Percolation in interdependent networks with groups and heterogeneous weak interdependency strength under targeted attack

Abstract Pairwise dependencies in interdependent networks exist not only between ordinary nodes but also between groups of nodes, where nodes cooperate and form groups to increase their robustness to risks and each group can be considered a ``supernode''. The interdependencies between supernodes are universal but not always complete and homogeneous. \textcolor{red}{In this paper, we study the robustness of interdependent networks with groups and heterogeneous weak interdependency strength under targeted attack, where the interdependency strength could vary between different supernodes.} We identify several types of percolation transitions, namely discontinuous, continuous, and hybrid, which are determined by the structure of the supernodes, the strengths of the interdependencies, and their distributions. We first apply this mathematical framework to the Erdös-Rényi and Scale-free interdependent networks with groups and homogeneous weak interdependency strength to calculate analytically and numerically the values of the phase transition thresholds and the critical interdependency strengths that distinguish different types of transitions. We then present examples of three distributions with different heterogeneous interdependency strengths. Our theoretical and numerical predictions are in close agreement, demonstrating that weaker interdependence strengths and higher levels of interdependence heterogeneity can enhance the robustness of interdependent networks under targeted attack.