points in both network layers, as well as the transition switching point, with strong numerical support from synthetic and empirical networks. Not only does our work shed light on the factors that determine the robustness of multilayer networks against cascading failures, but it also provides a scenario by which the system can be designed or controlled to reach a desirable level of resilience.

Individual diversity between interdependent networks promotes the evolution of cooperation by means of mixed coupling

Along with the rapid development of network-based information technology, such as cloud computing, big data, the IoT, and so on, human society has stepped into a new era of complex networks. People's life and production activities depend more and more on various complex networks to ensure security and reliability. The complex interrelationships between human and nature establish a link to explain the cooperation of individual behaviour, especially for individual diversity. However, existing researches mostly ignore the influence of individual diversity on networks involved in individual behaviour to strategy selection. Therefore, it needs further research on how to consider both individual diversity and independent networks in the evolution of cooperative behaviour. To address this issue, we extend a simple game model into the interdependent networks through the mixed coupling (i.e., utility and probability) in this work. Also, we divide the kinds of strategic behaviour of a player in one layer concerning individual diversity. Moreover, there exists an optimal region of mixed coupling between networks such that cooperation can be promoted. Finally, experimental results can open the path to understanding the emergence and maintenance of cooperation within various interconnected and interrelated real-world systems newly.

On the Rising Interdependency between the Power Grid, ICT Network, and E-Mobility: Modeling and Analysis

Boosting critical infrastructures’ (CIs) preparedness to threats, including natural disasters and manmade attacks, is a global imperative. The intrinsic dependencies and interdependencies between CIs hinder their resiliency. Moreover, the evolution of CIs is, in many cases, en routè to tighten those interdependencies. The goal of this paper is to uncover and analyze the rising interdependency between the electric power grid, information and communication technology (ICT) networks, and transportation systems that are heavily reliant on electric-power drivetrains, collectively referred to hereafter as electro-mobility (e-mobility). E-mobility includes electric vehicles (EVs) and electric railway systems. A new influence graph-based model is introduced, as a promising approach to model operational interdependencies between CIs. Each of the links of the influence graph represents the probability of failure of the sink node following a failure of the source node. A futuristic scenario has been analyzed assuming increased dependency of the power grid on ICT for monitoring and control, and high penetration levels of EVs and distributed energy resources (DERs) in an urban region. Inspecting the influence graph shows that the impact of interdependency between the power grid, the ICT network, and the transportation network, for the case study analyzed in this paper, does not lead to failures during normal operation with proper design; however, it is severe during emergency conditions since it leads to failure propagation among the three CIs. This paper sets the stage for more research on this topic, and calls for more attention to interdependency analysis.

Influence of Different Coupling Modes on the Robustness of Smart Grid under Targeted Attack

Many previous works only focused on the cascading failure of global coupling of one-to-one structures in interdependent networks, but the local coupling of dual coupling structures has rarely been studied due to its complex structure. This will result in a serious consequence that many conclusions of the one-to-one structure may be incorrect in the dual coupling network and do not apply to the smart grid. Therefore, it is very necessary to subdivide the dual coupling link into a top-down coupling link and a bottom-up coupling link in order to study their influence on network robustness by combining with different coupling modes. Additionally, the power flow of the power grid can cause the load of a failed node to be allocated to its neighboring nodes and trigger a new round of load distribution when the load of these nodes exceeds their capacity. This means that the robustness of smart grids may be affected by four factors, i.e., load redistribution, local coupling, dual coupling link and coupling mode; however, the research on the influence of those factors on the network robustness is missing. In this paper, firstly, we construct the smart grid as a two-layer network with a dual coupling link and divide the power grid and communication network into many subnets based on the geographical location of their nodes. Secondly, we define node importance (NI) as an evaluation index to access the impact of nodes on the cyber or physical network and propose three types of coupling modes based on NI of nodes in the cyber and physical subnets, i.e., Assortative Coupling in Subnets (ACIS), Disassortative Coupling in Subnets (DCIS), and Random Coupling in Subnets (RCIS). Thirdly, a cascading failure model is proposed for studying the effect of local coupling of dual coupling link in combination with ACIS, DCIS, and RCIS on the robustness of the smart grid against a targeted attack, and the survival rate of functional nodes is used to assess the robustness of the smart grid. Finally, we use the IEEE 118-Bus System and the Italian High-Voltage Electrical Transmission Network to verify our model and obtain the same conclusions: (I) DCIS applied to the top-down coupling link is better able to enhance the robustness of the smart grid against a targeted attack than RCIS or ACIS, (II) ACIS applied to a bottom-up coupling link is better able to enhance the robustness of the smart grid against a targeted attack than RCIS or DCIS, and (III) the robustness of the smart grid can be improved by increasing the tolerance α. This paper provides some guidelines for slowing down the speed of the cascading failures in the design of architecture and optimization of interdependent networks, such as a top-down link with DCIS, a bottom-up link with ACIS, and an increased tolerance α.

The "weak" interdependence of infrastructure systems produces mixed percolation transitions in multilayer networks

Previous studies of multilayer network robustness model cascading failures via a node-to-node percolation process that assumes "strong" interdependence across layers-once a node in any layer fails, its neighbors in other layers fail immediately and completely with all links removed. This assumption is not true of real interdependent infrastructures that have emergency procedures to buffer against cascades. In this work, we consider a node-to-link failure propagation mechanism and establish "weak" interdependence across layers via a tolerance parameter α which quantifies the likelihood that a node survives when one of its interdependent neighbors fails. Analytical and numerical results show that weak interdependence produces a striking phenomenon: layers at different positions within the multilayer system experience distinct percolation transitions. Especially, layers with high super degree values percolate in an abrupt manner, while those with low super degree values exhibit both continuous and discontinuous transitions. This novel phenomenon we call mixed percolation transitions has significant implications for network robustness. Previous results that do not consider cascade tolerance and layer super degree may be under- or over-estimating the vulnerability of real systems. Moreover, our model reveals how nodal protection activities influence failure dynamics in interdependent, multilayer systems.

Structural Robustness of Unidirectional Dependent Networks Based on Attack Strategies

Current works have been focused on the robustness of single network and interdependent networks. However, to be more correct, the dependence of many real systems should be described as unidirectional. To study the structural robustness of networks with unidirectional dependence, the dependent networks named UDN are proposed, the description of the propagation of failures in them is given, as well as the introduction of the attack strategies that the probability of a node being attacked depends on the degree (DP attack) or on the betweenness (BP attack) of this node. The simulated results show that UDN is more vulnerable to BP attack when is first attacked a node with high betweenness. Compared with the Interacting Networks (IN), the UDN is more fragile under the two attack’s strategies.

Cascading failures in coupled networks: The critical role of node-coupling strength across networks

The robustness of coupled networks against node failure has been of interest in the past several years, while most of the researches have considered a very strong node-coupling method, i.e., once a node fails, its dependency partner in the other network will fail immediately. However, this scenario cannot cover all the dependency situations in real world, and in most cases, some nodes cannot go so far as to fail due to theirs self-sustaining ability in case of the failures of their dependency partners. In this paper, we use the percolation framework to study the robustness of interdependent networks with weak node-coupling strength across networks analytically and numerically, where the node-coupling strength is controlled by an introduced parameter α. If a node fails, each link of its dependency partner will be removed with a probability 1-α. By tuning the fraction of initial preserved nodes p, we find a rich phase diagram in the plane p-α, with a crossover point at which a first-order percolation transition changes to a second-order percolation transition.

Impact of Degree Heterogeneity on Attack Vulnerability of Interdependent Networks

The study of interdependent networks has become a new research focus in recent years. We focus on one fundamental property of interdependent networks: vulnerability. Previous studies mainly focused on the impact of topological properties upon interdependent networks under random attacks, the effect of degree heterogeneity on structural vulnerability of interdependent networks under intentional attacks, however, is still unexplored. In order to deeply understand the role of degree distribution and in particular degree heterogeneity, we construct an interdependent system model which consists of two networks whose extent of degree heterogeneity can be controlled simultaneously by a tuning parameter. Meanwhile, a new quantity, which can better measure the performance of interdependent networks after attack, is proposed. Numerical simulation results demonstrate that degree heterogeneity can significantly increase the vulnerability of both single and interdependent networks. Moreover, it is found that interdependent links between two networks make the entire system much more fragile to attacks. Enhancing coupling strength between networks can greatly increase the fragility of both networks against targeted attacks, which is most evident under the case of max-max assortative coupling. Current results can help to deepen the understanding of structural complexity of complex real-world systems.

Cascading failures in coupled networks with both inner-dependency and inter-dependency links

We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.

Coupled effects of local movement and global interaction on contagion

By incorporating segregated spatial domain and individual-based linkage into the SIS (susceptible-infected-susceptible) model, we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model. The role of the individual-based linkage between different spatial domains is investigated. As we adjust the timescale parameter τ from 0 to unity, which represents the degree of activation of the individual-based linkage, three regions are found. Within the region of 0 < τ < 0.02 , the epidemic is determined by local movement and is sensitive to the timescale τ . Within the region of 0.02 < τ < 0.5 , the epidemic is insensitive to the timescale τ . Within the region of 0.5 < τ < 1 , the outbreak of the epidemic is determined by the structure of the individual-based linkage. As we keep an eye on the first region, the role of activating the individual-based linkage in the present model is similar to the role of the shortcuts in the two-dimensional small world network. Only activating a small number of the individual-based linkage can prompt the outbreak of the epidemic globally. The role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked. These two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed. A log-log relation between the change in the number of infected individuals and the timescale τ is found. By calculating the epidemic threshold and the mean first encounter time, we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model.

Multiple effect of social influence on cooperation in interdependent network games

The social influence exists widely in the human society, where individual decision-making process (from congressional election to electronic commerce) may be affected by the attitude and behavior of others belonging to different social networks. Here, we couple the snowdrift (SD) game and the prisoner's dilemma (PD) game on two interdependent networks, where strategies in both games are associated by social influence to mimick the majority rule. More accurately, individuals' strategies updating refers to social learning (based on payoff difference) and above-mentioned social influence (related with environment of interdependent group), which is controlled by social influence strength s. Setting s = 0 decouples the networks and returns the traditional network game; while its increase involves the interactions between networks. By means of numerous Monte Carlo simulations, we find that such a mechanism brings multiple influence to the evolution of cooperation. Small s leads to unequal cooperation level in both games, because social learning is still the main updating rule for most players. Though intermediate and large s guarantees the synchronized evolution of strategy pairs, cooperation finally dies out and reaches a completely dominance in both cases. Interestingly, these observations are attributed to the expansion of cooperation clusters. Our work may provide a new understanding to the emergence of cooperation in intercorrelated social systems.