Rich Club Coefficient

Computes the rich club curve across all prominence thresholds, measuring whether prominent nodes preferentially direct their strongest ties toward each other. Supports both unweighted (Colizza et al. 2006) and weighted (Opsahl et al. 2008) formulations.

Usage

rich_club(
  x,
  rich = c("k", "s"),
  weighted = TRUE,
  normalized = TRUE,
  n_random = 100,
  directed = NULL,
  seed = NULL,
  digits = NULL,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object.
rich: Character. Prominence definition: "k" (degree, default) or "s" (strength / weighted degree).
weighted: Logical. If TRUE (default), compute the weighted rich club coefficient. If FALSE, compute the unweighted version (density among rich nodes).
normalized: Logical. If TRUE (default), normalize against degree-preserving random graphs and include confidence intervals. The null graphs are drawn with igraph::sample_degseq() (which fixes the degree sequence); for a weighted rich club the observed edge weights are additionally reshuffled across the null edges, following Opsahl et al. (2008).
n_random: Integer. Number of random graphs for normalization. Default 100.
directed: Logical or NULL. Default NULL (auto-detect).
seed: Integer or NULL. Random seed for reproducibility. Default NULL.
digits: Integer or NULL. Round numeric output. Default NULL.
...: Currently unused; directed is already an explicit argument above and to_igraph accepts no others.

Value

A data frame with class "cograph_rich_club", one row per prominence threshold at which at least two nodes are "rich", and columns:

threshold: The prominence cut-off; nodes with prominence strictly greater than this value form the club. Thresholds range over the observed prominence values excluding the maximum.
n_rich: Number of club members at that threshold.
phi: Observed rich club coefficient.
phi_norm, phi_rand, ci_lo, ci_hi: Present only when normalized = TRUE: the observed coefficient divided by the null mean, the null mean itself, and the 2.5\ null distribution.

The data frame has zero rows for graphs that are too small, complete, or regular for any threshold to yield a club. The arguments rich, weighted, normalized and the original input ("network") are stored as attributes.

Details

Unweighted: \(\phi(k) = 2 E_{>k} / (N_{>k} (N_{>k} - 1))\)

Weighted: \(\phi^w(k) = W_{>k} / \sum_{l=1}^{E_{>k}} w_l^{ranked}\)

Normalization: \(\phi_{norm} = \phi_{obs} / \bar{\phi}_{rand}\). A value > 1 indicates rich club ordering beyond what the degree sequence alone explains.

References

Opsahl, T., Colizza, V., Panzarasa, P. & Ramasco, J.J. (2008). Prominence and control: The weighted rich-club effect. Physical Review Letters, 101, 168702.

Colizza, V., Flammini, A., Serrano, M.A. & Vespignani, A. (2006). Detecting rich-club ordering in complex networks. Nature Physics, 2, 110-115.

Examples

g <- igraph::sample_pa(50, m = 2, directed = FALSE)
rc <- cograph::rich_club(g, n_random = 20)
plot(rc)