Test Significance of Community Structure

Compares observed modularity against a null model distribution to assess whether the detected community structure is statistically significant.

Usage

cluster_significance(
  x,
  communities,
  n_random = 100,
  method = c("configuration", "gnm"),
  null = c("detect", "fixed"),
  seed = NULL
)

csig(
  x,
  communities,
  n_random = 100,
  method = c("configuration", "gnm"),
  null = c("detect", "fixed"),
  seed = NULL
)

Arguments

x

Network input: adjacency matrix, igraph object, or cograph_network.

communities

A communities object (from communities or igraph) or a membership vector (integer vector where communities[i] is the community of node i).

n_random

Number of random networks to generate for the null distribution. Default 100.

method

Null model type:

"configuration": (default) Undirected configuration model that preserves the total degree of each node.
"gnm": Erdos-Renyi G(n, m) model with the same number of nodes and edges and the same directedness.

null

Which null question to answer. Default "detect":

"detect": The null value is the modularity of the partition found by community detection on each null graph.
"fixed": The null value is the modularity of the supplied communities membership evaluated on each null graph.

seed

Random seed for reproducibility. Default NULL.

Value

A cograph_cluster_significance object with:

observed_modularity: Modularity of the input communities
null_mean: Mean modularity of random networks
null_sd: Standard deviation of null modularity
z_score: Standardized score (observed - null_mean) / null_sd, or NA when null_sd is zero
p_value: One-sided upper-tail p-value of z_score under the standard normal distribution; NA when null_sd is zero
null_values: Vector of modularity values from null distribution
method: Null model method used
null: Which null question was asked ("detect" or "fixed")
n_random: Number of random networks generated

Details

The function generates n_random random networks from the null model. With null = "detect", community detection (Louvain, or fast greedy when Louvain fails) is run on each null network and its modularity is recorded. A low p-value then indicates that the observed partition is stronger than the partitions detection recovers on random networks. With null = "fixed", the supplied membership is evaluated on each null network. A low p-value then indicates that the partition explains more structure in the observed network than in random networks, independently of any detection algorithm.

The observed modularity is computed with the edge weights of x. When x is weighted, each null network receives the observed edge weights, randomly reassigned to its edges, so the observed and null modularity are on the same scale.

Printing and plotting

Printing the result shows the null model, the observed and null modularity, the z-score and the p-value. plot() on the result plots a histogram of the null modularity values with the observed value marked.

References

Reichardt, J., & Bornholdt, S. (2006). Statistical mechanics of community detection. Physical Review E, 74, 016110.

Examples

comm <- communities(regulation_net, method = "walktrap")
cluster_significance(regulation_net, comm, n_random = 20, seed = 1)
#> Cluster Significance Test
#> =========================
#> 
#>   Null model:           configuration (n = 20 )
#>   Observed modularity:  0.2033 
#>   Null mean:            0.2828 
#>   Null SD:              0.0702 
#>   Z-score:              -1.13 
#>   P-value:              0.87136 
#> 
#>   Conclusion: No significant community structure (p >= 0.05)