  We consider the $k$-core decomposition of network models and Internet graphs
at the autonomous system (AS) level. The $k$-core analysis allows to
characterize networks beyond the degree distribution and uncover structural
properties and hierarchies due to the specific architecture of the system. We
compare the $k$-core structure obtained for AS graphs with those of several
network models and discuss the differences and similarities with the real
Internet architecture. The presence of biases and the incompleteness of the
real maps are discussed and their effect on the $k$-core analysis is assessed
with numerical experiments simulating biased exploration on a wide range of
network models. We find that the $k$-core analysis provides an interesting
characterization of the fluctuations and incompleteness of maps as well as
information helping to discriminate the original underlying structure.
