We study the mutual percolation of a system composed of two interdependent random regular networks. We introduce a notion of distance to explore the effects of the proximity of interdependent nodes on the cascade of failures after an initial attack. We find a non-trivial relation between the nature of the transition through which the networks disintegrate and the parameters of the system, which are the degree of the nodes and the maximum distance between interdependent nodes. We explain this relation by solving the problem analytically for the relevant set of cases.Previous studies of the robustness of interdependent networks have focused on networks in which there is no constraint on the distance between the interdependent nodes [1][2][3][4][5][6][7][8][9]. However, many dependency links in the real world connect nearby nodes. For example, the international network of seaports and the network of national highways form a complex system. As seen recently from the effects of Hurricane Sandy in New York City, if a seaport is damaged, the city that depends on it will become isolated from the highway network due to the lack of fuel. Similarly, a city without roads cannot supply a seaport properly. However, a city will depend on a nearby seaport, not on one across the world. Li et. al [10] investigated distance-limited interdependent lattice networks by computer simulations and found that allowing only local interdependency links changed the resilience properties of the system. Here, we study the analytically tractable random networks. We study the mutual percolation of two interdependent random regular (RR) graphs. We build two identical networks, A and B, each of whose nodes are labeled 1...N . Each node is randomly connected by edges to exactly k other nodes, in such a way that the two networks have identical topologies. We then create one-to-one bidirectional dependency links, requiring that the shortest path between the interdependent nodes does not exceed an integer constant ℓ. Formally, we establish two isomorphisms between networks A and B, a topological isomorphism and a dependency isomorphism. The topological isomorphism is defined for each node A i as T (A i ) = B i and T (B i ) = A i . If Following the mutual percolation model described in Buldyrev et al.[1], we destroy a fraction (1 − p) of randomly selected nodes in A. Any nodes that, as a result, lost their connectivity links to the largest cluster (as defined in classical, single-network percolation theory [11,12]) are also destroyed. In the next stage, nodes in B that have their interdependent nodes in the other network destroyed are also destroyed. Consequently, the nodes that are isolated from the largest cluster in B as a result of the destruction of nodes in B are also destroyed. The iteration of this process, which alternates between the two networks, leads to a cascade of failures. The cascade ends when no more nodes fail in either network. The pair of remaining largest interdependent clusters in both networks is called a largest mutual componen...

We study the mutual percolation of two interdependent lattice networks ranging from two to seven dimensions, denoted as D. We impose that the length of interdependent links connecting nodes in the two lattices be less than or equal to a certain value, r. For each value of D and r, we find the mutual percolation threshold, pc [D, r] below which the system completely collapses through a cascade of failures following an initial destruction of a fraction (1 − p) of the nodes in one of the lattices. We find that for each dimension, D < 6, there is a value of r = rI > 1 such that for r ≥ rI the cascading failures occur as a discontinuous first order transition, while for r < rI the system undergoes a continuous second order transition, as in the classical percolation theory. Remarkably, for D = 6, rI = 1 which is the same as in random regular (RR) graphs with the same degree (coordination number) of nodes. We also find that in all dimensions, the interdependent lattices reach maximal vulnerability (maximal pc [D, r]) at a distance r = rmax > rI , and for r > rmax the vulnerability starts to decrease as r → ∞. However the decrease becomes less significant as

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