Degree and connectivity of the Internet's scale-free topology

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 kmin-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.