![]() This provides routes to manipulate the percolation threshold and the level of conductivity in the final product. We argue that the host matrix in which the nanotubes are dispersed controls this threshold through the interactions it induces between them during processing and through the degree of connectedness that must be set by the tunneling distance of electrons, at least in the context of conductivity percolation. C., BigBrain: An ultrahigh-resolution 3D human brain model, Science 340 1472–1475.We apply continuum connectedness percolation theory to realistic carbon nanotube systems and predict how bending flexibility, length polydispersity, and attractive interactions between them influence the percolation threshold, demonstrating that it can be used as a predictive tool for designing nanotube-based composite materials. Amunts, K., Lepage, C., Borgeat, L., Mohlberg, H., Dickscheid, T., Rousseau, M., Bludau, S., Bazin, P.-L., Lewis, L. and Clauset, A., Scale-free networks are rare, Nat. Palka, Z., Extreme degrees in random graphs, J. and Vigna, S., Layered label propagation: A multiresolution coordinate-free ordering for compressing social networks, in Proc. K., Phase Transitions in Combinatorial Optimization Problems (Wiley-VCH, Weinheim, 2005). and Parisi, G., The Bethe lattice spin glass revisited, Eur. J., Random graphs with arbitrary degree distributions and their applications, Phys. Shang, Y., On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs, Open Math. and Barabási, A.-L., Control principles of complex systems, Rev. F., k -core organization of complex networks, Phys. and Pósfai, M., Core percolation on complex networks, Phys. and Golinelli, O., Core percolation in random graphs: A critical phenomena analysis, Eur. F., Core organization of directed complex networks, Phys. Shang, Y., Attack robustness and stability of generalized k -cores, New J. N., Generalization of core percolation on complex networks, Phys. Foundations of Computer Science (IEEE, Piscataway, 1981), pp. and Sipser, M., Maximum matching in sparse random graphs, in Proc. and Havlin, S., Percolation of localized attack on complex networks, New J. Shang, Y., Localized recovery of complex networks against failure, Sci. and Havlin, S., Localized attack on networks with clustering, New J. Dong, G., Xiao, H., Wang, F., Du, R., Shao, S., Tian, L., Stanley, H. and Bianconi, G., Redundant interdependencies boost the robustness of multiplex networks, Phys. and Havlin, S., Dynamic interdependence and competition in multilayer networks, Nat. and Havlin, S., Catastrophic cascade of failures in interdependent networks, Nature 464 1025–1028. and Moreno, Y., Multilayer networks in a nutshell, Annu. and Overink, F.-J., Priming and warnings are not effective to prevent social engineering attacks, Comput. and Madnick, S., Systematically understanding the cyber attack business: A survey, ACM Comput. and Gao, J., Local floods induce large-scale abrupt failures of road networks, Nat. and Vespignani, A., Resilience management during large-scale epidemic outbreaks, Sci. Massaro, E., Ganin, A., Perra, N., Linkov, I. Shang, Y., Unveiling robustness and heterogeneity through percolation triggered by random-link breakdown, Phys. and Wang, Z., Attack robustness and centrality of complex networks, PLoS ONE 8 e59613. and Arenas, A., Modeling structure and resilience of the dark network, Phys. and Barabási, A.-L., Error and attack tolerance of complex networks, Nature 406 378–382. J., Networks: An Introduction (Oxford University Press, Oxford, 2010). and Havlin, S., Complex Networks: Structure, Robustness and Function (Cambridge University Press, Cambridge, 2010). This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for G ( 2, 1 ) -core and discontinuous phase transition for G ( k, L ) -core with any other combination of k and L. The resulting subgraph is referred to as G ( k, L ) -core, extending the recently proposed G k -core and classical core of a network. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than k and their neighbors within L -hop distance are removed progressively from the network. Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown.
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