We study a problem of failure of two interdependent networks in the case of correlated degrees of mutually dependent nodes. We assume that both networks (A and B) have the same number of nodes N connected by the bidirectional dependency links establishing a one-to-one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links, i.e. their degrees coincide. This implies that both networks have the same degree distribution P (k). We call such networks correspondently coupled networks (CCN). We assume that the nodes in each network are randomly connected. We define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes which belong to the mutual giant component remain functional. We assume that initially a 1 − p fraction of nodes are randomly removed due to an attack or failure and find analytically, for an arbitrary P (k), the fraction of nodes µ(p) which belong to the mutual giant component. We find that the system undergoes a percolation transition at certain fraction p = p c which is always smaller than the p c for randomly coupled networks with the same P (k). We also find that the system undergoes a first order transition at p c > 0 if P (k) has a finite second moment. For the case of scale free networks with 2 < λ ≤ 3, the transition becomes a second order transition. Moreover, if λ < 3 we find p c = 0 as in percolation of a single network. For λ = 3 we find an exact analytical expression for p c > 0. Finally, we find that the robustness of CCN increases with the broadness of their degree distribution.

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The scattering of light from a disordered medium is considered. The effects of polarization and the transverse nature of the light on the coherent backscattering arising from weak localization phenomena are calculated. The angular line shape of the scattered light in an infinite medium and the reAected light from a half-space and a slab is found. The scattered light contains several components and the line shape is different from that found for scalar~aves. The results are in qualitative agreement with recent experiments.

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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

Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal and random lattices is studied. The adsorption process is modeled by using random sequential adsorption (RSA) algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdos-Renyi random graphs. The number of sites is M = L d for Euclidean and fractal lattices, where L is a characteristic length of the system. In the case of random graphs it does not exist such characteristic length, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. The paper concentrates on measuring (1) the probability W L(M ) (θ) that a lattice composed of L d (M ) elements reaches a coverage θ, and (2) the exponent νj characterizing the so-called "jamming transition". The results obtained for Euclidean, fractal and random lattices indicate that the main quantities derived from the jamming probability W L(M ) (θ) behave asymptotically as M 1/2 . In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M 1/2 = L d/2 = L 1/ν j , and νj = 2/d. arXiv:1907.02572v2 [cond-mat.stat-mech]

We study the cascading failure of networks due to overload, using the betweenness centrality of a node as the measure of its load following the Motter and Lai model. We study the fraction of survived nodes at the end of the cascade p_{f} as a function of the strength of the initial attack, measured by the fraction of nodes p that survive the initial attack for different values of tolerance α in random regular and Erdös-Renyi graphs. We find the existence of a first-order phase-transition line p_{t}(α) on a p-α plane, such that if pp_{t}, p_{f} is large and the giant component of the network is still present. Exactly at p_{t}, the function p_{f}(p) undergoes a first-order discontinuity. We find that the line p_{t}(α) ends at a critical point (p_{c},α_{c}), in which the cascading failures are replaced by a second-order percolation transition. We find analytically the average betweenness of nodes with different degrees before and after the initial attack, we investigate their roles in the cascading failures, and we find a lower bound for p_{t}(α). We also study the difference between localized and random attacks.