On the Fractality of Complex Networks: Covering Problem, Algorithms and Ahlfors Regularity

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Abstract

In this paper, we revisit the fractality of complex network by investigating three dimensions with respect to minimum box-covering, minimum ball-covering and average volume of balls. The first two dimensions are calculated through the minimum box-covering problem and minimum ball-covering problem. For minimum ball-covering problem, we prove its NP-completeness and propose several heuristic algorithms on its feasible solution, and we also compare the performance of these algorithms. For the third dimension, we introduce the random ball-volume algorithm. We introduce the notion of Ahlfors regularity of networks and prove that above three dimensions are the same if networks are Ahlfors regular. We also provide a class of networks satisfying Ahlfors regularity.

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Wang, L., Wang, Q., Xi, L., Chen, J., Wang, S., Bao, L., … Zhao, L. (2017). On the Fractality of Complex Networks: Covering Problem, Algorithms and Ahlfors Regularity. Scientific Reports, 7. https://doi.org/10.1038/srep41385

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