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

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 P ̃ \~A(k) and P ̃ \~B(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 {{o}} {o}{}s-R\'enyi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees a ̃ \~ and b ̃ \~ in networks A and B, respectively, ${{μ}}_{{∞}}A={R}A[1{-}exp({-}{a ̃ \~{}{}}{{μ}}_{{∞}}B)][1{-}exp({-}a{{μ}}_{{∞}}A)]$ and ${{μ}}_{{∞}}B={R}B[1{-}exp({-}{b ̃ \~{}{}}{{μ}}_{{∞}}A)][1{-}exp({-}b{{μ}}_{{∞}}B)]$. In the limit of ${a ̃ \~{}{}}{→}{∞}$ and ${b ̃ \~{}{}}{→}{∞}$, both networks become independent, and our model becomes equivalent to a random attack on a single Erd {{o}} {o}{}s-R\'enyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.