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

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.