The renormalization group approach to coexistence dynamics on complex networks with randomness

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We derived the analytical expressions for calculation of the percolation threshold and the giant component fraction of complex networks with strong degree-degree correlation against arbitrary types of degree based attack. With these expressions, we identified a network structure that consists of a set of weakly interconnected random regular graphs and that significantly improves the well-known vulnerability of scale-free networks against targeted high-degree node attack. We also investigated collective dynamics on complex networks with hierarchical or module structure. Our findings include instructive and useful results such that, in the prisoners' dilemma game, the connected cluster of"permanent cooperators"exhibits a percolation transition.