Robustness of a network formed of spatially embedded networks

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with (i) unconstrained dependency links and (ii) dependency links restricted to a maximum Euclidean length r. Analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations. For the case of two fully interdependent spatially embedded networks it was found [Li et al., Phys. Rev. Lett. 108, 228702 (2012)] that the system undergoes a first-order phase transition only for r>r(c) ≈ 8. We find here that for treelike networks of networks (composed of n networks) r(c) significantly decreases as n increases and rapidly (n ≥ 11) reaches its limiting value of 1. For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations that there is a certain fraction of dependent nodes, q(max), above which the entire network structure collapses even if a single node is removed. The value of q(max) decreases quickly with m, the degree of the random regular network of networks. Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse. The theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically.