  Many of the structural characteristics of a network depend on the
connectivity with and within the hubs. These dependencies can be related to the
degree of a node and the number of links that a node shares with nodes of
higher degree. In here we revise and present new results showing how to
construct network ensembles which give a good approximation to the
degree-degree correlations, and hence to the projections of this correlation
like the assortativity coefficient or the average neighbours degree. We present
a new bound for the structural cut--off degree based on the connectivity within
the hubs. Also we show that the connections with and within the hubs can be
used to define different networks cores. Two of these cores are related to the
spectral properties and walks of length one and two which contain at least on
hub node, and they are related to the eigenvector centrality. We introduce a
new centrality measured based on the connectivity with the hubs. In addition,
as the ensembles and cores are related by the connectivity of the hubs, we show
several examples how changes in the hubs linkage effects the degree--degree
correlations and core properties.
