Vertex Bootstrap for Network-Level Statistics

Non-parametric vertex bootstrap of a single observed network (Snijders & Borgatti 1999). Vertices are resampled with replacement and the weight matrix is rebuilt from the original entries of the resampled vertex pairs; network-level statistics computed on each replicate give bootstrap distributions, standard errors, and confidence intervals.

Unlike bootstrap_network, which resamples the underlying cases (sequences or rows) and therefore requires the raw data stored in the netobject, the vertex bootstrap needs only the weight matrix. It works on any netobject - including data-less ones such as build_mlvar constituents or as_tna(mcml) elements - and on plain weight matrices. The two procedures answer different questions: the case bootstrap quantifies sampling-of-subjects uncertainty in the edge weights; the vertex bootstrap quantifies structural uncertainty of whole-network descriptives given the one network you observed.

Usage

vertex_bootstrap(
  x,
  iter = 1000L,
  ci_level = 0.05,
  ci_method = c("percentile", "basic"),
  statistics = NULL,
  statistic_fn = NULL,
  directed = NULL,
  seed = NULL
)

Arguments

x: A netobject (from build_network or any builder), a cograph_network, or a square numeric weight matrix.
iter: Integer. Number of bootstrap replicates (default 1000).
ci_level: Numeric. Significance level for the confidence intervals (default 0.05 for 95% CIs).
ci_method: Character. "percentile" (default) for empirical quantile intervals, or "basic" for intervals reflected around the observed value (Davison & Hinkley 1997, eq. 5.6).
statistics: Character vector selecting built-in statistics (see Details). Default: all applicable to the network's directedness.
statistic_fn: Optional named list of functions, each taking the weight matrix and returning a single numeric value. Computed alongside the built-ins.
directed: Logical or NULL. Directedness of the network. NULL (default) reads x$directed when available, otherwise falls back to matrix symmetry.
seed: Integer or NULL. RNG seed for reproducibility.

Value

An object of class "net_vertex_bootstrap" containing:

summary: Tidy data frame, one row per statistic: statistic, observed, boot_mean, boot_sd, bias, ci_lower, ci_upper.
boot_stats: iter x n_statistics matrix of replicate values.
observed: Named vector of observed statistics.
iter, ci_level, ci_method, directed, n_nodes: Configuration.

Details

Each replicate draws n vertex indices with replacement and sets W_b[i, j] = W[idx_i, idx_j]. When the same original vertex is drawn for two different positions, the off-diagonal cell would be a structural self-pair; following Snijders & Borgatti, such cells are filled with the weight of a randomly chosen pair of distinct original vertices. Diagonal entries carry the original self-weight of the resampled vertex (W[idx_i, idx_i]) - self-loops are meaningful in transition networks and are never altered. For undirected networks the substitution is applied symmetrically so replicates stay symmetric.

Built-in statistics (all computed on the off-diagonal part of the weight matrix):

density: Proportion of non-zero off-diagonal cells.
mean_weight: Mean of the non-zero off-diagonal weights.
centralization: Freeman-type strength centralization: sum(max(s) - s) / ((n - 1) * max(s)) where s is total node strength on absolute weights. 0 when all nodes have equal strength, approaching 1 for a star.
reciprocity: Directed networks only. Weighted reciprocity sum(pmin(|W|, |t(W)|)) / sum(|W|) over off-diagonal cells: the proportion of total weight that is reciprocated.

References

Snijders, T. A. B., & Borgatti, S. P. (1999). Non-parametric standard errors and tests for network statistics. Connections, 22(2), 161-170.

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and their Application. Cambridge University Press.

Examples

seqs <- data.frame(
  T1 = c("plan", "code", "debug", "plan", "test", "code"),
  T2 = c("code", "debug", "code", "plan", "code", "test"),
  T3 = c("debug", "code", "plan", "code", "debug", "plan"),
  T4 = c("test", "plan", "test", "debug", "plan", "code")
)
net <- build_network(seqs, method = "relative")
vb <- vertex_bootstrap(net, iter = 100, seed = 1)
vb$summary
#>        statistic  observed boot_mean    boot_sd        bias   ci_lower
#> 1        density 0.8333333 0.8433333 0.13415153  0.01000000 0.58333333
#> 2    mean_weight 0.3800000 0.3734656 0.05149123 -0.00653441 0.27899554
#> 3 centralization 0.3589744 0.2255932 0.12632830 -0.13338118 0.04511327
#> 4    reciprocity 0.5939850 0.5726307 0.16772881 -0.02135424 0.13107527
#>    ci_upper
#> 1 1.0000000
#> 2 0.4725536
#> 3 0.4635149
#> 4 0.8545707
# \donttest{
plot(vb)

# }