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) over tilde (A)(k) and (P) over tilde (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 mu(A)(infinity) and mu(B)(infinity) 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-Renyi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees (a) over tilde and (b) over tilde in networks A and B, respectively, mu(A)(infinity) = R-A[1 - exp (-(a) over tilde mu(B)(infinity))][1 - exp (-a mu(A)(infinity))] and mu(B)(infinity) = R-B[1 - exp (-(b) over tilde mu(A)(infinity))][1 - exp (-b mu(B)(infinity))]. In the limit of (a) over tilde -> infinity and (b) over tilde -> infinity, both networks become independent, and our model becomes equivalent to a random attack on a single Erdos-Renyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.