Rich Club: Rich club calculations

Description

rich_club_coeff calculates the rich club of a graph, returning the rich-club coefficient, $\phi$, and the subgraph of rich club vertices.

rich_club_all is a wrapper for rich_club_coeff that calculates the rich-club coefficient for all degrees present in the graph. It returns a data.table with the coefficients and vertex and edge counts for each successive rich club.

rich_club_norm will (optionally) generate a number of random graphs, calculate their rich club coefficients ($\phi$), and return $\phi_{norm}$ 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

rich_club_coeff(g, k = 1, weighted = FALSE, A = NULL)
rich_club_all(g, weighted = FALSE, A = NULL)
rich_club_norm(g, N = 100, rand = NULL, ...)
rich_core(g, weighted = FALSE, A = NULL)

Value

rich_club_coeff - a list with components:

phi: The rich club coefficient, $\phi$.
graph: A subgraph containing only the rich club vertices.
Nk,Ek: The number of vertices/edges in the rich club graph.

rich_club_all - a data.table with components:

k: A vector of all vertex degrees present in the original graph
phi: The rich-club coefficient
Nk,Ek: The number of vertices/edges in the rich club for each successive k

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: P-values based on the distribution of rand
p.fdr: The FDR-adjusted P-values
density: The observed graph's density
threshold,Group,name

rich_core - a data table with columns:

density: The density of the graph.
rank: The rank of the boundary for the rich core.
k.r: The degree/strength of the vertex at the boundary.
core.size: The size of the core relative to the graph size.
weighted: Whether or not weights were used

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
A: Numeric matrix; the adjacency matrix of the input graph. Default: NULL
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)

Author

Christopher G. Watson, cgwatson@bu.edu

Details

If random graphs have already been generated, you can supply a list as an argument.

For weighted graphs, the degree is substituted by a normalized weight: $$ceiling(A / w_{min})$$ where $w_{min}$ is the minimum weight (that is greater than 0), and $ceiling()$ is the ceiling function that rounds up to the nearest integer.

References

Zhou, S. and Mondragon, R.J. (2004) The rich-club phenomenon in the internet topology. IEEE Comm Lett, 8, 180--182. tools:::Rd_expr_doi("10.4018/978-1-59140-993-9.ch066")

Opsahl, T. and Colizza, V. and Panzarasa, P. and Ramasco, J.J. (2008) Prominence and control: the weighted rich-club effect. Physical Review Letters, 101.16, 168702. tools:::Rd_expr_doi("10.1103/PhysRevLett.101.168702")

Colizza, V. and Flammini, A. and Serrano, M.A. and Vespignani, A. (2006) Detecting rich-club ordering in complex networks. Nature Physics, 2, 110--115. tools:::Rd_expr_doi("10.1038/nphys209")

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