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Many of the structural characteristics of a network depend on theconnectivity with and within the hubs. These dependencies can be related to thedegree of a node and the number of links that a node shares with nodes ofhigher degree. In here we revise and present new results showing how toconstruct network ensembles which give a good approximation to thedegree-degree correlations, and hence to the projections of this correlationlike the assortativity coefficient or the average neighbours degree. We presenta new bound for the structural cut--off degree based on the connectivity withinthe hubs. Also we show that the connections with and within the hubs can beused to define different networks cores. Two of these cores are related to thespectral properties and walks of length one and two which contain at least onhub node, and they are related to the eigenvector centrality. We introduce anew 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 showseveral examples how changes in the hubs linkage effects the degree--degreecorrelations and core properties.

Beyond the rich-club: Properties of networks related to the better connected nodes