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.
Highlights
In this paper, we revisit the fractality of complex network by investigating three dimensions dB5, dball[6] and df[7] with respect to minimum box-covering, minimum ball-covering and average volume of balls
#V /N l ∼ ldB and #V /Bl ∼ l, dball where dB is the fractal dimension defined by Song, Havlin and Makse[5], and dball is defined by Gao, Hu and Di6
Given a function f : V →, suppose we sort nodes according to values of f in nondecreasing order: If f is the degree function, we can obtain degree-order ball-covering algorithm (DOBC); If f (x) = #B(x, l) and, we obtain volume-order ball-covering algorithm (VOBC)
Summary
Networks: Covering Problem, Algorithms and Ahlfors Regularity received: 22 August 2016 accepted: 20 December 2016. 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. We revisit the fractality of complex network by investigating three dimensions dB5, dball[6] and df[7] with respect to minimum box-covering, minimum ball-covering and average volume of balls. The compact box burning algorithm (CBB)[8,9] and random ball-covering algorithm[6] are proposed to calculate dB and dball respectively. The minimum box-covering problem and minimum ball-covering problem are NP-complete, which are proved rigorously in Theorem 1 and Proposition 2 respectively. For Ahlfors regular networks, the random ball-volume algorithm is efficient to obtain the above three fractal dimensions
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