Accurately modeling the Internet topology

Based on measurements of the Internet topology data, we found out that there are two mechanisms which are necessary for the correct modeling of the Internet topology at the Autonomous Systems (AS) level: the Interactive Growth of new nodes and new internal links, and a nonlinear preferential attachment, where the preference probability is described by a positive-feedback mechanism. Based on the above mechanisms, we introduce the Positive-Feedback Preference (PFP) model which accurately reproduces many topological properties of the AS-level Internet, including: degree distribution, rich-club connectivity, the maximum degree, shortest path length, short cycles, disassortative mixing and betweenness centrality. The PFP model is a phenomenological model which provides a novel insight into the evolutionary dynamics of real complex networks.Comment: 20 pages and 17 figure

Ricci Curvature of the Internet Topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201

Understanding the internet topology evolution dynamics

The internet structure is extremely complex. The Positive-Feedback Preference (PFP) model is a recently introduced internet topology generator. The model uses two generic algorithms to replicate the evolution dynamics observed on the internet historic data. The phenomenological model was originally designed to match only two topology properties of the internet, i.e. the rich-club connectivity and the exact form of degree distribution. Whereas numerical evaluation has shown that the PFP model accurately reproduces a large set of other nontrivial characteristics as well. This paper aims to investigate why and how this generative model captures so many diverse properties of the internet. Based on comprehensive simulation results, the paper presents a detailed analysis on the exact origin of each of the topology properties produced by the model. This work reveals how network evolution mechanisms control the obtained topology properties and it also provides insights on correlations between various structural characteristics of complex networks.Comment: 15 figure

Fluctuation-driven dynamics of the Internet topology

We study the dynamics of the Internet topology based on the empirical data on the level of the autonomous systems. It is found that the fluctuations occurring in the stochastic process of connecting and disconnecting edges are important features of the Internet dynamics. The network's overall growth can be described approximately by a single characteristic degree growth rate $g_{\rm eff} \approx 0.016$ and the fluctuation strength $\sigma_{\rm eff} \approx 0.14$, together with the vertex growth rate $\alpha \approx 0.029$. A stochastic model which incorporate these values and an adaptation rule newly introduced reproduces several features of the real Internet topology such as the correlations between the degrees of different vertices.Comment: Final version appeared in Phys. Rev. Let

Understanding Internet topology: principles, models, and validation

Building on a recent effort that combines a first-principles approach to modeling router-level connectivity with a more pragmatic use of statistics and graph theory, we show in this paper that for the Internet, an improved understanding of its physical infrastructure is possible by viewing the physical connectivity as an annotated graph that delivers raw connectivity and bandwidth to the upper layers in the TCP/IP protocol stack, subject to practical constraints (e.g., router technology) and economic considerations (e.g., link costs). More importantly, by relying on data from Abilene, a Tier-1 ISP, and the Rocketfuel project, we provide empirical evidence in support of the proposed approach and its consistency with networking reality. To illustrate its utility, we: 1) show that our approach provides insight into the origin of high variability in measured or inferred router-level maps; 2) demonstrate that it easily accommodates the incorporation of additional objectives of network design (e.g., robustness to router failure); and 3) discuss how it complements ongoing community efforts to reverse-engineer the Internet