For large-scale power networks, the failure of particular transmission lines can offload power to other lines and cause self-protection trips to activate, instigating a cascade of line failures. In extreme cases, this can bring down the entire network. Learning where the vulnerabilities are and the expected timescales for which failures are likely is an active area of research. In this article we present a novel stochastic dynamics model for a large-scale power network along with a framework for efficient computer simulation of the model including long timescale events such as cascade failure. We build on an existing Hamiltonian formulation and introduce stochastic forcing and damping components to simulate small perturbations to the network. Our model and simulation framework allow assessment of the particular weaknesses in a power network that make it susceptible to cascade failure, along with the timescales and mechanism for expected failures.
Cascading failure as a systematic risk occurs in a wide range of real-world networks. Cascade size distribution is a basic and crucial characteristic of systemic cascade behaviors. Recent research works have revealed that the distribution of cascade sizes is a bimodal form indicating the existence of either very small cascades or large ones. In this paper, we aim to understand the properties and formation of such bimodal distribution of cascade sizes in complex networks, and further predict the final cascade size. We first find that the bimodal distribution of cascade sizes is ubiquitous in bo…
We propose a phase model to study cascade failure in power grids composed of generators and loads. If the power demand is below a critical value, the model system of power grids maintains the standard frequency by feedback control. On the other hand, if the power demand exceeds the critical value, an electric failure occurs via step out (loss of synchronization) or voltage collapse. The two failures are incorporated as two removal rules of generator nodes and load nodes. We perform direct numerical simulation of the phase model on a scale-free network and compare the results with a mean-field …
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…
Two node variables determine the evolution of cascades in random networks: a node's degree and threshold. Correlations between both fundamentally change the robustness of a network, yet, they are disregarded in standard analytic methods as local tree or heterogeneous mean field approximations because of the bad tractability of order statistics. We show how they become tractable in the thermodynamic limit of infinite network size. This enables the analytic description of node attacks that are characterized by threshold allocations based on node degree. Using two examples, we discuss possible im…
Electric power-systems are one of the most important critical infrastructures. In recent years, they have been exposed to extreme stress due to the increasing demand, the introduction of distributed renewable energy sources, and the development of extensive interconnections. We investigate the phenomenon of abrupt breakdown of an electric power-system under two scenarios: load growth (mimicking the ever-increasing customer demand) and power fluctuations (mimicking the effects of renewable sources). Our results indicate that increasing the system size causes breakdowns to become more abrupt; in…
Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perfor…
Modern network-like systems are usually coupled in such a way that failures in one network can affect the entire system. In infrastructures, biology, sociology, and economy, systems are interconnected and events taking place in one system can propagate to any other coupled system. Recent studies on such coupled systems show that the coupling increases their vulnerability to random failure. Properties for interdependent networks differ significantly from those of single-network systems. In this article, these results are reviewed and the main properties discussed.
How big is the risk that a few initial failures of nodes in a network amplify to large cascades that span a substantial share of all nodes? Predicting the final cascade size is critical to ensure the functioning of a system as a whole. Yet, this task is hampered by uncertain or changing parameters and missing information. In infinitely large networks, the average cascade size can often be well estimated by established approaches building on local tree approximations and mean field approximations. Yet, as we demonstrate, in finite networks, this average does not even need to be a likely outcome…
A characteristic property of networks is their ability to propagate influences, such as infectious diseases, behavioral changes, and failures. An especially important class of such contagious dynamics is that of cascading processes. These processes include, for example, cascading failures in infrastructure systems, extinctions cascades in ecological networks, and information cascades in social systems. In this review, we discuss recent progress and challenges associated with the modeling, prediction, detection, and control of cascades in networks.
In network systems, a local perturbation can amplify as it propagates, potentially leading to a large-scale cascading failure. Here we derive a continuous model to advance our understanding of cascading failures in power-grid networks. The model accounts for both the failure of transmission lines and the desynchronization of power generators, and incorporates the transient dynamics between successive steps of the cascade. In this framework, we show that a cascade event is a phase-space transition from an equilibrium state with high energy to an equilibrium state with lower energy, which can be…
The understanding of cascading failures in complex systems has been hindered by the lack of realistic large-scale modeling and analysis that can account for variable system conditions. Here, using the North American power grid, we identify, quantify, and analyze the set of network components that are vulnerable to cascading failures under any out of multiple conditions. We show that the vulnerable set consists of a small but topologically central portion of the network and that large cascades are disproportionately more likely to be triggered by initial failures close to this set. These result…
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…
Multilayer systems are coupled networks characterized by different contexts (layers) of interaction and have gained much attention recently due to their suitability to describe a broad spectrum of empirical complex systems. They are very fragile to percolation and first-neighbor failure propagation, but little is known about how they respond to non-local disruptions, as it occurs in failures induced by flow redistribution, for example. Acknowledging that many socio-technical and biological systems sustain a flow of some physical quantity, such as energy or information, across the their compone…
External or internal shocks may lead to the collapse of a system consisting of many agents. If the shock hits only one agent initially and causes it to fail, this can induce a cascade of failures among neighoring agents. Several critical constellations determine whether this cascade remains finite or reaches the size of the system, i.e. leads to systemic risk. We investigate the critical parameters for such cascades in a simple model, where agents are characterized by an individual threshold θ_i determining their capacity to handle a load αθ_i with 1-αbeing their safety margin. If agents fail,…
We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks whi…
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…
Cascade models on networks have been used extensively to study cascade failure in complex systems. However, most current models consider failure caused by node damage and neglect the possibility of link damage, which is relevant to transportation, social dynamics, biology, and medicine. In an attempt to generalize conventional cascade models to link damage, we propose a link cascade model based on the standard independent cascade model, which is then solved via both numerical simulation and analytic approximation. We find that the probability that a node loses all its links due to link damage …
Many critical infrastructure systems have network structure and are under stress. Despite their national importance, the complexity of large-scale transport networks means we do not fully understand their vulnerabilities to cascade failures. The research in this paper examines the interdependent rail networks in Greater London and surrounding commuter area. We focus on the morning commuter hours, where the system is under the most demand stress. There is increasing evidence that the topological shape of the network plays an important role in dynamic cascades. Here, we examine whether the diffe…
Cascading failures are a critical vulnerability of complex information or infrastructure networks. Here we investigate the properties of load-based cascading failures in real and synthetic spatially-embedded network structures, and propose mitigation strategies to reduce the severity of damages caused by such failures. We introduce a stochastic method for optimal heterogeneous distribution of resources (node capacities) subject to a fixed total cost. Additionally, we design and compare the performance of networks with N-stable and (N-1)-stable network-capacity allocations by triggering cascade…
We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependent relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support node in the other network. If both networks A and B are Erdős-Rényi networks, A and B, with (i) sizes $N^A$ and $N^B$, (ii) average degrees $a$ and $b$, and (iii) $c^{AB}_0N^B$ support links from network A to B and $c^{BA}_0N^B$ support links from network B to A, we find that under rand…
Epidemic spreading and cascading failure are two important dynamical processes over complex networks. They have been investigated separately for a long history. But in the real world, these two dynamics sometimes may interact with each other. In this paper, we explore a model combined with SIR epidemic spreading model and local loads sharing cascading failure model. There exists a critical value of tolerance parameter that whether the epidemic with high infection probability can spread out and infect a fraction of the network in this model. When the tolerance parameter is smaller than the crit…
Cascade Failures from Distributed Generation in Power Grids
ISC-CNR Physics Dept., Univ. ”La Sapienza”（意大利罗马大学 ISC-CNR 物理系）；IMT Alti Studi Lucca（卢卡高等研究所）；London Institute of Mathematical Sciences（伦敦数学科学研究所）；Department of Electrical and Computer Engineering, College of Engineering Kansas State University（堪萨斯州立大学工程学院电气与计算机工程系）
This paper considers a model for cascades on random networks in which the cascade propagation at any node depends on the load at the failed neighbor, the degree of the neighbor as well as the load at that node. Each node in the network bears an initial load that is below the capacity of the node. The trigger for the cascade emanates at a single node or a small fraction of the nodes from some external shock. Upon failure, the load at the failed node gets divided randomly and added to the existing load at those neighboring nodes that have not yet failed. Subsequently, a neighboring node fails if…
We study numerically the cascading failure problem by using artificially created scale-free networks and the real network structure of the power grid. The capacity for a vertex is assigned as a monotonically increasing function of the load (or the betweenness centrality). Through the use of a simple functional form with two free parameters, revealed is that it is indeed possible to make networks more robust while spending less cost. We suggest that our method to prevent cascade by protecting less vertices is particularly important for the design of more robust real-world networks to cascading …
How big is the risk that a few initial failures of networked nodes amplify to large cascades that endanger the functioning of the system? Common answers refer to the average final cascade size. Two analytic approaches allow its computation: a) (heterogeneous) mean field approximation and b) belief propagation. The former applies to (infinitely) large locally tree-like networks, while the latter is exact on finite trees. Yet, cascade sizes can have broad and multi-modal distributions that are not well represented by their average. Full distribution information is essential to identify likely ev…
Cascading failures constitute an important vulnerability of interconnected systems. Here we focus on the study of such failures on networks in which the connectivity of nodes is constrained by geographical distance. Specifically, we use random geometric graphs as representative examples of such spatial networks, and study the properties of cascading failures on them in the presence of distributed flow. The key finding of this study is that the process of cascading failures is non-self-averaging on spatial networks, and thus, aggregate inferences made from analyzing an ensemble of such networks…
Analyzing available FAO data from 176 countries over 21 years, we observe an increase of complexity in the international trade of maize, rice, soy, and wheat. A larger number of countries play a role as producers or intermediaries, either for trade or food processing. In consequence, we find that the trade networks become more prone to failure cascades caused by exogenous shocks. In our model, countries compensate for demand deficits by imposing export restrictions. To capture these, we construct higher-order trade dependency networks for the different crops and years. These networks reveal hi…
Many man-made networks support each other to provide efficient services and resources to the customers, despite that this support produces a strong interdependency between the individual networks. Thus an initial failure of a fraction $1-p$ of nodes in one network, exposes the system to cascade of failures and, as a consequence, to a full collapse of the overall system. Therefore it is important to develop efficient strategies to avoid the collapse by increasing the robustness of the individual networks against failures. Here, we provide an exact theoretical approach to study the evolution of …
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）
In a network, a local disturbance can propagate and eventually cause a substantial part of the system to fail, in cascade events that are easy to conceptualize but extraordinarily difficult to predict. Here, we develop a statistical framework that can predict cascade size distributions by incorporating two ingredients only: the vulnerability of individual components and the co-susceptibility of groups of components (i.e., their tendency to fail together). Using cascades in power grids as a representative example, we show that correlations between component failures define structured and often …
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…
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.
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…
Cascading Failures and Critical Infrastructures in Future Renewable European Power Systems
Forschungszentrum Jülich（于利希研究中心）；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院自动化与应用信息学研究所）；Open Energy Transition（开放能源转型）；TenneT TSO GmbH（TenneT输电系统运营有限公司）；Department of Digital Transformation in Energy Systems, Technische Universität Berlin（柏林工业大学能源系统数字化转型系）
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…
Epidemics, neural cascades, power failures, and many other phenomena can be described by a diffusion process on a network. To identify the causal origins of a spread, it is often necessary to identify the triggering initial node. Here we define a new morphological operator and use it to detect the origin of a diffusive front, given the final state of a complex network. Our method performs better than randomized trials as well as centrality based methods. More importantly, our method is applicable regardless of the specifics of the forward model, and therefore can be applied to a wide range of …
The behaviour of complex networks under failure or attack depends strongly on the specific scenario. Of special interest are scale-free networks, which are usually seen as robust under random failure but appear to be especially vulnerable to targeted attacks. In a recent study of public transport networks of 14 major cities of the world we have shown that these systems when represented by appropriate graphs may exhibit scale-free behaviour. In this paper we briefly review some of the recent results about the effects that defunct or removed nodes have on the properties of public transport netwo…
We study the cascading failure of networks due to overload, using the betweenness centrality of a node as the measure of its load following the Motter and Lai model. We study the fraction of survived nodes at the end of the cascade $p_f$ as function of the strength of the initial attack, measured by the fraction of nodes $p$, which survive the initial attack for different values of tolerance $α$ in random regular and Erdös-Renyi graphs. We find the existence of first order phase transition line $p_t(α)$ on a $p-α$ plane, such that if $p <p_t$ the cascade of failures lead to a very small fracti…
How do local topological changes affect the global operation and stability of complex supply networks? Studying supply networks on various levels of abstraction, we demonstrate that and how adding new links may not only promote but also degrade stable operation of a network. Intriguingly, the resulting overloads may emerge remotely from where such a link is added, thus resulting in nonlocal failure. We link this counter-intuitive phenomenon to Braess' paradox originally discovered in traffic networks. We use elementary network topologies to explain its underlying mechanism for different types …
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 complex networks, the failure of one or very few nodes may cause cascading failures. When this dynamical process stops in steady state, the size of the giant component formed by remaining un-failed nodes can be used to measure the severity of cascading failures, which is critically important for estimating the robustness of networks. In this paper, we provide a cascade of overload failure model with local load sharing mechanism, and then explore the threshold of node capacity when the large-scale cascading failures happen and un-failed nodes in steady state cannot connect to each other to f…
Power generation and distribution remains an important topic of discussion since the industrial revolution. As the system continues to grow, it needs to evolve both in infrastructure, robustness and its resilience to deal with failures. One such potential failure that we target in this work is the cascading failure. This avalanche effect propagates through the network and we study this propagation by Percolation Theory and implement some solutions for mitigation. We have extended the percolation theory as given in Mark Newman. Networks: an introduction,for random nodes to targeted nodes having…
Transmission line failures in power systems propagate and cascade non-locally. This well-known yet counter-intuitive feature makes it even more challenging to optimally and reliably operate these complex networks. In this work we present a comprehensive framework based on spectral graph theory that fully and rigorously captures how multiple simultaneous line failures propagate, distinguishing between non-cut and cut set outages. Using this spectral representation of power systems, we identify the crucial graph sub-structure that ensures line failure localization -- the network bridge-block dec…
