Rich Club Coefficient
Compute unnormalized rich-club coefficients for nodes with degree strictly above each supplied threshold; fewer-than-two-node cases carry an explicit undefined flag.
Description
A rich-club coefficient measures density among nodes above a chosen degree threshold.
A rich-club coefficient measures density among nodes above a chosen degree threshold. The threshold comparison excludes nodes exactly equal to that degree.
The tool returns eligibleNodeCounts (eligible node counts); richClubCoefficients (rich club coefficients); coefficientDefinedFlags (coefficient defined flags). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.
When to use Rich Club Coefficient
Use Rich Club Coefficient when you already have adjacency, degreeThresholds and want eligibleNodeCounts, richClubCoefficients, coefficientDefinedFlags. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.
How to use Rich Club Coefficient and understand its results
- Enter adjacency, degreeThresholds using the field table below. Match the units, ordering and data boundaries described in the calculation.
- Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
- Read eligibleNodeCounts, richClubCoefficients, coefficientDefinedFlags. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
- Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| adjacency | Adjacency | list of list of number | Required |
| degreeThresholds | Degree thresholds | list of number | Required |
Compute degrees in a binary undirected loop-free graph. Evaluate each nonnegative integer threshold; fewer than two eligible nodes returns coefficient zero and defined flag zero.
φ(k) = 2×edgesAmongDegree>k / (eligibleNodes×(eligibleNodes−1)) The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.
{
"input": {
"adjacency": [
[
0,
1,
1
],
[
1,
0,
1
],
[
1,
1,
0
]
],
"degreeThresholds": [
1,
2
]
},
"output": {
"eligibleNodeCounts": [
3,
0
],
"richClubCoefficients": [
1,
0
],
"coefficientDefinedFlags": [
1,
0
]
}
}Limitations and assumptions
Read defined flags before interpreting zero. No randomized reference normalization or significance test is performed.
Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.
Alternative or Complementary methods
Network Modularity: Evaluate resolution-parameter Newman–Girvan modularity for a supplied binary graph partition; this scores a partition, it does not discover or optimize communities. Network Clustering Coefficient: Compute local clustering, mean including degree<2 zero coefficients, and global transitivity for a binary undirected simple graph.
Methodological background: Complex network measures of brain connectivity: uses and interpretations.1
References
-
Complex network measures of brain connectivity: uses and interpretations — Rubinov and Sporns; NeuroImage
Similar or alternative tools
- Network Modularity
Evaluate resolution-parameter Newman–Girvan modularity for a supplied binary graph partition; this scores a partition, it does not discover or optimize communities.
- Network Clustering Coefficient
Compute local clustering, mean including degree<2 zero coefficients, and global transitivity for a binary undirected simple graph.