Cascade of failures in coupled network systems with multiple support-dependence relations

Jia Shao, Sergey V. Buldyrev, Shlomo Havlin, H. Eugene Stanley

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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 Erdos-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=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 Erdos-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.

Original languageEnglish
Article number036116
JournalPhysical Review E
Issue number3
StatePublished - 29 Mar 2011


FundersFunder number
Seventh Framework Programme231807


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