Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions

Hu, Yanqing; Ksherim, Baruch; Cohen, Reuven; Havlin, Shlomo

doi:10.1103/physreve.84.066116

Cited by 148 publications

(123 citation statements)

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“…Very recently a more realistic coupled network system with both dependence and connectivity links between the coupled networks was studied 83 . Using a percolation approach, rich and unusual phase transition phenomena were found, including a mixed first-order and second-order hybrid transition.…”

Section: Have Been Developedmentioning

confidence: 99%

“…A systematic study of the competing effects of a NON in which the interlinks are both dependence and connectivity interlinks is needed. Interesting results on a model containing both dependence and connectivity interlinks have been obtained 83 . Finally, we mention an early study of the Ising model on coupled networks 98 .…”

Section: Nature Physics Doi:101038/nphys2180mentioning

confidence: 99%

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Networks formed from interdependent networks

et al. 2011

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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...

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Section: Have Been Developedmentioning

confidence: 99%

Section: Nature Physics Doi:101038/nphys2180mentioning

confidence: 99%

Networks formed from interdependent networks

et al. 2011

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“…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.Previous studies of the robustness of interdependent networks have focused on networks in which there is no constraint on the distance between the interdependent nodes [1][2][3][4][5][6][7][8][9]. However, many dependency links in the real world connect nearby nodes.…”

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confidence: 99%

Cascading failures in networks with proximate dependent nodes

et al. 2014

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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.Previous studies of the robustness of interdependent networks have focused on networks in which there is no constraint on the distance between the interdependent nodes [1][2][3][4][5][6][7][8][9]. However, many dependency links in the real world connect nearby nodes. For example, the international network of seaports and the network of national highways form a complex system. As seen recently from the effects of Hurricane Sandy in New York City, if a seaport is damaged, the city that depends on it will become isolated from the highway network due to the lack of fuel. Similarly, a city without roads cannot supply a seaport properly. However, a city will depend on a nearby seaport, not on one across the world. Li et. al [10] investigated distance-limited interdependent lattice networks by computer simulations and found that allowing only local interdependency links changed the resilience properties of the system. Here, we study the analytically tractable random networks. We study the mutual percolation of two interdependent random regular (RR) graphs. We build two identical networks, A and B, each of whose nodes are labeled 1...N . Each node is randomly connected by edges to exactly k other nodes, in such a way that the two networks have identical topologies. We then create one-to-one bidirectional dependency links, requiring that the shortest path between the interdependent nodes does not exceed an integer constant ℓ. Formally, we establish two isomorphisms between networks A and B, a topological isomorphism and a dependency isomorphism. The topological isomorphism is defined for each node A i as T (A i ) = B i and T (B i ) = A i . If Following the mutual percolation model described in Buldyrev et al.[1], we destroy a fraction (1 − p) of randomly selected nodes in A. Any nodes that, as a result, lost their connectivity links to the largest cluster (as defined in classical, single-network percolation theory [11,12]) are also destroyed. In the next stage, nodes in B that have their interdependent nodes in the other network destroyed are also destroyed. Consequently, the nodes that are isolated from the largest cluster in B as a result of the destruction of nodes in B are also destroyed. The iteration of this process, which alternates between the two networks, leads to a cascade of failures. The cascade ends when no more nodes fail in either network. The pair of remaining largest interdependent clusters in both networks is called a largest mutual componen...

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“…Depending on the structure of these networks and the fraction of the initially removed vertices, this cascade may eliminate the networks completely or remain a finite fraction of nodes and edges undamaged [8][9][10]. The specific phase transition between these two situations is hybrid, which means that it combines a discontinuity and a critical singularity [11]. In the simplest representative situation, in which each vertex in interdependent networks has not more than one interdependence, the interdependent networks are actually equivalent to multiplex networks [12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

“…In recent few years, the focus of interest of the complex networks studies essentially shifted from single networks to coupled networks, networks of networks, etc., including interdependent and multiplex networks [5][6][7][8][9][10][11][12][13]. In the interdependent networks, each vertex in a network depends on a vertex or several vertices in other networks.…”

Section: Introductionmentioning

confidence: 99%

Giant components in directed multiplex networks

2014

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We describe the complex global structure of giant components in directed multiplex networks which generalizes the well-known bow-tie structure, generic for ordinary directed networks. By definition, a directed multiplex network contains vertices of one type and directed edges of m different types. In directed multiplex networks, we distinguish a set of different giant components based on the existence of directed paths of different types between their vertices, such that for each type of edges, the paths run entirely through only edges of that type. If, in particular, m = 2, we define a strongly viable component as a set of vertices, in which for each type of edges, each two vertices are interconnected by at least two directed paths in both directions, running through the edges of only this type. We show that in this case, a directed multiplex network contains, in total, 9 different giant components including the strongly viable component. In general, the total number of giant components is 3 m . For uncorrelated directed multiplex networks, we obtain exactly the size and the emergence point of the strongly viable component and estimate the sizes of other giant components.

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Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions

Cited by 148 publications

References 39 publications

Networks formed from interdependent networks

Networks formed from interdependent networks

Cascading failures in networks with proximate dependent nodes

Giant components in directed multiplex networks

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