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

TL;DR

This paper investigates the structural properties of networks related to highly connected nodes, introducing new models, bounds, and centrality measures to better understand their influence on network correlations and core structures.

Contribution

It presents novel methods for constructing network ensembles, bounds for structural cut-offs, and new centrality measures based on hub connectivity, advancing understanding of network core properties.

Findings

New bounds for the structural cut-off degree based on hub connectivity

Introduction of a centrality measure based on hub connections

Demonstration of how hub linkages influence degree correlations and network cores

Abstract

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.