  In a network, we define shell $\ell$ as the set of nodes at distance $\ell$
with respect to a given node and define $r_\ell$ as the fraction of nodes
outside shell $\ell$. In a transport process, information or disease usually
diffuses from a random node and reach nodes shell after shell. Thus,
understanding the shell structure is crucial for the study of the transport
property of networks. For a randomly connected network with given degree
distribution, we derive analytically the degree distribution and average degree
of the nodes residing outside shell $\ell$ as a function of $r_\ell$. Further,
we find that $r_\ell$ follows an iterative functional form
$r_\ell=\phi(r_{\ell-1})$, where $\phi$ is expressed in terms of the generating
function of the original degree distribution of the network. Our results can
explain the power-law distribution of the number of nodes $B_\ell$ found in
shells with $\ell$ larger than the network diameter $d$, which is the average
distance between all pairs of nodes. For real world networks the theoretical
prediction of $r_\ell$ deviates from the empirical $r_\ell$. We introduce a
network correlation function $c(r_\ell)\equiv r_{\ell+1}/\phi(r_\ell)$ to
characterize the correlations in the network, where $r_{\ell+1}$ is the
empirical value and $\phi(r_\ell)$ is the theoretical prediction. $c(r_\ell)=1$
indicates perfect agreement between empirical results and theory. We apply
$c(r_\ell)$ to several model and real world networks. We find that the networks
fall into two distinct classes: (i) a class of {\it poorly-connected} networks
with $c(r_\ell)>1$, which have larger average distances compared with randomly
connected networks with the same degree distributions; and (ii) a class of {\it
well-connected} networks with $c(r_\ell)<1$.
