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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 γ A and γ B 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 γ B ≥ 0. 67 , the robustness of which is gradually weakened with the increase of coupling strengths γ A and γ B. Nonetheless, significant variations in the edge-coupling strength γ still have limited impact on network robustness. Then a node protection strategy C o r o n a P r o t e c t 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 E R − E R and S F − S F 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. • A k-core percolation model for edge-coupled partially interdependent networks is put forward. • The relative sizes of k-core, corona clusters and percolation thresholds are gained. • The relationship of coupling strength on network robustness is obtained. • A node protection strategy based on the k-core percolation is proposed. [ABSTRACT FROM AUTHOR]

We study the transition in the functional networks that characterize the human brains' conscious-state to an unconscious subliminal state of perception by using k -core percolation. We find that the most inner core (i.e., the most connected kernel) of the conscious-state functional network corresponds to areas which remain functionally active when the brain transitions from the conscious-state to the subliminal-state. That is, the inner core of the conscious network coincides with the subliminal-state. Mathematical modeling allows to interpret the conscious to subliminal transition as driven by k -core percolation, through which the conscious state is lost by the inactivation of the peripheral k -shells of the conscious functional network. Thus, the inner core and most robust component of the conscious brain corresponds to the unconscious subliminal state. This finding imposes constraints to theoretical models of consciousness, in that the location of the core of the functional brain network is in the unconscious part of the brain rather than in the conscious state as previously thought. [ABSTRACT FROM AUTHOR]