Our review

Validates topological data analysis results by checking persistence diagram sanity, null-model consistency, Wasserstein statistics, and cross-era replication against known benchmarks.

Strengths

Provides a concrete checklist of internal consistency checks for TDA outputs.
Uses known benchmarks from the trajectory_tda/P01 study to compare p-values and metrics.
Includes both null-model negative controls and Wasserstein-specific validation rules.
Covers cross-era replication between BHPS and USoc datasets.

Limitations

Only as reliable as the provided benchmark tables; outside this domain the expected values may not apply.
Does not offer automated statistical testing, just a manual or assisted checklist.
Benchmarks are specific to the trajectory_tda/P01 experiment and may not generalize.

When to use it

When you need to verify that topological data analysis or null-model results are mathematically and statistically plausible before reporting them.

When not to use it

When you are doing exploratory TDA without established benchmarks or when you need to generate new statistical tests rather than validate existing outputs.

Examples

Validate a Wasserstein null model result

/validate-topology trajectory_tda results/trajectory_tda_integration/04_nulls_wasserstein.json

Run validation checklist on TDA output

Run /validate-topology on the latest trajectory TDA results and check all the persistence diagram and null-model consistency items.

Check cross-era replication

Use /validate-topology to compare the BHPS and USoc order-shuffle and Markov-1 p-values against the known P01 benchmarks.

/validate-topology — Validate Topological Results

Check mathematical correctness of TDA results against known benchmarks and internal consistency checks.

Usage

/validate-topology [domain] [result-file]

Example: /validate-topology trajectory_tda results/trajectory_tda_integration/04_nulls_wasserstein.json

Validation checklist

Persistence diagram sanity

[ ] All birth values ≥ 0
[ ] All death values > birth (finite features)
[ ] H₀ has exactly one infinite feature (the final connected component)
[ ] Feature counts are plausible for landmark count L: H₀ has L-1 finite features; H₁ count varies

Total persistence scaling

[ ] Total persistence scales approximately linearly with L (±15% across a 2× range)
[ ] Maximum persistence is stable across L values (should vary < 5%)

Null model consistency

[ ] Label/cohort shuffle p-values are non-significant (negative control)
[ ] Markov-2 null generates more total persistence than Markov-1 (higher-order Markov → more structured surrogates)
[ ] Null distribution standard deviations are plausible (not near-zero, not huge)

Wasserstein-specific

[ ] W(obs↔null) and W(null↔null) are of comparable magnitude (within ~3×)
[ ] p-value = proportion of null-null distances ≥ mean(obs-null distances)
[ ] 500 null-null pairs is sufficient for stable p-value at 3 decimal places

Cross-era replication

[ ] BHPS-era order-shuffle H₀ p-value ≈ USoc order-shuffle direction (both significant or both not)
[ ] BHPS-era Markov-1 direction ≈ USoc (both non-significant under total persistence)

Known benchmarks (trajectory_tda / P01)

| Test | Expected | Source | |---|---|---| | USoc order-shuffle H₀ (total persistence, L=5000) | p < 0.005 | P01 v5 Table 2 | | USoc Markov-1 H₀ (total persistence) | p = 1.000 | P01 v5 Table 2 | | USoc Markov-1 H₀ (Wasserstein, L=2000) | p = 0.002 | P01 v5 Table 2b | | BHPS order-shuffle H₀ | p = 0.000 | P01 v5 §4.7 | | BHPS Markov-1 H₀ | p = 1.000 | P01 v5 §4.7 | | GMM bootstrap ARI | 0.646 ± 0.086 | P01 v5 §3.5 |

Validate Topological Results

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