The extreme vulnerability of interdependent spatially embedded networks
TL;DR: Analysis of real-world interdependent networks shows that randomly positioned networks, where nodes are positioned according to geographical constraints, might not be so resilient.
Abstract: Networks of networks are vulnerable: a failure in one sub-network can bring the rest crashing down. Previous simulations have suggested that randomly positioned networks might offer some limited robustness under certain circumstances. Analysis now shows, however, that real-world interdependent networks, where nodes are positioned according to geographical constraints, might not be so resilient.
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TL;DR: This work offers a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.
2,669 citations
Cites background from "The extreme vulnerability of interd..."
...[189] partial interdependence was considered, and the Authors found that, in contrast to unembedded networks, for any fraction of dependency links the system collapses in an abrupt transition....
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TL;DR: In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications.
Abstract: In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others We also survey and discuss existing data sets that can be represented as multilayer networks We review attempts to generalize single-layer-network diagnostics to multilayer networks We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks We conclude with a summary and an outlook
1,934 citations
TL;DR: In this article, a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.
Abstract: In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.
401 citations
TL;DR: The recent advances in this forefront and rapidly evolving field of reconstructing nonlinear and complex dynamical systems from measured data or time series are reviewed, aiming to cover topics such as compressive sensing, noised-induced dynamical mapping, perturbations, reverse engineering, synchronization, inner composition alignment, global silencing and Granger Causality.
227 citations
Cites background from "The extreme vulnerability of interd..."
...A variety of models were introduced to gain insights into binary-state dynamics on complex networks [42], such as the votermodels for competition of twoopinions [437], stochastic propagationmodels for epidemic spreading [438],models of rumor diffusion and adoption of new technologies [439], cascading failuremodels [440], Ising spin models for ferromagnetic phase transition [441], and evolutionary games for cooperation and altruism [442]....
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TL;DR: In this article, the authors review the most relevant works that have investigated robustness in power grids using complex networks (CN) concepts, and propose strategies to improve robustness such as intentional islanding, restricted link addition, and more efficient electrical metrics such as electrical betweenness, net-ability and others.
Abstract: This paper reviews the most relevant works that have investigated robustness in power grids using Complex Networks (CN) concepts. In this broad field there are two different approaches. The first one is based solely on topological concepts, and uses metrics such as mean path length, clustering coefficient, efficiency and betweenness centrality, among many others. The second, hybrid approach consists of introducing (into the CN framework) some concepts from Electrical Engineering (EE) in the effort of enhancing the topological approach, and uses novel, more efficient electrical metrics such as electrical betweenness, net-ability, and others. There is however a controversy about whether these approaches are able to provide insights into all aspects of real power grids. The CN community argues that the topological approach does not aim to focus on the detailed operation, but to discover the unexpected emergence of collective behavior, while part of the EE community asserts that this leads to an excessive simplification. Beyond this open debate it seems to be no predominant structure (scale-free, small-world) in high-voltage transmission power grids, the vast majority of power grids studied so far. Most of them have in common that they are vulnerable to targeted attacks on the most connected nodes and robust to random failure. In this respect there are only a few works that propose strategies to improve robustness such as intentional islanding, restricted link addition, microgrids and Energies 2015, 8 9212 smart grids, for which novel studies suggest that small-world networks seem to be the best topology.
208 citations
References
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Book•
01 Jan 1985
TL;DR: In this paper, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques
9,830 citations
Book•
01 Jan 1992
TL;DR: In this article, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques
7,349 citations
TL;DR: Network motifs, patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks, are defined and may define universal classes of networks.
Abstract: Complex networks are studied across many fields of science. To uncover their structural design principles, we defined “network motifs,” patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks. We found such motifs in networks from biochemistry, neurobiology, ecology, and engineering. The motifs shared by ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomyces cerevisiae or from those found in the World Wide Web. Similar motifs were found in networks that perform information processing, even though they describe elements as different as biomolecules within a cell and synaptic connections between neurons in Caenorhabditis elegans. Motifs may thus define universal classes of networks. This
6,992 citations
TL;DR: In this paper, the authors develop a framework for understanding the robustness of interacting networks subject to cascading failures and present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks.
Abstract: Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.
3,651 citations
TL;DR: A conceptual framework for addressing infrastructure interdependencies is presented that could serve as the basis for further understanding and scholarship in this important area and is used to explore the challenges and complexities of interdependency.
Abstract: The notion that our nation's critical infrastructures are highly interconnected and mutually dependent in complex ways, both physically and through a host of information and communications technologies (so-called "cyberbased systems"), is more than an abstract, theoretical concept. As shown by the 1998 failure of the Galaxy 4 telecommunications satellite, the prolonged power crisis in California, and many other recent infrastructure disruptions, what happens to one infrastructure can directly and indirectly affect other infrastructures, impact large geographic regions and send ripples throughout the national a global economy. This article presents a conceptual framework for addressing infrastructure interdependencies that could serve as the basis for further understanding and scholarship in this important area. We use this framework to explore the challenges and complexities of interdependency. We set the stage for this discussion by explicitly defining the terms infrastructure, infrastructure dependencies, and infrastructure interdependencies and introducing the fundamental concept of infrastructures as complex adaptive systems. We then focus on the interrelated factors and system conditions that collectively define the six dimensions. Finally, we discuss some of the research challenges involved in developing, applying, and validating modeling and simulation methodologies and tools for infrastructure interdependency analysis.
2,341 citations
"The extreme vulnerability of interd..." refers background or methods in this paper
...The left-hand side and right-hand side of equation (3) are plotted as a straight (red) line and a (blue) curve respectively....
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...For any values of p and q, the solution of equation (3) can be graphically presented as the intersection between the curve y = pqP∞(x)+p(1−q) and the straight line y = x representing the right-hand side and the left-hand side of equation (3) respectively, as demonstrated in Fig....
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...From the solution of equation (3) we obtain P∞(p) as a function of p for several values...
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...Therefore, the critical dependency qc below which the discontinuous transition becomes continuous must satisfy equations (3) and (4) for x→ pc given by...
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...x = pqP∞(x)+p(1−q) (3) where the size of the giant component at steady state is P∞(x)....
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