rich_club: Rich Club Coefficient

In cograph: Analysis and Visualization of Complex Networks

Rich Club Coefficient

Description

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.
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.
...	Additional arguments passed to to_igraph.

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.

Value

A data frame with class "cograph_rich_club" and columns: threshold, n_rich, phi, and if normalized: phi_norm, phi_rand, ci_lo, ci_hi.

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.

See Also

rich_club_local, robustness, centrality

Examples

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

cograph documentation built on May 31, 2026, 5:06 p.m.