Networks formed from interdependent networks

Abstract

Complex networks appear in almost every aspect of science and technology. Although most results in the field have been obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of 'networks of networks' formed by interdependent random networks.T he interdisciplinary field of network science has attracted a great deal of attention in recent years . This development is based on the enormous number of data that are now routinely being collected, modelled and analysed, concerning social 31-39 , economic 14,36,40,41 , technological 40,42-48 and biological 9,13,49,50 systems. The investigation and growing understanding of this extraordinary volume of data will enable us to make the infrastructures we use in everyday life more efficient and more robust.The original model of networks, random graph theory, was developed in the 1960s by ErdAEs and Rényi, and is based on the assumption that every pair of nodes is randomly connected with the same probability, leading to a Poisson degree distribution. In parallel, in physics, lattice networks, where each node has exactly the same number of links, have been studied to model physical systems. Although graph theory is a well-established tool in the mathematics and computer science literature, it cannot describe well modern, real-life networks. Indeed, the pioneering 1999 observation by Barabasi 2 , that many real networks do not follow the ErdAEs-Rényi model but that organizational principles naturally arise in most systems, led to an overwhelming accumulation of supporting data, new models and computational and analytical results, and to the emergence of a new science, that of complex networks.Complex networks are usually non-homogeneous structures that in many cases obey a power-law form in their degree (that is, number of links per node) distribution. These systems are called scale-free networks. Real networks that can be approximated as scale-free networks include the Internet 3 , the World Wide Web 4 , social networks 31-39 representing the relations between individuals, infrastructure networks such as those of airlines 51 , networks in biology 9,13,49,50 , in particular networks of proteinprotein interactions 10 , gene regulation and biochemical pathways, and networks in physics, such as polymer networks or the potentialenergy-landscape network. The discovery of scale-free networks led to a re-evaluation of the basic properties of networks, such as their robustness, which exhibit a drastically different character than those of ErdAEs-Rényi networks. For example, whereas homogeneous ErdAEs-Rényi networks are extremely vulnerable to random fa...

Multi-Universality and Localized Attacks in Spatially Embedded Networks

“…Therefore, there is a need of modeling these systems as interdependent for understanding their structure, function and robustness [19][20][21][22][23][24][25][26]. Studies on spatially embedded interdependent networks found that in many cases they are significantly more vulnerable then non-embedded systems [27].…”

Correlated loss of ecosystem services in coupled mutualistic networks

“…Together with the existing examples, our study highlights that complex systems, be it infrastructural 33 , biological 34 or climate systems 35 , do not exist in isolation but form parallel, interdependent networks that respond collectively to perturbation. Our study demonstrates that strong interdependence among coupled networks can cause predictable transmission of perturbations across these networks.…”

“…Our study provides one of very few empirical examples of coupled, interdependent networks 7,[32][33][34][35][36] . Together with the existing examples, our study highlights that complex systems, be it infrastructural 33 , biological 34 or climate systems 35 , do not exist in isolation but form parallel, interdependent networks that respond collectively to perturbation.…”

An Ensemble Perspective on Multi-layer Networks

“…Note that λ = 1, −1 hence λ 2 < 1 which implies that v (1) does not correspond to the largest eigenvalue and therefore its entries sum up to zero. The same holds for v (2) and the matrix T I 21 T I 12 . For the multiplication of v with T we deduce that…”

“…Today we understand that this assumption is a rough simplification, since real networks usually have complex patterns of interaction with other networks. In order to study more realistic systems, network theory extended its perspective to account for these network to network interactions, and to investigate their influence on various processes of interest that may use the network topology as substrate [1,2,3,4].…”