Integrated cyber-physical systems (CPSs), such as the smart grid, are increasingly becoming the underpinning technology for major industries. A major concern regarding such systems are the seemingly unexpected large-scale failures, which are often attributed to a small initial shock getting escalated due to intricate dependencies within and across the individual counterparts of the system. In this paper, we develop a novel interdependent system model to capture this phenomenon, also known as cascading failures. Our framework consists of two networks that have inherently different characteristics governing their intra-dependency: i) a cyber-network where a node is functional as long as it belongs to the largest connected (i.e., giant) component; and ii) a physical network where nodes are given an initial flow and a capacity, and failure of a node results with redistribution of its flow to the remaining nodes, upon which further failures might take place due to overloading (i.e., the flow of a node exceeding its capacity). Furthermore, it is assumed that these two networks are inter-dependent. For simplicity, we consider a one-to-one interdependency model where every node in the cyber-network is dependent upon and supports a single node in the physical network, and vice versa. We provide a thorough analysis of the dynamics of cascading failures in this interdependent system initiated with a random attack. The system robustness is quantified as the surviving fraction of nodes at the end of cascading failures, and is derived in terms of all network parameters involved (e.g., degree distribution, load/capacity distribution, failure size, etc.). Analytic results are supported through an extensive numerical study. Among other things, these results demonstrate the ability of our model to capture the unexpected nature of large-scale failures and provide insights on improving system robustness.
Cascading failures and epidemic dynamics, as two successful application realms of network science, are usually investigated separately. How do they affect each other is still one open, interesting problem. In this letter, we couple both processes and put them into the framework of interdependent networks, where each network only supports one dynamical process. Of particular interest, they spontaneously form a feedback loop: virus propagation triggers cascading failures of systems while cascading failures suppress virus propagation. Especially, there exists crucial threshold of virus transmissi…
We present a cascading failure model of two interdependent networks in which functional nodes belong to components of size greater than or equal to $s$. We find theoretically and via simulation that in complex networks with random dependency links the transition is first-order for $s\geq 3$ and second-order for $s=2$. We find for two square lattices with a distance constraint $r$ in the dependency links that increasing $r$ moves the system from a regime without a phase transition to one with a second-order transition. As $r$ continues to increase the system collapses in a first-order transitio…
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 deg…
The structure of real-world multilayer infrastructure systems usually exhibits anisotropy due to constraints of the embedding space. For example, geographical features like mountains, rivers and shores influence the architecture of critical infrastructure networks. Moreover, such spatial networks are often non-homogeneous but rather have a modular structure with dense connections within communities and sparse connections between neighboring communities. When the networks of the different layers are interdependent, local failures and attacks may propagate throughout the system. Here we study th…
We study the cascading failures in a system composed of two interdependent square lattice networks A and B placed on the same Cartesian plane, where each node in network A depends on a node in network B randomly chosen within a certain distance $r$ from the corresponding node in network A and vice versa. Our results suggest that percolation for small $r$ below $r_{\rm max}\approx 8$ (lattice units) is a second-order transition, and for larger $r$ is a first-order transition. For $r<r_{\rm max}$, the critical threshold increases linearly with $r$ from 0.593 at $r=0$ and reaches a maximum, 0.738…
Large cascades are a common occurrence in many natural and engineered complex systems. In this paper we explore the propagation of cascades across networks using realistic network topologies, such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. We find that three properties, scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections, are all necessary components to significantly reduce the size of large cascades in the Bak-Tang-Wiesenfeld sandpile model. We demonstrate that correlations present in the st…
We study the influence of clustering, more specifically triangles, on cascading failures in interdependent networks or systems, in which we model the dependence between comprising systems using a dependence graph. First, we propose a new model that captures how the presence of triangles in the dependence graph alters the manner in which failures transmit from affected systems to others. Unlike existing models, the new model allows us to approximate the failure propagation dynamics using a multi-type branching process, even with triangles. Second, making use of the model, we provide a simple co…
We study the mutual percolation of a system composed of two interdependent random regular networks. We introduce a notion of distance to explore the effects of the proximity of interdependent nodes on the cascade of failures after an initial attack. We find a non-trivial relation between the nature of the transition through which the networks disintegrate and the parameters of the system, which are the degree of the nodes and the maximum distance between interdependent nodes. We explain this relation by solving the problem analytically for the relevant set of cases.
Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain…
In this paper, the multi-type branching process is applied to describe the statistics and interdependencies of line outages, the load shed, and isolated buses. The offspring mean matrix of the multi-type branching process is estimated by the Expectation Maximization (EM) algorithm and can quantify the extent of outage propagation. The joint distribution of two types of outages is estimated by the multi-type branching process via the Lagrange-Good inversion. The proposed model is tested with data generated by the AC OPA cascading simulations on the IEEE 118-bus system. The largest eigenvalues o…
Robustness and cascading failures in interdependent systems has been an active research field in the past decade. However, most existing works use percolation-based models where only the largest component of each network remains functional throughout the cascade. Although suitable for communication networks, this assumption fails to capture the dependencies in systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines. Here, we consider a model consisting of systems $A$ and…
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 anot…
ISC-CNR Physics Dept., Univ. dir Roma "La Sapienza"（罗马第一大学）；IMT Alti Studi Lucca（卢卡高等研究院）；LIMS London Institute of Mathematical Sciences（伦敦数学科学研究所）；Gubkin Russian State University of Oil and Gas（古比雪夫俄罗斯国立石油天然气大学）；ENEA: Italian National Agency for New Technologies, Energy and Sustainable Economic Development（意大利国家新能源与可持续发展技术署）
Interdependent networks are ubiquitous in our society, ranging from infrastructure to economics, and the study of their cascading behaviors using percolation theory has attracted much attention in the recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed including generating functions, eigenvalues among some others. These different frameworks approach the phase transition behaviors from different angles, and have been very successful in shaping the different quantities of interest including critical threshold, size of the gian…
Modern networks are becoming increasingly interdependent. As a prominent example, the smart grid is an electrical grid controlled through a communications network, which in turn is powered by the electrical grid. Such interdependencies create new vulnerabilities and make these networks more susceptible to failures. In particular, failures can easily spread across these networks due to their interdependencies, possibly causing cascade effects with a devastating impact on their functionalities. In this paper we focus on the interdependence between the power grid and the communications network, a…
In a system of interdependent networks, an initial failure of nodes invokes a cascade of iterative failures that may lead to a total collapse of the whole system in a form of an abrupt first order transition. When the fraction of initial failed nodes $1-p$ reaches criticality, $p=p_c$, the abrupt collapse occurs by spontaneous cascading failures. At this stage, the giant component decreases slowly in a plateau form and the number of iterations in the cascade, $τ$, diverges. The origin of this plateau and its increasing with the size of the system remained unclear. Here we find that simultaneou…
It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman [Phys. Rev. Lett. {\bf 103}, 058701 (2009)], we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system's robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold $p_c$ in interdependent networks.
Coupling cyber and physical systems gives rise to numerous engineering challenges and opportunities. An important challenge is the contagion of failure from one system to another, which can lead to large-scale cascading failures. However, the self-healing ability emerges as a valuable opportunity where the overlaying cyber network can cure failures in the underlying physical network. To capture both self-healing and contagion, this paper considers a graphical model representation of an interdependent cyber-physical system, in which nodes represent various cyber or physical functionalities, and…
Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering.Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak-Tang-Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, inte…
We study the robustness of interdependent networks where two networks are said to be interdependent if the operation of one network depends on the operation of the other one, and vice versa. In this paper, we propose a model for analyzing bidirectional interdependent networks with known topology. We define the metric MR(D) to be the minimum number of nodes that should be removed from one network to cause the failure of D nodes in the other network due to cascading failures. We prove that evaluating this metric is not only NP-complete, but also inapproximable. Next, we propose heuristics for ev…
Interdependent networks have been shown to be extremely vulnerable based on the percolation model. Parshani et. al further indicated that the more inter-similar networks are, the more robust they are to random failure. Our understanding of how coupling patterns shape and impact the cascading failures of loads in interdependent networks is limited, but is essential for the design and optimization of the real-world interdependent networked systems. This question, however, is largely unexplored. In this paper, we address this question by investigating the robustness of interdependent ER random gr…
Increased coupling between critical infrastructure networks, such as power and communication systems, will have important implications for the reliability and security of these systems. To understand the effects of power-communication coupling, several have studied interdependent network models and reported that increased coupling can increase system vulnerability. However, these results come from models that have substantially different mechanisms of cascading, relative to those found in actual power and communication networks. This paper reports on two sets of experiments that compare the ne…
It was recently recognized that interdependencies among different networks can play a crucial role in triggering cascading failures and hence system-wide disasters. A recent model shows how pairs of interdependent networks can exhibit an abrupt percolation transition as failures accumulate. We report on the effects of topology on failure propagation for a model system consisting of two interdependent networks. We find that the internal node correlations in each of the two interdependent networks significantly changes the critical density of failures that triggers the total disruption of the tw…
For interdependent networks with identity dependency map, percolation is exactly the same with that on a single network and follows a second-order phase transition, while for random dependency, percolation follows a first-order phase transition. In real networks, the dependency relations between networks are neither identical nor completely random. Thus in this paper, we study the influence of randomness for dependency maps on the robustness of interdependent lattice networks. We introduce approximate entropy($ApEn$) as the measure of randomness of the dependency maps. We find that there is cr…
Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that may abruptly fragment the system, while below this "critical dependency" (CD) a failure of few nodes leads only to small damage to the system. So far, the research has been focused on interdependent random networks without space limitations. However, many real systems, such as power grids and the Internet, are not random but are spatially embedded. Here we a…
We consider a cyber-physical system consisting of two interacting networks, i.e., a cyber-network overlaying a physical-network. It is envisioned that these systems are more vulnerable to attacks since node failures in one network may result in (due to the interdependence) failures in the other network, causing a cascade of failures that would potentially lead to the collapse of the entire infrastructure. The robustness of interdependent systems against this sort of catastrophic failure hinges heavily on the allocation of the (interconnecting) links that connect nodes in one network to nodes i…
Cascading failure is a potentially devastating process that spreads on real-world complex networks and can impact the integrity of wide-ranging infrastructures, natural systems, and societal cohesiveness. One of the essential features that create complex network vulnerability to failure propagation is the dependency among their components, exposing entire systems to significant risks from destabilizing hazards such as human attacks, natural disasters or internal breakdowns. Developing realistic models for cascading failures as well as strategies to halt and mitigate the failure propagation can…
Understanding and analyzing cascading failures in power grids have been the focus of many researchers for years. However, the complex interactions among the large number of components in these systems and their contributions to cascading failures are not yet completely understood. Therefore, various techniques have been developed and used to model and analyze the underlying interactions among the components of the power grid with respect to cascading failures. Such methods are important to reveal the essential information that may not be readily available from power system physical models and …
Interdependencies are ubiquitous throughout the world. Every real-world system interacts with and is dependent on other systems, and this interdependency affects their performance. In particular, interdependencies among networks make them vulnerable to failure cascades, the effects of which are often catastrophic. Failure propagation fragments network components, disconnects them, and may cause complete systemic failure. We propose a strategy of avoiding or at least mitigating the complete destruction of a system of interdependent networks experiencing a failure cascade. Starting with a fracti…
Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy of nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of re…
In reality, many real-world networks interact with and depend on other networks. We develop an analytical framework for studying interacting networks and present an exact percolation law for a network of $n$ interdependent networks (NON). We present a general framework to study the dynamics of the cascading failures process at each step caused by an initial failure occurring in the NON system. We study and compare both $n$ coupled Erdős-Rényi (ER) graphs and $n$ coupled random regular (RR) graphs. We found recently [Gao et. al. arXive:1010.5829] that for an NON composed of $n$ ER networks each…
Interdependent networks are characterized by two kinds of interactions: The usual connectivity links within each network and the dependency links coupling nodes of different networks. Due to the latter links such networks are known to suffer from cascading failures and catastrophic breakdowns. When modeling these phenomena, usually one assumes that a fraction of nodes gets damaged in one of the networks, which is followed possibly by a cascade of failures. In real life the initiating failures do not occur at once and effort is made replace the ties eliminated due to the failing nodes. Here we …
Cascading failures may lead to dramatic collapse in interdependent networks, where the breakdown takes place as a discontinuity of the order parameter. In the cascading failure (CF) model with healing there is a control parameter which at some value suppresses the discontinuity of the order parameter. However, up to this value of the healing parameter the breakdown is a hybrid transition, meaning that, besides this first order character, the transition shows scaling too. In this paper we investigate the question of universality related to the scaling behavior. Recently we showed that the hybri…
Cascading failures in complex systems have been studied extensively using two different models: $k$-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically and validate our theoretical results through extensive simulations. We also study the complete phase diagram of the percolation transition as we tune the average local $k$-core threshold and the coupling between networks. We find that the phase diagram of the combined processes is very rich and includes novel features that do not appear in the models studying each of the processes …
Many real-world networks are coupled together to maintain their normal functions. Here we study the robustness of multiplex networks with interdependent and interconnected links under k-core percolation, where a node fails when it connects to a threshold of less than k neighbors. By deriving the self-consistency equations, we solve the key quantities of interests such as the critical threshold and size of the giant component analytically and validate the theoretical results with numerical simulations. We find a rich phase transition phenomenon as we tune the inter-layer coupling strength. Spec…
We model smart grids as complex interdependent networks, and study targeted attacks on smart grids for the first time. A smart grid consists of two networks: the power network and the communication network, interconnected by edges. Occurrence of failures (attacks) in one network triggers failures in the other network, and propagates in cascades across the networks. Such cascading failures can result in disintegration of either (or both) of the networks. Earlier works considered only random failures. In practical situations, an attacker is more likely to compromise nodes selectively. We study c…
Many real infrastructure networks, such as power grids and communication networks, are not only depend on one another to function, but also embedded in space. A lot of works have been devoted to reveal the vulnerability of interdependent spatially embedded networks considers random failures. However, recently research show that they are susceptible to geographically localized attacks or failures cased by natural disasters or terrorist attacks, which affect all nodes within a given radius. In particular, small localized attacks may lead to catastrophic consequences. As a remedy of the collapse …
We study the mutual percolation of two interdependent lattice networks ranging from two to seven dimensions, denoted as $D$. We impose that the length of interdependent links connecting nodes in the two lattices be less than or equal to a certain value, $r$. For each value of $D$ and $r$, we find the mutual percolation threshold, $p_c[D,r]$ below which the system completely collapses through a cascade of failures following an initial destruction of a fraction $ (1-p)$ of the nodes in one of the lattices. We find that for each dimension, $D<6$, there is a value of $r=r_I>1$ such that for $r\geq…
Networks are often made up of several layers that exhibit diverse degrees of interdependencies. A multilayer interdependent network consists of a set of graphs $G$ that are interconnected through a weighted interconnection matrix $ B $, where the weight of each inter-graph link is a non-negative real number $ p $. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix $ Q $ characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each …
Modern large network systems normally work in cooperation and incorporate dependencies between their components for purposes of efficiency and regulation. Such dependencies may become a major risk since they can cause small scale failures to propagate throughout the system. Thus, the dependent nodes could be a natural target for malicious attacks that aim to exploit these vulnerabilities. Here, we consider for the first time a new type of targeted attacks that are based on the dependency between the networks. We study strategies of attacks that range from dependency-first to dependency-last, w…
Real-world complex systems always interact with each other, which causes these systems to collapse in an avalanche or cascading manner in the case of random failures or malicious attacks. The robustness of multilayer networks has attracted great interest, where the modeling and theoretical studies of which always rely on the concept of multilayer networks and percolation methods. A straightforward and tacit assumption is that the interdependence across network layers is strong, which means that a node will fail entirely with the removal of all links if one of its interdependent neighbours fail…
In this paper we study the Motter-Lai model of cascading failures induced by overloads in both isotropic and anisotropic spatial networks, generated by placing nodes in a square lattice and using various distributions of link lengths and angles. Anisotropy has not been earlier considered in the Motter-Lai model and is a real feature that may affect the cascading failures. This could reflect the existence of a preferred direction in which a given attribute of the system manifests, such as power lines that follow a city built parallel to the coast. We show that the anisotropy causes a greater sp…
Percolation theory is an approach to study vulnerability of a system. We develop analytical framework and analyze percolation properties of a network composed of interdependent networks (NetONet). Typically, percolation of a single network shows that the damage in the network due to a failure is a continuous function of the fraction of failed nodes. In sharp contrast, in NetONet, due to the cascading failures, the percolation transition may be discontinuous and even a single node failure may lead to abrupt collapse of the system. We demonstrate our general framework for a NetONet composed of $…
An adversary looking to disrupt a power grid may look to target certain substations and sources of power generation to initiate a cascading failure that maximizes the number of customers without electricity. This is particularly an important concern when the enemy has the capability to launch cyber-attacks as practical concerns (i.e. avoiding disruption of service, presence of legacy systems, etc.) may hinder security. Hence, a defender can harden the security posture at certain power stations but may lack the time and resources to do this for the entire power grid. We model a power grid as a …
Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as $k$-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Much effort has been devoted to describe the properties of HPTs of individual systems. Yet the fundamental question about the possible universal mechanism underlying those HPTs has not been investigated at a microscopic level. Here, we find that the discontinuity in the order parameter in such HPTs results from two steps: a durable critical branching (CB) and an explosive,…
Natural hazards including floods can trigger catastrophic failures in interdependent urban transport network-of-networks (NoNs). Population growth has enhanced transportation demand while urbanization and climate change have intensified urban floods. However, despite the clear need to develop actionable insights for improving the resilience of critical urban lifelines, the theory and methods remain underdeveloped. Furthermore, as infrastructure systems become more intelligent, security experts point to the growing threat of targeted cyber-physical attacks during natural hazards. Here we develo…
The self-consistent probabilistic approach has proven itself powerful in studying the percolation behavior of interdependent or multiplex networks without tracking the percolation process through each cascading step. In order to understand how directed dependency links impact criticality, we employ this approach to study the percolation properties of networks with both undirected connectivity links and directed dependency links. We find that when a random network with a given degree distribution undergoes a second-order phase transition, the critical point and the unstable regime surrounding t…
With a steadily growing population and rapid advancements in technology, the global economy is increasing in size and complexity. This growth exacerbates global vulnerabilities and may lead to unforeseen consequences such as global pandemics fueled by air travel, cyberspace attacks, and cascading failures caused by the weakest link in a supply chain. Hence, a quantitative understanding of the mechanisms driving global network vulnerabilities is urgently needed. Developing methods for efficiently monitoring evolution of the global economy is essential to such understanding. Each year the World …
