ncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue in understanding their fabric and dynamical properties 1-5 . The 'rich-club' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities, and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences 4-8 . Here, we provide the analytical expression and the correct null models that allow for a quantitative discussion of the rich-club phenomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains. The informatics revolution has made it possible for a wide range of large-scale, rapidly evolving networks, such as transportation, technological, social and biological networks, to be analysed [1] [2] [3] [4] [5] . Although these networks are extremely different in their function and attributes, the analysis of their fabric provides evidence of several shared regularities, suggesting general and common selforganizing principles beyond the specific details of the individual systems. In this context, the statistical physics approach has been exploited as a very convenient strategy because of its deep connection with statistical graph theory and because of its power to quantitatively characterize macroscopic phenomena in terms of the microscopic dynamics of the various systems [1] [2] [3] [4] 9 . As an initial discriminant of structural ordering, attention has been focused on the networks' degree distribution; that is, the probability P(k) that any given node in the network shares an edge with k neighbouring nodes. This function is, however, only one of the many statistics characterizing the structural and hierarchical ordering of a network; a full account of the connectivity pattern calls for the detailed study of the multipoint degree correlation functions and/or an opportune combination of these. In this letter, we tackle a main structural property of complex networks, the so-called 'rich-club' phenomenon. This property has been discussed in several instances in both social and computer sciences and refers to the tendency of high-degree nodes, the hubs of the network, to be very well-connected to each other. Essentially, nodes with a large number of links, usually referred to as rich nodes, are much more likely to form tight and well-interconnected subgraphs (clubs) than low-degree nodes. A first quantitative definition of the rich-club phenomenon is given by the rich-club coefficient φ, introduced in the context of the Internet in ref. 7. Denoting by E >k the number of edges among the N >k nodes having degree higher than a given value k, the rich-club coefficient is expressed as where N >k (N >k − 1)/2 represents the maximum possible number of edges among the N >k nodes. Therefore, φ(k) measures the fraction of edges actually connecting those nodes out of the maximum number of edges they might possibly share. The behaviour of the rich-club coefficient as a function of the degree k is a probe for the topological correlations in a complex network, and it yields important information about its underlying architecture. Structural properties, in turn, have immediate consequences on a network's features and tasks, such as robustness, performance of biological functions or selection of traffic backbones, depending on the system at hand. In a social context, for example, a rich-club coefficient increasing with the degree k indicates the dominance of an 'oligarchy' of highly connected and mutually communicating individuals, as opposed to a structure consisting of many loosely connected and relatively independent subcommunities. In the Internet, such a feature would point to an architecture in which important hubs are much more densely interconnected than peripheral nodes to provide the transit backbone of the network 7 . It is also worth stressing that the rich-club phenomenon is not trivially related to the mixing properties of networks, which enable the distinction between assortative networks, where high-degree nodes preferentially attach to high-degree nodes, and disassortative networks, showing the opposite tendency [10] [11] [12] . Indeed, the rich-club phenomenon is not necessarily associated with assortative mixing. In the top panel of Fig. 1 , we sketch a simple construction in which a disassortative network is showing the rich-club phenomenon. In other words, the rich-club phenomenon and the mixing properties
Despite the different formulations of the rich-club coefficient, the fundamental contribution of REF derives from exploiting a null model used to detect the presence of the rich-club.
Detecting rich-club ordering in complex networks
Nature Physics 2, 110-115 (2006)
Physics