The Asymptotic Mandelbrot Law of Some Evolution Networks

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 $p_k \propto (k + c)^{-\gamma}$ in certain conditions. We also analytically derive the best fitting values for the scaling exponent $\gamma$ and the shifting coefficient $c$.