Robustness of a Network of Networks

By Jianxi Gao, Sergey V. Buldyrev, Shlomo Havlin, H. Eugene Stanley. Physical Review Letters, 2010

595 citations recorded by the source corpus.

Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of n interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of n fully dependent Erdős-Rényi (ER) networks, each of average degree k, we find that the giant component is P∞ =p[1-exp(-kP∞)](n) where 1-p is the initial fraction of removed nodes. This general result coincides for n = 1 with the known second-order phase transition for a single network. For any n>1 cascading failures occur and the percolation becomes an abrupt first-order transition. (ii) For a starlike network of n partially interdependent ER networks, P∞ depends also on the topology-in contrast to case (i). (iii) For a looplike network formed by n partially dependent ER networks, P∞ is independent of n.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/78b06209937b54006fdb0d8b97f972f35e90859e

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Citations in the corpus

Network resilience (2020, background): “limiting case of the general situation of n interdependent networks [646, 647].”
The structure and dynamics of multilayer networks (2014, background): “either all the layers percolate or none of them does [153].”
Suppressing cascades of load in interdependent networks (2011, background): “[22] Gao J, Buldyrev S, Havlin S, Stanley HE (2011) Robustness of a Network of Networks.”
Networks formed from interdependent networks (2011, background): “Here we review the generalization of the theory of a pair of interdependent networks 73,76 to a system of n interacting networks 96 , which can be graphically represented (Fig.”
Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization (2016, background): “The suppression principle plays a fundamental role in achieving discontinuous percolation on random graphs, inter-dependent networks [95, 100], adaptive networks [32, 116], and fractal networks [103].”
Robustness and resilience of complex networks (2024, unknown): “The theory is flexible and admits generalizations to an arbitrary number of coupled subsystems 19,155 , partial and asymmetric interdependency relations 156,157 , and so on.”
Multilayer networks (2013, background): “This yields a so-called network of networks [46, 47,109,110,127,128,130].”
Mathematical Formulation of Multilayer Networks (2013, unknown): “[16] J.”
Fantastic Generalization Measures and Where to Find Them (2019, methodology): “Our base model is inspired by the Network-in-Network (Gao et al., 2011).”
The extreme vulnerability of interdependent spatially embedded networks (2012, background): “1= (pc+1)(qc+δ)P ′ ∞(pc+ )=Aβ(pc+1)(qc+δ) β−1 (8)”
A quantitative method for assessing resilience of interdependent infrastructures (2017, unknown): “In order to quantify this capability, robustness (R) can be used, which is defined as the strength of the system to resist disruption [24, 25].”