title: Cascade of failures in coupled network systems with multiple support-dependence relations.
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 N{A} and N{B}, (ii) degree distributions of connectivity links P{A}(k) and P{B}(k), (iii) degree distributions of support links P̃{A}(k) and P̃{B}(k), and (iv) random attack removes (1-R{A})N{A} and (1-R{B})N{B} 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 Erdős-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}=R{A}[1-exp(-ãμ{∞}{B})][1-exp(-aμ{∞}{A})] and μ{∞}{B}=R{B}[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 Erdős-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: Jia
authors.initials: J
authors.lastname: Shao
authors.forename: Sergey V
authors.initials: SV
authors.lastname: Buldyrev
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley
mesh_terms: D008962:Models, Theoretical; D013269:Stochastic Processes