Ensembles of reference networks based on the rich–club structure for non–evolving networks

In networks the rich nodes are the subset of nodes with large numbers of links, or high degrees. The rich nodes and the connectivity between themselves (rich–club connectivity) tend to dominate the organisation of network structure. Recently there has been a considerable effort to characterise and model the rich–club connectivity in a variety of complex networks. In this paper we firstly clarify a number of terms: the rich-club coefficient quantifies the density of connectivity between a subset of rich nodes; the rich-club structure is the rich-club coefficient measured across the hierarchies of nodes; the rich-club phenomenon refers to the dynamic behaviour responsible for the formation of the rich–club connectivity in evolving networks; and the rich-club ordering discerns whether the connectivity between rich nodes in a nonevolving network (i.e. closed network or network snapshot) is higher than a reference network obtained by network randomisation. We then evaluate a recently proposed null model which is based on an ensemble of reference networks conserving the degree distribution of the original network. We remark that one should not confuse the rich-club structure of a network with the rich-club ordering detected by the null model. We also demonstrate that the null model cannot identify the dynamical mechanism that generates the rich–club connectivity. The main contribution of the paper is that we introduce two new ensembles of reference networks based on the rich-club structure for non–evolving networks. The first ensemble preserves the rich-club coefficient (as a function of the rank of a node) of the original network. Members of the ensemble exhibit similar degree distribution as the original network, and for assortative networks, similar assortative mixing as well. We propose that this ensemble can be used to study networks where assortativeness is a fundamental property, e.g. to detect the community structure in social networks. Analysis on the ensemble also provides a different way to interpret and model the evolution of social networks. The second ensemble preserves both the degree distribution and the rich–club coefficient (as a function of node degree). The reference networks in this ensemble have a similar structure as the original network. We use them to quantify the correlation profile between the rich nodes and pinpoint which links between the nodes are the backbone of network structure.