Vertex-degree sequences in complex networks: New characteristics and applications

Many complex networks exhibit a scale-free vertex-degree distribution in a power-law form ck−γ, where k is the vertex-degree variable and c and γ are constants. To better understand the mechanism of power-law formation in real-world networks, it is effective to explore and analyze their vertex-degree sequences. We had shown before that, for a scale-free network of size N, if its vertex-degree sequence is k11, then the length l of the vertex-degree sequence is of order logN. In the present paper, we further study complex networks with an exponential vertex-degree distribution and prove that the same conclusion also holds. In addition, we verify our claim by showing many real-world examples. We finally discuss some applications of the new finding in various fields of science and technology.

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  M. Newman,et al.  Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Francesc Comellas,et al.  Modeling complex networks with self-similar outerplanar unclustered graphs ☆ , 2009 .

[4]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[6]  Wenjun Xiao,et al.  Characterizing vertex-degree sequences in scale-free networks ☆ , 2014 .

[7]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[8]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[9]  S. N. Dorogovtsev,et al.  Pseudofractal scale-free web. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  S. Bornholdt,et al.  Scale-free topology of e-mail networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  J. S. Andrade,et al.  Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. , 2004, Physical review letters.

[12]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[13]  Ramon Ferrer i Cancho,et al.  The small world of human language , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[14]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[15]  Wenjun Xiao,et al.  On Necessary Conditions for Scale-Freedom in Complex Networks with Applications to Computer Communication Systems , 2008, 2008 First International Conference on Intelligent Networks and Intelligent Systems.

[16]  Guanrong Chen,et al.  A Small-World Model of Scale-Free Networks: Features and Verifications , 2011 .

[17]  M. Markus,et al.  On-off intermittency and intermingledlike basins in a granular medium. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.