WJM  Vol.2 No.3 , June 2012
An Analytical Approach for Degree Correlations in Complex Network
Author(s) Kosuke Takagi*
We investigate correlations between neighbor degrees in the scale-free network. According to the empirical studies, it is known that the degree correlations exhibit nontrivial statistical behaviors. With using an analytical approach, we show that the scale-freeness and one of statistical laws for degree correlations can be reproduced consistently in a unified framework. Our result would have its importance in understanding the mechanisms which generate the complex network.

Cite this paper
nullK. Takagi, "An Analytical Approach for Degree Correlations in Complex Network," World Journal of Mechanics, Vol. 2 No. 3, 2012, pp. 171-174. doi: 10.4236/wjm.2012.23020.
[1]   A.-L. Barabási and R. Albert, “Emergence of Scaling in Random Networks,” Science, Vol. 286, No. 5439, 1999, pp. 509-512. doi:10.1126/science.286.5439.509

[2]   A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins and J. Wiener, “Graph Structure in the Web,” Computer Networks, Vol. 33, No. 1-6, 2000, pp. 309-320. doi:10.1016/S1389-1286(00)00083-9

[3]   M. Faloutsos, P. Faloutsos and C. Faloutsos, “On Power- Law Relationships of the Internet Topology,” ACM SIG- COMM Computer Communication Review, Vol. 29, No. 4, 1999, pp. 251-262. doi:10.1145/316194.316229

[4]   S. N. Dorogovtsev and J. F. F. Mendes, “Evolution of Networks,” Advances in Physics, Vol. 51 No. 4, 2002, pp. 1079-1187. doi:10.1080/00018730110112519

[5]   R. Albert and A.-L. Barabási, “Statistical Mechanics of Complex Networks,” Reviews of Modern Physics, Vol. 74, No. 1, 2002, pp. 47-97. doi:10.1103/RevModPhys.74.47

[6]   M. E. J. Newman, “The Structure and Function of Complex Networks,” SIAM Review, Vol. 45, No. 2, 2003, pp. 167-256 doi:10.1137/S003614450342480

[7]   R. Albert, H. Jeong and A.-L. Barabási, “Internet: Diameter of the World-Wide Web,” Nature, Vol. 401, No. 6749, 1999, pp. 130-131. doi:10.1038/43601

[8]   M. E. J. Newman, “Scientific Collaboration Networks. I. Network Construction and Fundamental Results,” Physical Review E, Vol. 64, No. 1, 2001, pp. 1-8. doi:10.1103/PhysRevE.64.016131

[9]   H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai and A.-L. Barabási, “The Large-Scale Organization of Metabolic Networks,” Nature, Vol. 407, No. 6804, 2000, pp. 651- 654. doi:10.1038/35036627

[10]   H. Jeong, S. Mason, A.-L. Barabási and Z. N. Oltvai, “Lethality and Centrality in Protein Networks,” Nature, Vol. 411, No. 6833, 2001, pp. 41-42. doi:10.1038/35075138

[11]   A.-L. Barab′asi, R. Albert and H. Jeong, “Mean-Field Theory for Scale-Free Random Networks,” Physica A, Vol. 272, No. 1-2, 1999, pp. 173-187. doi:10.1016/S0378-4371(99)00291-5

[12]   P. L. Krapivsky, S. Redner and F. Leyvraz, “Connectivity of Growing Random Networks,” Physical Review Letters, Vol. 85, No. 21, 2000, pp. 4629-4632. doi:10.1103/PhysRevLett.85.4629

[13]   S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, “Structure of Growing Networks with Preferential Linking,” Physical Review Letters, Vol. 85, No. 21, 2000, pp. 4633-4636. doi:10.1103/PhysRevLett.85.4633

[14]   C. Song, S. Havlin and H. A. Makse, “Self-Similarity of Complex Networks,” Nature, Vol. 433, No. 7024, 2005, pp. 392-395. doi:10.1038/nature03248

[15]   C. Song, S. Havlin and H. A. Makse, “Origins of Fractality in the Growth of Complex Networks,” Nature Physics, Vol. 2, 2006, pp. 275-281. doi:10.1038/nphys266

[16]   S. Maslov and K. Sneppen, “Specificity and Stability in Topology of Protein Networks,” Science, Vol. 296 no. 5569 2002, pp. 910-913. doi:10.1126/science.1065103

[17]   R. Pastor-Satorras A. Vázquez and A. Vespignani, “Dynamical and Correlation Properties of the Internet,” Phy- sical Review Letters, Vol. 87, No. 25, 2001, p. 258701. doi:10.1103/PhysRevLett.87.258701

[18]   M. E. J. Newman, “Assortative Mixing in Networks,” Phy- sical Review Letters, Vol. 89, No. 20, 2002, p. 208701. doi:10.1103/PhysRevLett.89.208701

[19]   K. Takagi, “Scale Free Distribution in an Analytical Approach,” Physica A, Vol. 389, No. 10, 2010, pp. 2143- 2146. doi:10.1016/j.physa.2010.01.034