RESUMEN
Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.
RESUMEN
In this paper we study the scaling behavior of the interface fluctuations (roughness) for a discrete model with conservative noise on complex networks. Conservative noise is a noise which has no external flux of deposition on the surface and the whole process is due to the diffusion. It was found that in Euclidean lattices the roughness of the steady state W(s) does not depend on the system size. Here, we find that for scale-free networks of N nodes, characterized by a degree distribution P(k) approximately k(-lambda), W(s) is independent of N for any lambda. This behavior is very different than the one found by Pastore y Piontti [Phys. Rev. E 76, 046117 (2007)] for a discrete model with nonconservative noise, which implies an external flux, where W(s) approximately ln N for lambda<3 , and was explained by nonlinear terms in the analytical evolution equation for the interface [La Rocca, Phys. Rev. E 77, 046120 (2008)]. In this work we show that in these processes with conservative noise the nonlinear terms are not relevant to describe the scaling behavior of W(s).
RESUMEN
In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution P(k) approximately k(-lambda) for lambda<3 [Pastore y Piontti, Phys. Rev. E 76, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks nonlinear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti for lambda<3.
