Rich-club coefficient

The rich-club coefficient was first introduced as an unscaled metric parametrized by node degree ranks.[1] More recently, this has been updated to be parameterized in terms of node degrees k , indicating a degree cut-off. The rich-club coefficient for a given network N is then defined as:

${\displaystyle \phi (k)={\frac {2E_{>k}}{N_{>k}(N_{>k}-1)}}}$

(1)

[3][4][5]

where ${\displaystyle E_{>k}}$ is the number of edges between the nodes of degree greater than or equal to k, and ${\displaystyle N_{>k}}$ is the number of nodes with degree greater than or equal to k. This measures how many edges are present between nodes of degree at least k, normalized by how many edges there could be between these nodes in a complete graph. When this value is close to 1 for values of k close to ${\displaystyle k_{max}}$, it is interpreted that high degree nodes of the network are well connected. 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. In certain degree distributions, it is not possible to avoid connecting high degree hubs. To account for this, it is necessary to compare the above metric to the same metric on a degree distribution preserving randomized version of the network. This updated metric is defined as:

${\displaystyle \rho _{rand}(k)={\frac {\phi (k)}{\phi _{rand}(k)}}}$

(2)

where ${\displaystyle \phi _{rand}(k)}$ is the rich-club metric on a maximally randomized network with the same degree distribution ${\displaystyle P(k)}$ of the network under study. 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.

For this metric, if for certain values of k we have ${\displaystyle \rho _{rand}(k)>1}$ , this denotes the presence of the rich-club effect.

The natural definition of a node's "richness" is its number of neighbours. If instead we replace this with a generic richness metric on nodes r, then we can rewrite the unscaled Rich-Club coefficient as:

${\displaystyle \phi (r)={\frac {2E_{>r}}{N_{>r}(N_{>r}-1)}}}$

(3)

Where we are instead considering the sub graph on only nodes with a richness measure of at least r. For example, on scientific collaboration networks, replacing the degree richness (number of coauthors) with a strength richness (number of published papers), the topology of the rich club graph changes dramatically.

The Assortativity of a network is a measurement of how connected similar nodes are, where similarity is typically viewed in terms of node degree. 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, if a network consisted of a collection of hub and spokes, where the hubs were well connected, such a network would be considered disassortative. However, due to the strong connectedness of the hubs in the network, the network would demonstrate the rich-club effect.

An example of a network which is both disassortative and demonstrates the Rich Club effect. The red nodes are hubs and form the "Rich Club."