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.

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Catastrophic cascade of failures in interdependent networks

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

Multilayer Networks

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

Reliability Engineering and System Safety

Aspects concerning the structure and behaviours of individual networks have been studied intensely in the past decade, but the exploration of interdependent systems in the context of complex networks has started only recently. This article reviews a general framework for modelling the percolation properties of interacting networks and the first results drawn from its study.

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Networks formed from interdependent networks

Today's strongly connected, global networks have produced highly interdependent systems that we do not understand and cannot control well. These systems are vulnerable to failure at all scales, posing serious threats to society, even when external shocks are absent. As the complexity and interaction strengths in our networked world increase, man-made systems can become unstable, creating uncontrollable situations even when decision-makers are well-skilled, have all data and technology at their disposal, and do their best. To make these systems manageable, a fundamental redesign is needed. A 'Global Systems Science' might create the required knowledge and paradigm shift in thinking.

Globally networked risks and how to respond

We investigate the consequence of failures, occurring on the electrical grid, on a telecommunication network We have focused on the Italian electrical transmission network and the backbone of the internet network for research (GARR) Electrical network has been simulated using the DC power flow method; data traffic on GARR by a model of the TCP/IP basic features The status of GARR nodes has been related to the power level of the (geographically) neighbouring electrical nodes (if the power level of a node is lower than a threshold, all communication nodes depending on it are switched off) The electrical network has been perturbed by lines removal: the consequent re-dispatching reduces the power level in all nodes This reduces the number of active GARR nodes and, thus, its Quality of Service (QoS) Averaging over many configurations of perturbed electrical network, we have correlated the degradation of the electrical network with that of the communication network Results point to a sizeable amplification of the effects of faults on the electrical network on the communication network, also in the case of a moderate coupling between the two networks

Modelling interdependent infrastructures using interacting dynamical models

A model of the Italian 380 kV electrical transmission network has been analyzed under the topological and the functional viewpoints. The DC power flow model used to evaluate the power flux has been solved on the basis of input conditions (injected power - extracted power, line's reactances and the maximum flux capacity of each line) taken from real data. The vulnerability of the network under load conditions has been estimated by evaluating the power flux redistribution along the lines subsequent to line's removal. When the perturbed network cannot sustain a given input--output demand, the maximum power sustainable by the network has been evaluated to optimize the \texttt{Quality of Service}, defined as the difference between the expected and the effective dispatched power. The functional relevance of the different lines of the network has been classified according to the amount of power that the network must reduce, to keep alive, upon their removal. Results show that topological and functional relevances are related to different lines; lines having a strong topological relevance may have a bare relevance in the flow distribution and thus their removal does not affect the functioning of the network.

Influence of the topology on the power flux of the Italian high-voltage electrical network

The Internet is one of the most interesting realizations of a “complex” network. As a non-supervised growing object, it allows the study of the selective pressure which drives the network to assume its current structure. The DIMES and the ROUTEVIEWS projects are ongoing projects aimed at evaluating the topological structure of the Internet (at the Autonomous System or AS grain-level) on the basis of different types of measurements. The topological analysis of the networks produced by the two projects has allowed us to infer a growth mechanism which has been used to build up synthetic networks with similar properties. These networks have been used as test-beds for the implementation of a model of traffic dynamics, with the aim of assessing the ability of the Internet’s topology to support the basic actions for data traffic handling. Results have been compared with those obtained by using a random network of similar size. The effects of some structural perturbations (arcs and nodes’ removal, traffic localization) have been also evaluated in terms of the induced variations of the network’s efficiency. The resulting scenario is consistent with the hypothesis that the structure of the Internet is only partially fit to host communication processes and that the intelligence of the TCP/IP protocol is partly needed to overcome some “structural” deficiencies.

Is the topology of the internet network really fit to sustain its function

The electricity network represents an example of an evolving complex system. The first local networks contained only a few nodes, but within several decades, they have evolved into a highly connected continental system. The growth of these networks was influenced by various factors such as economic, demographic, political and technological developments. In this paper, we analyse the growth of the French 400 kV electricity transmission network from its establishment in 1960 until the year 2000. We study the different topological characteristics that describe the intensity of the growth process, such as the number of nodes, the number of lines, the average node connectivity and the overall length of wires. We compare these results with several economic and demographic indicators in order to identify the factors which correlate with the growth rate of the electricity network. Apart from this, we evaluate how the topological efficiency and vulnerability measures (clustering coefficient, information centrality, betweenness centrality) evolve in the course of time. The decisions regarding the power grid topology are influenced by many, very often contradictory factors, such as costs, the size of the covered area, demand, fault tolerance, reliability and quality of service. Our results yield a deeper insight into the process of Critical Infrastructure (CI) construction.

Limor Issacharoff

Papers

Modelling interdependent infrastructures using interacting dynamical models

Influence of the topology on the power flux of the Italian high-voltage electrical network

Is the topology of the internet network really fit to sustain its function

The evolution of the topology of high-voltage electricity networks

Critical Infrastructures Vulnerability: The Highway Networks