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{Rich-Club Effect}

Rich-Club Effect is a measure used on networks and graphs. It was designed to quantify the extent to which well connected nodes in a network are also connected to each other. Networks with a strong Rich-Club Effect will have many connections between nodes of high degree, while networks which do not demonstrate the Rich-Club Effect will lack these connections. Scientific collaboration networks have been shown to exhibit higher than expected values on the Rich-Club Effect measurement.

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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. Networks which have a relatively high Rich-Club Coefficient are said to demonstrate the Rich-Club Effect. This effect has been measured and noted on scientific collaboration networks and air transportation networks. It has been shown to be significantly lacking on protein interaction networks.

Definition[edit]

Non-normalized Form[edit]

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)

^[2]

where ${\displaystyle E_{>k}}$ is the number of edges in N between nodes of degree greater than or equal to k, and ${\displaystyle N_{>k}}$ is the number of nodes in N 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.

Normalized for topology randomization[edit]

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.