March  2012, 7(1): 59-69. doi: 10.3934/nhm.2012.7.59

A sufficient condition for classified networks to possess complex network features

1. 

College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China, China, China

2. 

College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210046, China

Received  March 2011 Revised  December 2011 Published  February 2012

We investigate network features for complex networks. A sufficient condition for the limiting random variable to possess the scale free property and the high clustering property is given. The uniqueness and existence of the limit of a sequence of degree distributions for the process is proved. The limiting degree distribution and a lower bound of the limiting clustering coefficient of the graph-valued Markov process are obtained as well.
Citation: Xianmin Geng, Shengli Zhou, Jiashan Tang, Cong Yang. A sufficient condition for classified networks to possess complex network features. Networks & Heterogeneous Media, 2012, 7 (1) : 59-69. doi: 10.3934/nhm.2012.7.59
References:
[1]

R. Albert and A.-L. Barabási, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.  Google Scholar

[2]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Mordern Physics, 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.  Google Scholar

[3]

D. J. Aldous, A tractable complex network model based on the stochastic mean-field model of distance, in "Complex Networks," Lecture Notes in Physics, 650, Springer, Berlin, (2004), 51-87.  Google Scholar

[4]

D. J. Aldous and W. S. Kendall, Short-length routes in low-cost networks via Poisson line patterns, Advances in Applied Probability, 40 (2008), 1-21. doi: 10.1239/aap/1208358883.  Google Scholar

[5]

H. G. Bartel, H. J. Mucha and J. Dolata, On a modified graph-theoretic partitioning method of cluster analysis, Match-Communications in Mathematical and in Computer Chemistry, 48 (2003), 209-233. Google Scholar

[6]

B. Bollobás, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs, Random Structures and Algorithms, 31 (2007), 3-122. doi: 10.1002/rsa.20168.  Google Scholar

[7]

A. Diaz-Guilera, Complex networks: Statics and dynamics, Advanced Summer School in Physics 2006, 885 (2007), 107-128. Google Scholar

[8]

Z. P. Fan, G. R. Chen and Y. N. Zhang, A comprehensive multi-local-world model for complex networks, Physics Letters A, 373 (2009), 1601-1605. doi: 10.1016/j.physleta.2009.02.072.  Google Scholar

[9]

A. Ganesh and F. Xue, On the connectivity and diameter of small-world networks, Advances in Applied Probability, 39 (2007), 853-863. doi: 10.1239/aap/1198177228.  Google Scholar

[10]

P. Holme and B. J. Kim, Growing scale-free networks with tunable clustering, Physical Review E, 65 (2002), 026107. Google Scholar

[11]

G. Lee and G. I. Kim, Degree and wealth distribution in a network, Physica A-Statistical Mechanics and its Applications, 383 (2007), 677-686. Google Scholar

[12]

N. Miyoshi, T. Shigezumi, R. Uehara and O. Watanabe, Scale free interval graphs, Theoretical Computer Science, 410 (2009), 4588-4600. doi: 10.1016/j.tcs.2009.08.012.  Google Scholar

[13]

Y. Ou and C.-Q. Zhang, A new multimembership clustering method, Journal of Industrial and Management Optimization, 3 (2007), 619-624. doi: 10.3934/jimo.2007.3.619.  Google Scholar

[14]

M. M. Sørensen, b-tree facets for the simple graph partitioning polytope, Journal of Combinatorial Optimization, 8 (2004), 151-170. doi: 10.1023/B:JOCO.0000031417.96218.26.  Google Scholar

[15]

J. Szymański, Concentration of vertex degrees in a scale-free random graph process, Random Structures and Algorithms, 26 (2005), 224-236. doi: 10.1002/rsa.20065.  Google Scholar

[16]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.  Google Scholar

show all references

References:
[1]

R. Albert and A.-L. Barabási, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.  Google Scholar

[2]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Mordern Physics, 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.  Google Scholar

[3]

D. J. Aldous, A tractable complex network model based on the stochastic mean-field model of distance, in "Complex Networks," Lecture Notes in Physics, 650, Springer, Berlin, (2004), 51-87.  Google Scholar

[4]

D. J. Aldous and W. S. Kendall, Short-length routes in low-cost networks via Poisson line patterns, Advances in Applied Probability, 40 (2008), 1-21. doi: 10.1239/aap/1208358883.  Google Scholar

[5]

H. G. Bartel, H. J. Mucha and J. Dolata, On a modified graph-theoretic partitioning method of cluster analysis, Match-Communications in Mathematical and in Computer Chemistry, 48 (2003), 209-233. Google Scholar

[6]

B. Bollobás, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs, Random Structures and Algorithms, 31 (2007), 3-122. doi: 10.1002/rsa.20168.  Google Scholar

[7]

A. Diaz-Guilera, Complex networks: Statics and dynamics, Advanced Summer School in Physics 2006, 885 (2007), 107-128. Google Scholar

[8]

Z. P. Fan, G. R. Chen and Y. N. Zhang, A comprehensive multi-local-world model for complex networks, Physics Letters A, 373 (2009), 1601-1605. doi: 10.1016/j.physleta.2009.02.072.  Google Scholar

[9]

A. Ganesh and F. Xue, On the connectivity and diameter of small-world networks, Advances in Applied Probability, 39 (2007), 853-863. doi: 10.1239/aap/1198177228.  Google Scholar

[10]

P. Holme and B. J. Kim, Growing scale-free networks with tunable clustering, Physical Review E, 65 (2002), 026107. Google Scholar

[11]

G. Lee and G. I. Kim, Degree and wealth distribution in a network, Physica A-Statistical Mechanics and its Applications, 383 (2007), 677-686. Google Scholar

[12]

N. Miyoshi, T. Shigezumi, R. Uehara and O. Watanabe, Scale free interval graphs, Theoretical Computer Science, 410 (2009), 4588-4600. doi: 10.1016/j.tcs.2009.08.012.  Google Scholar

[13]

Y. Ou and C.-Q. Zhang, A new multimembership clustering method, Journal of Industrial and Management Optimization, 3 (2007), 619-624. doi: 10.3934/jimo.2007.3.619.  Google Scholar

[14]

M. M. Sørensen, b-tree facets for the simple graph partitioning polytope, Journal of Combinatorial Optimization, 8 (2004), 151-170. doi: 10.1023/B:JOCO.0000031417.96218.26.  Google Scholar

[15]

J. Szymański, Concentration of vertex degrees in a scale-free random graph process, Random Structures and Algorithms, 26 (2005), 224-236. doi: 10.1002/rsa.20065.  Google Scholar

[16]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.  Google Scholar

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