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 N^A and N^B, (ii) degree distributions of connectivity links P^A (k) and P^B (k), (iii) degree distributions of support links P \~{}^A (k) and P \~{}^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 _^A and _^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 os-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, _^A=R^A1-exp (-a \~_^B) 1-exp (-a_^A) and _^B=R^B1-exp (-b \~_^A) 1-exp (-b_^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 os-R\'enyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.