| Degree and connectivity of the Internet’s scale-free topology |
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| 引用本文： | 张连明,邓晓衡,余建平,伍祥生. Degree and connectivity of the Internet’s scale-free topology[J]. 中国物理 B, 2011, 20(4): 48902-048902. DOI: 10.1088/1674-1056/20/4/048902 |
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| 作者姓名： | 张连明 邓晓衡 余建平 伍祥生 |
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| 作者单位： | (1)College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China; (2)College of Physics and Information Science, Hunan Normal University, Changsha 410081, China; (3)Institute of Information Science and Engineering, Central South University, Changsha 410083, China |
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| 基金项目： | Project supported by the National Natural Science Foundation of China (Grant Nos. 60973129, 60903058 and 60903168), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200805331109), the China Postdoctoral Science Foundation (Grant No. 200902324) and the Program for Excellent Talents in Hunan Normal University, China (Grant No. ET10902). |
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| 摘 要： | This paper theoretically and empirically studies the degree and connectivity of the Internet's scale-free topology at an autonomous system (AS) level. The basic features of scale-free networks influence the normalization constant of degree distribution p(k). It develops a new mathematic model for describing the power-law relationships of Internet topology. From this model we theoretically obtain formulas to calculate the average degree, the ratios of the kmin-degree (minimum degree) nodes and the kmax-degree (maximum degree) nodes, and the fraction of the degrees (or links) in the hands of the richer (top best-connected) nodes. It finds that the average degree is larger for a smaller power-law exponent λ and a larger minimum or maximum degree. The ratio of the kmin-degree nodes is larger for larger λ and smaller kmin or kmax. The ratio of the kmax-degree ones is larger for smaller λ and kmax or larger kmin. The richer nodes hold most of the total degrees of Internet AS-level topology. In addition, it is revealed that the increased rate of the average degree or the ratio of the k_min-degree nodes has power-law decay with the increase of kmin. The ratio of the kmax-degree nodes has a power-law decay with the increase of kmax, and the fraction of the degrees in the hands of the richer 27% nodes is about 73% (the '73/27 rule'). Finally, empirically calculations are made, based on the empirical data extracted from the Border Gateway Protocol, of the average degree, ratio and fraction using this method and other methods, and find that this method is rigorous and effective for Internet AS-level topology.
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| 关 键 词： | scale-free networks power-law distribution Internet topology average degree |
| 收稿时间： | 2010-08-14 |
Degree and connectivity of the Internet's scale-free topology |
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| Zhang Lian-Ming,Deng Xiao-Heng,Yu Jian-Ping and Wu Xiang-Sheng. Degree and connectivity of the Internet's scale-free topology[J]. Chinese Physics B, 2011, 20(4): 48902-048902. DOI: 10.1088/1674-1056/20/4/048902 |
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| Authors: | Zhang Lian-Ming Deng Xiao-Heng Yu Jian-Ping Wu Xiang-Sheng |
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| Affiliation: | College of Physics and Information Science, Hunan Normal University, Changsha 410081, China; Institute of Information Science and Engineering, Central South University, Changsha 410083, China; College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China |
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| Abstract: | This paper theoretically and empirically studies the degree and connectivity of the Internet's scale-free topology at an autonomous system (AS) level. The basic features of scale-free networks influence the normalization constant of degree distribution p(k). It develops a new mathematic model for describing the power-law relationships of Internet topology. From this model we theoretically obtain formulas to calculate the average degree, the ratios of the kmin-degree (minimum degree) nodes and the kmax-degree (maximum degree) nodes, and the fraction of the degrees (or links) in the hands of the richer (top best-connected) nodes. It finds that the average degree is larger for a smaller power-law exponent λ and a larger minimum or maximum degree. The ratio of the kmin-degree nodes is larger for larger λ and smaller kmin or kmax. The ratio of the kmax-degree ones is larger for smaller λ and kmax or larger kmin. The richer nodes hold most of the total degrees of Internet AS-level topology. In addition, it is revealed that the increased rate of the average degree or the ratio of the k_min-degree nodes has power-law decay with the increase of kmin. The ratio of the kmax-degree nodes has a power-law decay with the increase of kmax, and the fraction of the degrees in the hands of the richer 27% nodes is about 73% (the '73/27 rule'). Finally, empirically calculations are made, based on the empirical data extracted from the Border Gateway Protocol, of the average degree, ratio and fraction using this method and other methods, and find that this method is rigorous and effective for Internet AS-level topology. |
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| Keywords: | scale-free networks power-law distribution Internet topology average degree |
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