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.
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
.
DOI: 10.13306/j.1672-3813.2014.01.001
[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.
