[formula omitted]-core percolation and node protection strategy for edge-coupling partially interdependent networks

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In the real-world, there exist mutual dependencies among not only nodes but also edges. Consequently, the study on the robustness of edge-coupling interdependent networks has emerged as a highly focused area. By analyzing the characteristics of interdependent networks, a relevant k-core percolation model is proposed. Given that for any degree distributions of edge-coupling interdependent networks, the relative sizes of k-core and corona clusters, as well as percolation thresholds are obtained by the derivation of self-consistent equations, and then the phase transition for k-core percolation is analyzed. It is found that when k≥3, with any edge-coupling strengths γA and γB in the networks, the transitions for k-core percolation always occur as first-order discontinuous one. By integrating the theoretical findings with experimental simulation, the significant influence of corona clusters on both the k-core structure and networks robustness is discovered. In view of the 3-core structure upon edge-coupling partially interdependent networks, when γB≥0.67, the robustness of which is gradually weakened with the increase of coupling strengths γA and γB. Nonetheless, significant variations in the edge-coupling strength γ still have limited impact on network robustness. Then a node protection strategy CoronaProtect is put forward, it can be found that with the increase of protection intensity α, the robustness of network is enhanced, while the phase transition of k-core may be changed as well. The effectiveness of the given protection strategy is finally verified through the ER−ER and SF−SF networks. Nevertheless, the given k-core percolation model and protection strategy can not only give help in understanding the hierarchy structure of networks but also provide some guidance in enhancing the resilience of networks against attacks.