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.
rich_club_coeff(g, k = 1, weighted = FALSE)rich_club_all(g, weighted = FALSE)
rich_club_norm(g, N = 100, rand = NULL, ...)
rich_core(g, weighted = FALSE)
An igraph graph object
Integer; the minimum degree for including a vertex (default: 1)
Logical indicating whether or not edge weights should be
used (default: FALSE)
Integer; the number of random graphs to generate (default: 100)
A list of igraph graph objects, if random graphs have
already been generated (default: NULL)
Other parameters (passed to sim.rand.graph.par)
rich_club_coeff - a list with components:
The rich club coefficient, \(\phi\).
A subgraph containing only the rich club vertices.
The number of vertices in the rich club graph.
The number of edges in the rich club graph.
rich_club_all - a data.table with components:
A vector of all vertex degrees present in the original graph
The rich-club coefficient
The number of vertices in the rich club for each successive k
The number of edges in the rich club for each successive k
rich_club_norm - a data table with columns:
Sequence of degrees
Rich-club coefficients for the random graphs
Rich-club coefficients for the original graph.
Normalized rich-club coefficients.
The P-values based on the distribution of rich-club coefficients from the random graphs.
The FDR-adjusted P-values
The observed graph's density
rich_core - a data table with columns:
The density of the graph.
The rank of the boundary for the rich core.
The degree/strength of the vertex at the boundary.
The size of the core relative to the graph size.
Whether or not weights were used
If random graphs have already been generated, you can supply a list as an argument (since graph generation is time consuming).
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.
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