RichClubCoefficients - Maple Help

GraphTheory

 RichClubCoefficients
 compute rich club coefficients

 Calling Sequence RichClubCoefficients(G,opts)

Parameters

 G - graph opts - (optional) one or more options; see below

Options

 • datatype = one of double, float[4], float[8], rational
 Specifies the datatype of the generated Array. The default is rational.
 • mixingmultiplier = posint
 When normalized is true, mixingmultiplier*numedges edge swaps are performed to generate a random graph for purposes of normalization, where numedges is the number of edges of G, as described in McAuley (see References). If normalized is false, this parameter is ignored. The default is 100.
 • normalized = truefalse
 If the option normalized is specified, then the computed coefficients are normalized against a random graph with the same degree sequence as the input graph G. The default is true.
 • seed = integer or none
 Seed for the random number generator. If an integer is specified, this is equivalent to calling randomize(seed) immediately before invoking this function. If normalized is false this parameter is ignored. The default is none.

Description

 • RichClubCoefficients(G,opts) computes an Array of rich club coefficients for the given graph G. By default, the coefficients are normalized using a random graph with the same degree sequence as G.
 • The random number generator used can be seeded using the randomize function or the seed option.

Definition

 • For each non-negative integer k, the (non-normalized) rich club coefficient $\mathrm{\phi }\left(k\right)$ is defined to be $\mathrm{\phi }\left(k\right)=\frac{2{E}_{k}}{{N}_{k}\left({N}_{k}-1\right)}$ where ${N}_{k}$ is the number of vertices with degree greater than k and ${E}_{k}$ is the number of edges between these vertices.
 • The quantity $\mathrm{\phi }\left(k\right)$ is intended to measure the extent to which well-connected vertices also connect to each other.

Examples

 > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$

Compute the rich club coefficients for a cycle graph

 > $G≔\mathrm{CycleGraph}\left(10\right)$
 ${G}{≔}{\mathrm{Graph 1: an undirected graph with 10 vertices and 10 edge\left(s\right)}}$ (1)
 > $\mathrm{RichClubCoefficients}\left(G\right)$

Compute the rich club coefficients for a complete graph

 > $G≔\mathrm{CompleteGraph}\left(10\right)$
 ${G}{≔}{\mathrm{Graph 2: an undirected graph with 10 vertices and 45 edge\left(s\right)}}$ (2)
 > $\mathrm{RichClubCoefficients}\left(G\right)$

Compute the rich club coefficients for a random Barabasi-Albert graph

 > $G≔\mathrm{RandomGraphs}:-\mathrm{BarabasiAlbertGraph}\left(25,10,\mathrm{seed}=1024\right)$
 ${G}{≔}{\mathrm{Graph 3: an undirected graph with 25 vertices and 150 edge\left(s\right)}}$ (3)
 > $\mathrm{RichClubCoefficients}\left(G\right)$

References

 J. J. McAuley, L. da Fontoura Costa, and T. S. Caetano, “The rich-club phenomenon across complex network hierarchies”, Applied Physics Letters Vol 91 Issue 8, August 2007. https://arxiv.org/abs/physics/0701290, doi:10.1063/1.2773951 R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, and U. Alon (2006), cond-mat/0312028

Compatibility

 • The GraphTheory[RichClubCoefficients] command was introduced in Maple 2022.