Rich-club coefficient

The rich-club coefficient is a metric on graphs and networks, designed to measure the extent to which well-connected nodes also connect to each other.

The rich-club coefficient was first introduced in 2004 in a paper studying Internet topology.

The rich-club coefficient was first introduced as an unscaled metric parametrized by node degree ranks.

The associated subgraph of nodes with degree at least k is also called the "Rich Club" graph.

A criticism of the above metric is that it does not necessarily imply the existence of the rich-club effect, as it is monotonically increasing even for random networks.

is the rich-club metric on a maximally randomized network with the same degree distribution

This new ratio discounts unavoidable structural correlations that are a result of the degree distribution, giving a better indicator of the significance of the rich-club effect.

Rich-club can be viewed as a more specific notation of assortativity, where we are only concerned with the connectivity of nodes beyond a certain richness metric.

For example, the consistent observation of high rich-club coefficients for scientific collaboration networks adds evidence to the theory that within social groups, the elite tend to associate with one another.

The rich-club coefficient has been implemented in NetworkX, a Python library for network analysis.