Analytic certainty of network edges (Bayesian Dirichlet-Multinomial)

Closed-form alternative to bootstrap_network for transition networks. Models the outgoing transitions from each state as a Dirichlet-Multinomial process: with a Jeffreys prior the posterior for state \(i\) is \(\mathrm{Dirichlet}(c_i + \mathrm{prior})\), so each edge is marginally Beta and its posterior mean, standard deviation, credible interval and stability decision are available analytically. No resampling, so it runs in microseconds.

The return value has the same structure as bootstrap_network (same slots and summary columns) and carries class c("net_certainty", "net_bootstrap"), so summary() and any code that consumes a net_bootstrap object work unchanged.

Usage

certainty(
  x,
  prior = 0.5,
  ci_level = 0.05,
  inference = c("stability", "threshold"),
  consistency_range = c(0.75, 1.25),
  edge_threshold = NULL
)

Arguments

x: A netobject from build_network using a transition-probability method ("relative" / "tna"), or a netobject_group.
prior: Numeric. Dirichlet prior concentration added to every cell (default 0.5, the Jeffreys prior).
ci_level: Numeric in (0,1). Tail level for credible intervals and the stability decision (default 0.05, i.e. a 95\ match bootstrap_network().
inference: Character. "stability" (default) tests whether the posterior keeps the edge within a multiplicative consistency_range of its weight; "threshold" tests whether the edge exceeds edge_threshold.
consistency_range: Numeric vector of length 2. Multiplicative bounds for stability inference (default c(0.75, 1.25)).
edge_threshold: Numeric or NULL. Fixed threshold for inference = "threshold". If NULL, defaults to the 10th percentile of non-zero edge weights.

Value

An object of class c("net_certainty", "net_bootstrap") with the same fields as bootstrap_network: original, mean, sd, p_values, significant, ci_lower, ci_upper, cr_lower, cr_upper, summary, model, method, params, ci_level, inference, consistency_range, edge_threshold, plus prior and iter = NA (no iterations).

Details

Certainty (this function), stability (bootstrap_network) and reliability (reliability) answer different questions about an edge: how precisely it is pinned down by the observed counts, whether it survives resampling the sequences, and whether it is consistent across split-halves. Certainty and stability agree on homogeneous data; certainty is over-confident when the data are a mixture of latent classes, because it treats transitions clustered within a sequence as independent.

References

Johnston, L. & Jendoubi, T. (2026). How Delivery Mode Reshapes Resource Engagement: A Bayesian Differential Network Analysis. TNA Workshop 2026.

Examples

seqs <- data.frame(V1 = c("A","B","A","C","B"), V2 = c("B","C","B","A","C"),
                   V3 = c("C","A","C","B","A"))
net <- build_network(seqs, method = "relative")
cert <- certainty(net)
cert
#> Certainty (Dirichlet)  [Transition Network (relative) | directed]
#>   Prior      : Dirichlet(0.50)  |  Nodes : 3
#>   Edges      : 0 certain / 3 total
#>   CI         : 95%  |  Inference: stability  |  CR [0.75, 1.25]
summary(cert)
#>   from to weight      mean        sd   p_value   sig  ci_lower  ci_upper
#> 1    A  B      1 0.7777778 0.1772720 0.3653545 FALSE 0.3485528 0.9927924
#> 2    B  C      1 0.8181818 0.1512819 0.2740159 FALSE 0.4405413 0.9943896
#> 3    C  A      1 0.7777778 0.1772720 0.3653545 FALSE 0.3485528 0.9927924
#>   cr_lower cr_upper
#> 1     0.75     1.25
#> 2     0.75     1.25
#> 3     0.75     1.25