  The micro-structure of the giant component of the Erd{\H o}s-R\'enyi network
and other configuration model networks is analyzed using generating function
methods. While configuration model networks are uncorrelated, the giant
component exhibits a degree distribution which is different from the overall
degree distribution of the network and includes degree-degree correlations of
all orders. We present exact analytical results for the degree distributions as
well as higher order degree-degree correlations on the giant components of
configuration model networks. We show that the degree-degree correlations are
essential for the integrity of the giant component, in the sense that the
degree distribution alone cannot guarantee that it will consist of a single
connected component. To demonstrate the importance and broad applicability of
these results, we apply them to the study of the distribution of shortest path
lengths on the giant component, percolation on the giant component and the
spectra of sparse matrices defined on the giant component. We show that by
using the degree distribution on the giant component, one obtains high quality
results for these properties, which can be further improved by taking the
degree-degree correlations into account. This suggests that many existing
methods, currently used for the analysis of the whole network, can be adapted
in a straightforward fashion to yield results conditioned on the giant
component.
