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The models of k -core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in k -core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering k -core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, $d^{^{\prime}}_\textrm{f}$ , and correlation length exponent, $\nu^{^{\prime}}$ ) observed in d -dimensional IN spatial networks also exist with the same values in k -core percolation. That is, the fractality of the k -core giant component fluctuations is manifested by a fractal fluctuation dimension, $\widetilde d_\textrm{f} = 3/4$ , within a correlation size N ʹ that scales as $N^{^{\prime}} \propto (p-p_c)^{-\widetilde\nu}$ , with $\widetilde\nu = 2$ . Here we define, $\widetilde\nu \equiv d\cdot \nu^{^{\prime}}$ and $\widetilde{d}_\textrm{f} \equiv d^{^{\prime}}_\textrm{f}/d$ . This implies that both models, IN and k -core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k -core we find here that for k = 1 and k = 2 the influences are SR and in contrast for $k\unicode{x2A7E}3$ the influence is LR. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the k -core giant component and for IN as well as for any mixed-order transition are established. Our analysis enhances the comprehension of k -core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.

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]

The integrity and functionality of many real-world complex systems hinge on a small set of pivotal nodes, or influencers. In different contexts, these influencers are defined as either structurally important nodes that maintain the connectivity of networks, or dynamically crucial units that can disproportionately impact certain dynamical processes. In practice, identification of the optimal set of influencers in a given system has profound implications in a variety of disciplines. In this review, we survey recent advances in the study of influencer identification developed from different perspectives, and present state-of-the-art solutions designed for different objectives. In particular, we first discuss the problem of finding the minimal number of nodes whose removal would breakdown the network (i.e. the optimal percolation or network dismantle problem), and then survey methods to locate the essential nodes that are capable of shaping global dynamics with either continuous (e.g. independent cascading models) or discontinuous phase transitions (e.g. threshold models). We conclude the review with a summary and an outlook., (© The authors 2019. Published by Oxford University Press. All rights reserved.)