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Complex networks are now the focus of many branches of research. Particularly, the scale-free property of some networks is of great interest, due to their importance and pervasiveness. Recent studies have shown that in some complex networks, e.g., transportation networks and social collaboration networks, the degree distribution follows the so-called “shifted power law” (or Mandelbrot law) . This study analyzes some evolving networks that grow with linear preferential attachments. Recent results for the quotient Gamma function are used to prove the asymptotic Mandelbrot law for the degree distribution in certain conditions. The best fit values for the scaling exponent, γ, and the shifting coefficient, c, can be directly calculated using Bernoulli polynomial functions. The study proves that the degree distribution of some complex networks follows an asymptotic Mandelbrot law with linear preferential attachment depicted .