In this paper we investigate the evolution of the IPv4 and IPv6 Internettopologies at the autonomous system (AS) level over a long period of time.Weprovide abundant empirical evidence that there is a phase transition in thegrowth trend of the two networks. For the IPv4 network, the phase changeoccurred in 2001. Before then the network's size grew exponentially, andthereafter it followed a linear growth. Changes are also observed around thesame time for the maximum node degree, the average node degree and the averageshortest path length. For the IPv6 network, the phase change occurred in late2006. It is notable that the observed phase transitions in the two networks aredifferent, for example the size of IPv6 network initially grew linearly andthen shifted to an exponential growth. Our results show that following decadesof rapid expansion up to the beginning of this century, the IPv4 network hasnow evolved into a mature, steady stage characterised by a relatively slowgrowth with a stable network structure; whereas the IPv6 network, after a slowstartup process, has just taken off to a full speed growth. We also provideinsight into the possible impact of IPv6-over-IPv4 tunneling deployment schemeon the evolution of the IPv6 network. The Internet topology generators so farare based on an inexplicit assumption that the evolution of Internet followsnon-changing dynamic mechanisms. This assumption, however, is invalidated byour results.Our work reveals insights into the Internet evolution and providesinputs to future AS-Level Internet models.
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