We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes NA and NB, (ii) degree distributions of connectivity links PA(k) and PB(k), (iii) degree distributions of support links P̃A(k) and P̃B(k), and (iv) random attack removes (1-RA)NA and (1-RB)NB nodes form the networks A and B, respectively. We find the fractions of nodes μ∞A and μ∞B which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erd\Hos-Rényi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees ã and b̃ in networks A and B, respectively, μ∞A=RA1-exp(-ãμ∞B)1-exp(-aμ∞A) and μ∞B=RB1-exp(-b̃μ∞A)1-exp(-bμ∞B). In the limit of ã→∞ and b̃→∞, both networks become independent, and our model becomes equivalent to a random attack on a single Erd\Hos-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
%0 Journal Article
%1 Shao2011Cascade
%A Shao, Jia
%A Buldyrev, Sergey V.
%A Havlin, Shlomo
%A Stanley, H. Eugene
%D 2011
%I American Physical Society
%J Physical Review E
%K resilience cascades interdependent-networks
%P 036116+
%R 10.1103/physreve.83.036116
%T Cascade of failures in coupled network systems with multiple support-dependence relations
%U http://dx.doi.org/10.1103/physreve.83.036116
%V 83
%X We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes NA and NB, (ii) degree distributions of connectivity links PA(k) and PB(k), (iii) degree distributions of support links P̃A(k) and P̃B(k), and (iv) random attack removes (1-RA)NA and (1-RB)NB nodes form the networks A and B, respectively. We find the fractions of nodes μ∞A and μ∞B which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erd\Hos-Rényi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees ã and b̃ in networks A and B, respectively, μ∞A=RA1-exp(-ãμ∞B)1-exp(-aμ∞A) and μ∞B=RB1-exp(-b̃μ∞A)1-exp(-bμ∞B). In the limit of ã→∞ and b̃→∞, both networks become independent, and our model becomes equivalent to a random attack on a single Erd\Hos-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
@article{Shao2011Cascade,
abstract = {{We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes NA and NB, (ii) degree distributions of connectivity links PA(k) and PB(k), (iii) degree distributions of support links P̃A(k) and P̃B(k), and (iv) random attack removes (1-RA)NA and (1-RB)NB nodes form the networks A and B, respectively. We find the fractions of nodes μ∞A and μ∞B which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erd\H{o}s-R\'{e}nyi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees ã and b̃ in networks A and B, respectively, μ∞A=RA[1-exp(-ãμ∞B)][1-exp(-aμ∞A)] and μ∞B=RB[1-exp(-b̃μ∞A)][1-exp(-bμ∞B)]. In the limit of ã→∞ and b̃→∞, both networks become independent, and our model becomes equivalent to a random attack on a single Erd\H{o}s-R\'{e}nyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Shao, Jia and Buldyrev, Sergey V. and Havlin, Shlomo and Stanley, H. Eugene},
biburl = {https://www.bibsonomy.org/bibtex/2d1f3bd2c312356d89b1af98f8d250fc5/nonancourt},
citeulike-article-id = {9746292},
citeulike-linkout-0 = {http://dx.doi.org/10.1103/physreve.83.036116},
citeulike-linkout-1 = {http://link.aps.org/abstract/PRE/v83/i3/e036116},
citeulike-linkout-2 = {http://link.aps.org/pdf/PRE/v83/i3/e036116},
doi = {10.1103/physreve.83.036116},
interhash = {97f4f919aa87808745f25249cb30c9d5},
intrahash = {d1f3bd2c312356d89b1af98f8d250fc5},
journal = {Physical Review E},
keywords = {resilience cascades interdependent-networks},
month = mar,
pages = {036116+},
posted-at = {2011-09-07 09:46:15},
priority = {2},
publisher = {American Physical Society},
timestamp = {2019-08-01T15:38:00.000+0200},
title = {{Cascade of failures in coupled network systems with multiple support-dependence relations}},
url = {http://dx.doi.org/10.1103/physreve.83.036116},
volume = 83,
year = 2011
}