The Asymptotic Mandelbrot Law of Some Evolution Networks

arXiv:1106.3740v2 [physics.data-an] 21 Jun 2011 The Asymptotic Mandelbrot Law of Some Evolution Networks Li Li November 8, 2018 Abstract In this letter, we study some evolution networks that grow with linear preferential attachment. Based upon some recent results on the quotient Gamma function, we give a rigorous proof of the asymptotic Mandelbrot law for the degree distribution pk ∝(k + c)−γ in certain conditions. We also analytically derive the best fitting values for the scaling exponent γ and the shifting coefficient c. Complex networks are now the joint focus of many branches of research[1−3]. Particularly, the scale-free property of some networks attracts continuous inter- ests, due to their importance and pervasiveness[4−6]. In short, this property means that the degree distribution of a network obeys a power law P(k) ∝k−γ, where k is the degree and P(k) is the corresponding probability density, and the scaling exponent γ is a constant. A pioneering model that generates power-law degree distribution was presented by Barab´asi and Albert (BA)[4]. In recent studies, it was found that in some complex networks, e.g. trans- portation networks[7] and social collaboration networks[8], the degree distribu- tion follows the so-called “shifted power law”[9] P(k) ∝(k + c)−γ, where the shifting coefficient c is another constant. This property is also called “Mandel- brot law”[10]. To understand the origins of such Mandelbrot law, Ren, Yang and Wang[11] proposed a interesting growing network that is generated with linear preferential attachment. In such networks, there exits a recursive dependence relationship between every two consecutive degrees p(k)  k + 2 + 2mβ 1 −β  = p(k −1)  k + 2mβ 1 −β −1  (1) where where k = 2, ..., n, n is the number of nodes. m is a positive integer constant and β ∈[0, 1] is another constant. Defining a = 2mβ 1−β −1, b = 2+2mβ 1−β , we can abbreviate Eq.(1) as pk [k + b] = pk−1 [k + a] (2) To derive the asymptotic of the degree distribution, Ren, Yang and Wang[11] studied the following three kinds of approximations: 1 I) forward-difference approximation, assuming dp(k) dk ≈p(k) −p(k −1) = p(k) −k + b k + ap(k) = a −b k + ap(k) (3) we have an estimation of the power-law as p(k) ∝(k + a)−(b−a) (4) II) backward-difference approximation, assuming dp(k) dk ≈p(k + 1) −p(k) = k + 1 + a k + 1 + b p(k) −p(k) = a −b k + 1 + bp(k) (5) we have another estimation of the power-law as p(k) ∝(k + b + 1)−(b−a) (6) III) Suppose we must have a Mandelbrot law p(k) ∝(k + c)−γ. As a result, we have p(k −1) ∝(k −1 + c)−γ. Substitute these two approximations in the logarithm type of Eq.(2), we have ln k + a k + b = ln p(k) p(k −1) = −γ ln(k + c) + γ ln(k −1 + c) (7) Rewrite Eq.(7) as ln 1 + a 1 k 1 + b 1 k = γ ln 1 + (c −1) 1 k 1 + c 1 k (8) and apply the second order Taylor expansion of 1 k in Eq.(8), we have p(k) ∝  k + b + a + 1 2 −(b−a) (9) All these three estimations indicates that the scaling exponent of the degree distribution should be −(b −a). Simulation results[11] show that Eq.(9) gives the best approximation accuracy of the empirical distributions. However, we still need a rigorous proof of this interesting finding. Indeed, further assuming Pn k=1 p(k) = 1, we have the following matrix equa- tion   2 + a −(2 + b) 0 ... 0 0 3 + a −(3 + b) ... 0 ... 0 0 ... n + a −(n + b) 1 1 ... 1 1     p(1) p(2) ... p(n −1) p(n)   =   0 0 ... 0 1   (10) 2 Using Gaussian elimination algorithm, we can directly solve p(n) from Eq.(10) as p(n) =  1 + n + b n + a + ... + n Y j=2 j + b j + a   −1 =  1 + n X i=2 n Y j=i j + b j + a   −1 (11) Based on the recursive relationship Eq.(2), for a given n, we have p(k) = p(n)   n Y j=k+1 j + b j + a  = p(n) Qn j=1 j+b j+a Qk j=1 j+b j+a ! = p(n)   n Y j=1 j + a j + b     k Y j=1 j + a j + b   (12) where k = 1, ..., n −1. It is well known that for Gamma function Γ(z), we have Γ(z + 1) = zΓ(z). So, we get (j + b) = Γ(j + 1 + b) Γ(j + b) , (j + a) = Γ(j + 1 + a) Γ(j + a) (13) where j = 1, ..., n −1. From Eq.(12), we have p(k) = p(n)   n Y j=1 j + a j + b     k Y j=1 Γ(j + 1 + a) Γ(j + a)     k Y j=1 Γ(j + b) Γ(j + 1 + b)   = p(n)   n Y j=1 j + a j + b  Γ(k + 1 + a) Γ(1 + a) Γ(1 + b) Γ(k + 1 + b) = λ · Γ(k + 1 + a) Γ(k + 1 + b) (14) where λ = p(n) Qn j=1 j+a j+b  Γ(1+b) Γ(1+a) is a constant. Eq.(14) indicates that p(k) has the same asymptotic behavior of Γ(k+1+a) Γ(k+1+b) . Actually, the quotient of two Gamma functions is a difficult problem that re- ceived consistent attentions[12−15]. There are numbers of approximation formu- las which are not accurate enough for the above applications. Fortunately, an important results had been obtained very recently[15] as Lemma 1[15] Given two constants s and t, when x →∞, we have  Γ(x + t) Γ(x + s)  1 t−s ∼ ∞ X k=0 Fk(t, s)x−n+1 (15) 3 where Fk(t, s) are the polynomials of order n defined by F0(t, s) = 1 (16) Fn(t, s) = 1 n n X k=1 (−1)k+1 Bk+1(t) −Bk+1(s) (k + 1)(t −s)

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