  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.
