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.
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