Our review

Validates topological data analysis results by running a checklist of mathematical sanity checks and comparing them against known benchmarks.

Strengths

Covers persistence diagram sanity, total persistence scaling, null model consistency, Wasserstein p-value interpretation, and cross-era replication.
Includes concrete benchmark values from a real study (P01) for trajectory TDA.
Provides a reusable validation checklist that can be applied to different domains and result files.
Catches common errors in TDA output such as invalid persistence pairs and unstable p-values.

Limitations

Depends on the user providing the correct result-file path and domain context.
Benchmarks listed are specific to trajectory_tda/P01 and may not generalize to other datasets.
Does not perform the actual computation, only validates existing results.

When to use it

Use when you need to check whether a TDA output file is mathematically sound and consistent with established null-model expectations.

When not to use it

Do not use for generating new topological summaries or for datasets without known benchmarks or null distributions.

Examples

Validate integration Wasserstein nulls

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

Validate BHPS-era order shuffle

/validate-topology bhps results/bhps_era/order_shuffle_pvalues.json

Check USoc persistence diagram sanity

/validate-topology usoc results/usoc_tda/persistence_diagram.json

/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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