Phase Changes in the Evolution of the IPv4 and IPv6 AS-Level Internet Topologies

In this paper we investigate the evolution of the IPv4 and IPv6 Internet topologies at the autonomous system (AS) level over a long period of time.We provide abundant empirical evidence that there is a phase transition in the growth trend of the two networks. For the IPv4 network, the phase change occurred in 2001. Before then the network's size grew exponentially, and thereafter it followed a linear growth. Changes are also observed around the same time for the maximum node degree, the average node degree and the average shortest path length. For the IPv6 network, the phase change occurred in late 2006. It is notable that the observed phase transitions in the two networks are different, for example the size of IPv6 network initially grew linearly and then shifted to an exponential growth. Our results show that following decades of rapid expansion up to the beginning of this century, the IPv4 network has now evolved into a mature, steady stage characterised by a relatively slow growth with a stable network structure; whereas the IPv6 network, after a slow startup process, has just taken off to a full speed growth. We also provide insight into the possible impact of IPv6-over-IPv4 tunneling deployment scheme on the evolution of the IPv6 network. The Internet topology generators so far are based on an inexplicit assumption that the evolution of Internet follows non-changing dynamic mechanisms. This assumption, however, is invalidated by our results.Our work reveals insights into the Internet evolution and provides inputs to future AS-Level Internet models.Comment: 12 pages, 21 figures; G. Zhang et al.,Phase changes in the evolution of the IPv4 and IPv6 AS-Level Internet topologies, Comput. Commun. (2010

A critical look at power law modelling of the Internet

This paper takes a critical look at the usefulness of power law models of the Internet. The twin focuses of the paper are Internet traffic and topology generation. The aim of the paper is twofold. Firstly it summarises the state of the art in power law modelling particularly giving attention to existing open research questions. Secondly it provides insight into the failings of such models and where progress needs to be made for power law research to feed through to actual improvements in network performance.Comment: To appear Computer Communication

Chinese Internet AS-level Topology

We present the first complete measurement of the Chinese Internet topology at the autonomous systems (AS) level based on traceroute data probed from servers of major ISPs in mainland China. We show that both the Chinese Internet AS graph and the global Internet AS graph can be accurately reproduced by the Positive-Feedback Preference (PFP) model with the same parameters. This result suggests that the Chinese Internet preserves well the topological characteristics of the global Internet. This is the first demonstration of the Internet's topological fractality, or self-similarity, performed at the level of topology evolution modeling.Comment: This paper is a preprint of a paper submitted to IEE Proceedings on Communications and is subject to Institution of Engineering and Technology Copyright. If accepted, the copy of record will be available at IET Digital Librar

Graph Annotations in Modeling Complex Network Topologies

The coarsest approximation of the structure of a complex network, such as the Internet, is a simple undirected unweighted graph. This approximation, however, loses too much detail. In reality, objects represented by vertices and edges in such a graph possess some non-trivial internal structure that varies across and differentiates among distinct types of links or nodes. In this work, we abstract such additional information as network annotations. We introduce a network topology modeling framework that treats annotations as an extended correlation profile of a network. Assuming we have this profile measured for a given network, we present an algorithm to rescale it in order to construct networks of varying size that still reproduce the original measured annotation profile. Using this methodology, we accurately capture the network properties essential for realistic simulations of network applications and protocols, or any other simulations involving complex network topologies, including modeling and simulation of network evolution. We apply our approach to the Autonomous System (AS) topology of the Internet annotated with business relationships between ASs. This topology captures the large-scale structure of the Internet. In depth understanding of this structure and tools to model it are cornerstones of research on future Internet architectures and designs. We find that our techniques are able to accurately capture the structure of annotation correlations within this topology, thus reproducing a number of its important properties in synthetically-generated random graphs

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

Systematic Topology Analysis and Generation Using Degree Correlations

We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.Comment: Final versio

Structural constraints in complex networks

We present a link rewiring mechanism to produce surrogates of a network where both the degree distribution and the rich--club connectivity are preserved. We consider three real networks, the AS--Internet, the protein interaction and the scientific collaboration. We show that for a given degree distribution, the rich--club connectivity is sensitive to the degree--degree correlation, and on the other hand the degree--degree correlation is constrained by the rich--club connectivity. In particular, in the case of the Internet, the assortative coefficient is always negative and a minor change in its value can reverse the network's rich--club structure completely; while fixing the degree distribution and the rich--club connectivity restricts the assortative coefficient to such a narrow range, that a reasonable model of the Internet can be produced by considering mainly the degree distribution and the rich--club connectivity. We also comment on the suitability of using the maximal random network as a null model to assess the rich--club connectivity in real networks.Comment: To appear in New Journal of Physics (www.njp.org

Dynamic Exploration of Networks: from general principles to the traceroute process

Dynamical processes taking place on real networks define on them evolving subnetworks whose topology is not necessarily the same of the underlying one. We investigate the problem of determining the emerging degree distribution, focusing on a class of tree-like processes, such as those used to explore the Internet's topology. A general theory based on mean-field arguments is proposed, both for single-source and multiple-source cases, and applied to the specific example of the traceroute exploration of networks. Our results provide a qualitative improvement in the understanding of dynamical sampling and of the interplay between dynamics and topology in large networks like the Internet.Comment: 13 pages, 6 figure