# rich_club: Rich club calculations In brainGraph: Graph Theory Analysis of Brain MRI Data

## Description

`rich_club_coeff` calculates the rich club of a graph, returning the rich-club coefficient, φ, and the subgraph of rich club vertices.

`rich_club_norm` will (optionally) generate a number of random graphs, calculate their rich club coefficients (φ), and return φ_{norm} of the graph of interest, which is the observed rich-club coefficient divided by the mean across the random graphs.

`rich_core` finds the boundary of the rich core of a graph, based on the decreasing order of vertex degree. It also calculates the degree that corresponds to that rank, and the core size relative to the total number of vertices in the graph.

## Usage

 ```1 2 3 4 5``` ```rich_club_coeff(g, k = 1, weighted = FALSE) rich_club_norm(g, N = 100, rand = NULL, ...) rich_core(g) ```

## Arguments

 `g` An `igraph` graph object `k` Integer; the minimum degree for including a vertex (default: 1) `weighted` Logical indicating whether or not edge weights should be used (default: `FALSE`) `N` Integer; the number of random graphs to generate (default: 100) `rand` A list of `igraph` graph objects, if random graphs have already been generated (default: `NULL`) `...` Other parameters (passed to `sim.rand.graph.par`)

## Details

If random graphs have already been generated, you can supply a list as an argument (since graph generation is time consuming).

## Value

`rich_club_coeff` - a list with components:

 `phi` The rich club coefficient, φ. `graph` A subgraph containing only the rich club vertices. `Nk` The number of vertices in the rich club graph. `Ek` The number of edges in the rich club graph.

`rich_club_norm` - a data table with columns:

 `k` Sequence of degrees `rand` Rich-club coefficients for the random graphs `orig` Rich-club coefficients for the original graph. `norm` Normalized rich-club coefficients. `p` The P-values based on the distribution of rich-club coefficients from the random graphs. `p.fdr` The FDR-adjusted P-values `density` The observed graph's density `threshold` `Group` `name`

`rich_core` - a data frame with columns:

 `density` The density of the graph. `rank` The rank of the boundary for the rich core. `k.r` The degree of the vertex at the boundary. `core.size` The size of the core relative to the graph size.

## Author(s)

Christopher G. Watson, [email protected]

## References

Zhou S., Mondragon R.J. (2004) The rich-club phenomenon in the internet topology. IEEE Comm Lett, 8:180-182.

Opsahl T., Colizza V., Panzarasa P., Ramasco J.J. (2008) Prominence and control: the weighted rich-club effect. Physical Review Letters, 101.16:168702.

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

Ma A & Mondragon R.J. (2015) Rich-cores in networks. PLoS One, 10(3): e0119678. doi: 10.1371/journal.pone.0119678

Other Rich-club functions: `plot_rich_norm`, `rich_club_attrs`
Other Random graph functions: `RandomGraphs`, `analysis_random_graphs`