Modeling the IPv6 internet AS-level topology

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Abstract

To measure the IPv6 internet AS-level topology, a network topology discovery system, called Dolphin, was developed. By comparing the measurement result of Dolphin with that of CAIDA’s Scamper, it was found that the IPv6 Internet at AS level, similar to other complex networks, is also scale-free but the exponent of its degree distribution is 1.2, which is much smaller than that of the IPv4 Internet and most other scale-free networks. In order to explain this feature of IPv6 Internet we argue that the degree exponent is a measure of uniformity of the degree distribution. Then, for the purpose of modeling the networks, we propose a new model based on the two major factors affecting the exponent of the EBA model. It breaks the lower bound of degree exponent which is 2 for most models. To verify the validity of this model, both theoretical and experimental analyses have been carried out. Finally, we demonstrate how this model can be successfully used to reproduce the topology of the IPv6 Internet.

Introduction

As one of the most significant inventions in the last century, the Internet has provided human beings with a brand new information society. Now the Internet is undergoing a gradual change. As core parts of Internet, TCP/IP protocol stacks perform the tasks of packaging and transmitting information. On the IP layer, the Internet Protocol version 4 (IPv4) has the limitation of address space and some other problems such as the QoS, security and performance. Under this circumstance, the Internet Protocol version 6 (IPv6) has been put into practical use, leading to the coexistence of the IPv6 Internet and the IPv4 Internet. Currently, the IPv6 networks usually connect to each other through IPv4 tunnels. For the purpose of predicting how new technologies, policies, or economic conditions will impact the Internet’s connectivity structure at different layers, the topology of global IPv6 Internet is necessary. It made use of those tunnels and treated them as the links between IPv6 domains [1], [2], [3], [4]. The research can be either at the router level or at the autonomous system (AS) level. In fact, more interest boomed in AS topologies, because from a macroscopic view AS topologies are the “skeletons” of this complex system and are more representative. The topology model is the outcome of theoretically modeling the real networks from the view of systematical evolution or the aspect of reproducing some important topology metrics [5], [6]. Therefore, topology modeling can explain the origins of the existing properties of network topologies and the model also contributes to research on network simulations and structural analysis.

In recent years, considerable research has been done on complex networks which describe a wide variety of systems in nature and society including Internet, World Wide Web (WWW), social relationship networks, economy networks, power networks, transportation networks and neural networks [7]. It is also known that some of the networks can be represented as scale-free networks [8], whose degree distribution follows the power-law form p(k)kγ where p(k) is the probability that a randomly selected node has exactly k edges and γ is called the degree exponent which characterizes the degree distribution of a scale-free network. To understand the evolving mechanisms of scale-free networks, a number of evolving topology models have been proposed. A simple model of a growing network was introduced by Barabási and Albert (the BA model [8]) in which they found that the growth of networks and the preferential attachment were the origins of the power-law degree distribution and proposed the concept of “scale-free” networks. The BA model produces networks with the degree exponent γ=3. Based on the BA model, a lot of other evolving models have been introduced to obtain degree distributions with variable degree exponents [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].

For many observed scale-free networks with 2<γ<3, these models fit their degree exponent feature very well. However, a major challenge arises when using these models to reproduce the IPv6 Internet topology with quite a small degree exponent because most of them have the limitation of γ>2 [7]. In this paper we propose our model to break this limitation and reproduce the IPv6 Internet AS-level topology.

The paper is organized as follows. In Section 2, we first review the evolving models for network topologies. And then we briefly introduce the IPv6 Internet topology discovery and present a new feature of the AS-level topology in Section 3. The degree exponent and its implications are discussed in Section 4. Then, based on the two major factors affecting the exponent and the EBA model, we propose our model and make a theoretical analysis in Section 5. It is shown in Section 6 that our model breaks the bound of 2 for the degree exponent γ, and reproduces the topology of the IPv6 Internet. Finally, we conclude our work in Section 7.

Section snippets

Evolving models for network topologies

In 1999, several power laws were observed in the IPv4 Internet topology [19], [20]. The discovery of the power law of degree distribution has significant impact on the network topology research: the Internet is neither as “flat” as the random models (such as the ER model [21], [22]) describe, nor as “hierarchical” as the structural ones (such as the Tires and Transit-Stub models [23], [24]) describe. To model this scale-free property in the Internet, many efforts have been brought forward to

The small degree exponent: A new feature of the IPv6 Internet topology

In this section we first analyzed the topology data from CAIDA’s Scamper and Dolphin. We found that the IPv6 Internet at AS level, similarly with other complex networks, is also scale-free but its exponent of the degree distribution is 1.2, which is much smaller than that of the IPv4 Internet and most other scale-free networks.

Explanation of small degree exponents and challenges for network evolving models

For degree distribution which follows the power law, the slope of the regression line in a log–log plot can represent the degree exponent. For the networks of the same size, it is easy to obtain that if the degree exponent is bigger, then the degree range is more concentrative, as shown in Fig. 2. Especially when the degree exponent is equal to infinity, all nodes have the same degree and it is a uniform topology. On the opposite side, for the smaller degree exponent, especially when the degree

Our model

As a measure for uniformity of degree distribution of scale-free networks, the degree exponent is affected by many factors. Based on current evolving network models, we make use of two major factors to reproduce the feature of small degree exponent: the preferential attachment and the edge rewiring in network evolving process. For the preferential attachment, the linear relationship may be the most basic one. Although reality is far more complicated than that, statistics of many real networks

Model verifications

When analyzing statistics on the network power-law degrees, we adopt a usual method for topology analysis to capture “the power-law tail” better, which is to ignore some (no more than 5%) nodes whose frequencies are small but degrees are big [19], [20].

In our model the preferential attachment probability is π(k)k+εk̄. To make sure it has physical meaning, π(k) needs to be above zero. However in the case of ε<0, if the percentage of nodes whose π(k)<0 is less than 15%, the experimental results

Conclusion

Based on the analysis of the AS topologies of the IPv6 Internet from two data sources, we find that the IPv6 Internet is also a scale-free network but with a far smaller degree exponent than that of IPv4. As a measure of uniformity of the degree distribution, the degree exponent is affected by many factors. Based on the most important two of them: the probability of preferential attachment and edge rewiring in network evolving process, we propose our model which is a generalization of the EBA

Acknowledgements

The authors would like to thank Professor Guanrong Chen for his helpful comments and suggestions and Matthew Luckie for his help on Scamper data analysis and CAIDA for the topology data of Skitter and Scamper. They are also very grateful to the anonymous reviewers whose valuable comments and constructive criticism helped in improving the paper significantly.

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    This research is supported by National 973 Program of China (Grant No. 2005CB321901) and Beijing Nova Program (Grant. No. 2005B12).

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