cograph (version 2.7.2)

rich_club: 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,
  ...
)

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.

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.

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.

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)