Small-World Coefficients

Computes the small-world coefficient sigma, which compares a graph's clustering and path length against a degree-matched random ensemble.

Usage

small_worldness(
  graph,
  n = 100,
  model = c("rewire", "configuration", "erdos_renyi"),
  weights = NULL,
  weight_type = c("strength", "distance"),
  seed = NULL
)

Arguments

graph

An igraph object or a data frame edge list.

n

Integer. Number of null graphs. Default is 100.

model

Passed to null_model(). Default is "rewire".

weights, weight_type

Passed to summarize_graph_metrics(). See netkit-weights.

seed

Integer or NULL. Seed for reproducibility.

Value

A list with:

result

A one-row tibble: sigma, C, C_rand, L, L_rand and n_null.

graph

The input graph, unchanged.

method

A human-readable description.

Details

A network is "small-world" when it is much more clustered than a random graph with the same degree sequence while having a comparable average path length. sigma expresses that as a single ratio: (C/C_rand) / (L/L_rand), where values appreciably above 1 indicate small-world organization.

Only sigma is reported. The companion coefficient omega of Telesford et al. (2011) additionally requires a lattice reference, and igraph provides no degree-preserving latticization; implementing one approximately would make omega quietly dependent on how well that approximation worked, so it is omitted rather than shipped unreliable.

sigma is known to grow with network size, so it is not comparable across graphs of different size – use it to ask whether one graph is small-world, not which of two is more so.

References

Humphries, M. D., & Gurney, K. (2008). Network "small-world-ness": a quantitative method for determining canonical network equivalence. PLoS ONE, 3(4), e0002051. doi:10.1371/journal.pone.0002051

Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of "small-world" networks. Nature, 393(6684), 440-442. doi:10.1038/30918

See also

null_model(), metric_significance()

Examples

g <- igraph::sample_smallworld(1, 60, 4, 0.05)

# `n` is small here to keep the example fast.
sw <- small_worldness(g, n = 10, seed = 1)
sw$result
#> # A tibble: 1 × 6
#>   sigma     C C_rand     L L_rand n_null
#>   <dbl> <dbl>  <dbl> <dbl>  <dbl>  <int>
#> 1  3.40 0.470  0.114  2.61   2.15     10