Competition and Adaptation in an Internet Evolution Model

We model the evolution of the Internet at the autonomous system level as a process of competition for users and adaptation of bandwidth capability. From a weighted network formalism, where both nodes and links are weighted, we find the exponent of the degree distribution as a simple function of the growth rates of the number of autonomous systems and connections in the Internet, both empirically measurable quantities. Our approach also accounts for a high level of clustering as well as degree-degree correlations, both with the same hierarchical structure present in the real Internet. Further, it also highlights the interplay between bandwidth, connectivity, and traffic of the network.

Figure 1

Temporal evolution of the number of hosts, autonomous systems, and connections among them from November 1997 to May 2002. Solid lines are the best fit estimates which give the values for the rate growths of $\alpha =0.036\pm 0.001$, $\beta =0.0304\pm 0.0003$, and $\delta =0.0330\pm 0.0002$ (units are ${\mathrm{\text{month}}}^{-1}$). Each point for the number of ASs and connections is an average over one month of daily measurements. Error bars are of the order of the symbol size.

Figure 2

Cumulative degree distribution [$P_c(k)=\sum _{k}P(k^{\prime })$] for the extended AS map compared to simulations of the model, $r=0.8$. Inset: simulation results of the AS's degree as a function of the AS's bandwidth. The solid line stands for the scaling relation Eq. (7) with $\mu =\beta /{\delta }^{\prime }=0.75$.

Figure 3

Clustering coefficient, $c_k$ (top) and normalized average nearest neighbors degree, ${\overline{k}}_{nn}(k)⟨k⟩/⟨k^2⟩$ (bottom), as functions of the node's degree for the extended autonomous system map (circles) and for the model with and without distance constraints (red squares and dashed line, respectively).