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Many man-made networks support each other to provide efficient services and resources to the customers, despite that this support produces a strong interdependency between the individual networks. Thus an initial failure of a fraction $1-p$ of nodes in one network, exposes the system to cascade of failures and, as a consequence, to a full collapse of the overall system. Therefore it is important to develop efficient strategies to avoid the collapse by increasing the robustness of the individual networks against failures. Here, we provide an exact theoretical approach to study the evolution of the cascade of failures on interdependent networks when a fraction $alpha$ of the nodes with higher connectivity in each individual network are autonomous. With this pattern of interdependency we found, for pair of heterogeneous networks, two critical percolation thresholds that depend on $alpha$, separating three regimes with very different networks final sizes that converge into a triple point in the plane $p-alpha$. Our findings suggest that the heterogeneity of the networks represented by high degree nodes is the responsible of the rich phase diagrams found in this and other investigations.
Many real-world networks depend on other networks, often in non-trivial ways, to maintain their functionality. These interdependent networks of networks are often extremely fragile. When a fraction $1-p$ of nodes in one network randomly fails, the da
Modern world builds on the resilience of interdependent infrastructures characterized as complex networks. Recently, a framework for analysis of interdependent networks has been developed to explain the mechanism of resilience in interdependent netwo
Taking into account the fact that overload failures in real-world functional networks are usually caused by extreme values of temporally fluctuating loads that exceed the allowable range, we study the robustness of scale-free networks against cascadi
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 su
We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distribution