  As the Internet AS-level topology grows over time, some of its structural
properties remain unchanged. Such time- invariant properties are generally
interesting, because they tend to reflect some fundamental processes or
constraints behind Internet growth. As has been shown before, the
time-invariant structural properties of the Internet include some most basic
ones, such as the degree distribution or clustering. Here we add to this
time-invariant list a non-trivial property - k-dense decomposition. This
property is derived from a recursive form of edge multiplicity, defined as the
number of triangles that share a given edge. We show that after proper
normalization, the k- dense decomposition of the Internet has remained stable
over the last decade, even though the Internet size has approximately doubled,
and so has the k-density of its k-densest core. This core consists mostly of
content providers peering at Internet eXchange Points, and it only loosely
overlaps with the high-degree or high-rank AS core, consisting mostly of tier-1
transit providers. We thus show that high degrees and high k-densities reflect
two different Internet-specific properties of ASes (transit versus content
providers). As a consequence, even though degrees and k-densities of nodes are
correlated, the relative fluctuations are strong, and related to that, random
graphs with the same degree distribution or even degree correlations as in the
Internet, do not reproduce its k-dense decomposition. Therefore an interesting
open question is what Internet topology models or generators can fully explain
or at least reproduce the k-dense properties of the Internet.
