Abstract：An important concept in complex networks is the degree distribution has a power-law tail. In order to determine the degree exponent of geometric growth networks, people need to use complementary degree distribution. Then, a theoretical problem is proposed: for discrete distributions, if complementary distribution has a power-law tail, distribution has a power-law tail, and vice versa. We found that this conclusion is not true in general. A necessary and sufficient condition that distribution and complementary distribution also has a power-law tail is given in this paper.
阎春宁, 山石, 史定华. 幂律思考系列文章1——论Barabási律与Pareto律互不包含[J]. 复杂系统与复杂性科学, 2014, 11(1): 1-4.
YAN Chun-ning, SHAN Shi, SHI Ding-hua. Power Law Thinking Series 1—The Barabási Law and Pareto Law are Not Mutually Included[J]. Complex Systems and Complexity Science, 2014, 11(1): 1-4.
[1] Barabási A L, Albert R. Emergence of scaling in random networks[J]. Science, 1999, 286:509-512. [2] Newmana M E J. The structure and function of complex networks[J]. SIAM Reviw, 2003, 45:167-256. [3] Pareto V. Cours d'Economie politique professé à l'Université de Lausanne[M]. Lausanne, François Rouge, 1896. [4] Dorogovtsev S N, Gotsev A V, Mendes J F F. Pseudofractal scale-free web[J]. Physical Review E, 2002, 65（6):066122. [5] Durrett R. Probability:Theory and Examples[M]. New York:Combridge University Press, 2010. [6] Mikosch T. Regular Variation, Subexponentiality and Their Applications in Probability Theory[M]. Eindhoven:Eindhoven University of Technology, 1999 [7] Shi D H. Theory of Network Degree Distribution[M]. Beijing:Higher Education Press, 2011. [8] Andrade J S, Herrmann H J, Andrade R F S, et al. Apollonian networks[J]. Physical Review Letters, 2005, 94（1):018702. [9] Andrade J S, Herrmann H J, Andrade R F S, et al. Erratum:apollonian network[J]. Physical Review Letters, 2009, 102（7):079901. [10] Mungan M. Comment on “apollonian networks”[J]. Physical Review Letters, 2011, 106（2):029802.
