Small World Index
Normalize observed clustering and characteristic path length against caller-generated matched reference graphs; no random-graph null or universal small-world label is invented.
Description
Small-world normalization compares observed clustering and path length with reference ensembles.
Small-world normalization compares observed clustering and path length with reference ensembles. The ensemble definition matters as much as the resulting ratio.
The tool returns normalizedClustering (normalized clustering); normalizedPathLength (normalized path length); smallWorldSigma (small world sigma). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.
When to use Small World Index
Use Small World Index when you already have observedClustering, observedPathLength, referenceClusterings, referencePathLengths and want normalizedClustering, normalizedPathLength, smallWorldSigma. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.
How to use Small World Index and understand its results
- Enter observedClustering, observedPathLength, referenceClusterings, referencePathLengths 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 normalizedClustering, normalizedPathLength, smallWorldSigma. 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 |
|---|---|---|---|
| observedClustering | Observed clustering | number | Required |
| observedPathLength | Observed path length | number | Required |
| referenceClusterings | Reference clusterings | list of number | Required |
| referencePathLengths | Reference path lengths | list of number | Required |
Use clustering fractions and positive path lengths with aligned nonempty references; mean reference clustering must be positive.
γ = C/mean(Cref); λ = L/mean(Lref); σ = γ/λ The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.
{
"input": {
"observedClustering": 0.6,
"observedPathLength": 2,
"referenceClusterings": [
0.2,
0.4
],
"referencePathLengths": [
1,
3
]
},
"output": {
"normalizedClustering": 1.9999999999999996,
"normalizedPathLength": 1,
"smallWorldSigma": 1.9999999999999996
}
}Limitations and assumptions
Reference networks are supplied, not generated. This does not test significance or ensure comparable density and connectivity conventions.
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 Clustering Coefficient: Compute local clustering, mean including degree<2 zero coefficients, and global transitivity for a binary undirected simple graph. Network Path Length: Calculate mean shortest path over connected unordered pairs and report excluded disconnected pairs; zero mean with zero connected pairs is an explicit empty-pair result.
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 Clustering Coefficient
Compute local clustering, mean including degree<2 zero coefficients, and global transitivity for a binary undirected simple graph.
- Network Path Length
Calculate mean shortest path over connected unordered pairs and report excluded disconnected pairs; zero mean with zero connected pairs is an explicit empty-pair result.