TopoGeoML

A preregistered, self-falsifying investigation into topology-aware graph classification — plus a differentiable-TDA toolkit.

TopoGeoML asks one question and tries hard to answer it honestly: does encoding topological structure via the Hodge Laplacian improve graph classification beyond what node features already provide? The question was tested across 14 preregistered hypotheses (53 falsifiable sub-predictions), with the hypothesis documents committed to git before each experiment ran.

This site is the navigable record of that investigation. It is written to inform, not to sell. Every accuracy figure on it is bounded by the regime caveat below.

---

Primary finding — negative

Encoding topological structure via the Hodge Laplacian does not confer a unique advantage for graph classification on any tested dataset. Once an external residual connection is present, a plain normalised-adjacency operator matches or slightly exceeds the Hodge Laplacian; without that residual, both collapse to the class prior. The operative architectural factor is the residual connection, not the topology (hypothesis H008c).

This refutes the strong “topology helps graph classification” hypothesis at the tested configuration. It is reported with the same prominence as any positive result — that is the point of the project.

Secondary finding — positive, narrow

On NCI1 (4110 chemical-compound graphs), a one-layer message-passing classifier with an external residual outperforms a matched-capacity MLP baseline by 8–10 percentage points (paired Wilcoxon p_BH = 4.83 × 10⁻³; survives investigation-wide Benjamini–Hochberg correction but not Bonferroni).

Read before citing any accuracy number. All results are obtained under a deliberately constrained matched-capacity protocol (1 layer, hidden_dim = 32, 10–20 epochs, no batch normalisation, ~1.4–2.3k parameters per arm). Under this protocol the standard GNN baselines (GIN, GAT) collapse to the class prior (0.500) on NCI1, and the best arm reaches only ~0.61–0.63 — roughly 20 percentage points below the ~0.80+ that properly-trained GNNs achieve on this benchmark in the literature. These comparisons isolate architectural mechanism at fixed capacity; they are not statements about leaderboard performance, and a phrase like “outperforms GIN/GAT” means only “at equal, severely-limited capacity.”

What this is — and is not

• It is a rigorous, reproducible research investigation and a small (~7K LOC) toolkit of correct, citable, type-checked topology-aware layers and a statistical harness.
• It is not a production training framework, a competitive benchmark entry, or evidence that topological methods beat well-tuned GNNs. No claim of generality beyond the tested configuration is made.

---

What is in the toolkit

Component	Module	Notes
Differentiable persistent homology (Rips)	topogeoml.nn.diff_ph	autograd through critical-edge indexing (Hofer 2017 / Carrière 2021)
Differentiable cubical PH + topology loss	topogeoml.nn.cubical_diff_ph	CubicalTopologyLoss for image-segmentation training
Hodge Laplacian message-passing layer	topogeoml.nn.hodge	one round of σ(L̃ₖ x W + b); a minimal building block, not a SOTA architecture
Persistence vectorizers + pipeline	topogeoml.core, topogeoml.pipelines	persistence images, Betti curves; sklearn-compatible
Statistical machinery	benchmarks.stats	BCa + block + percentile bootstrap; Wilcoxon, Mann–Whitney, BH-FDR

Quality floor: 497 tests, 100% line coverage with full dependencies, mypy strict, ruff clean.

How the investigation was run

• Preregistration. Each hypothesis was committed to git with falsifiable sub-predictions and a pre-specified outcome decision tree before the experiment ran.
• Statistics. 30 seeds per experiment, paired Wilcoxon signed-rank with Benjamini–Hochberg FDR control, BCa bootstrap confidence intervals; investigation-wide correction across 76 comparisons.
• Negative results shipped. 37% of comparisons are non-significant and are reported with identical formatting to the positive ones.

---

Start here

• Research report — the full structured technical write-up (H001–H007 arc).
• Hypotheses (H001–H011b) — every preregistration and its resolved outcome.
• Statistical summary — multiple-testing burden, power, and FDR across the whole investigation.
• Claims → evidence — each claim mapped to the artefact that backs it.
• Limitations & scope — what does not work, and the regime bounds.
• Mathematical foundations — the underlying TDA / Hodge theory.

Code, reproduction commands, and the full leaderboard live in the GitHub repository.

---

Citation

@software{maniches_topogeoml_2026,
  author  = {Maniches, Santiago},
  title   = {TopoGeoML: A Preregistered Investigation into Topology-Aware Graph Classification},
  year    = {2026},
  version = {0.0.2},
  doi     = {10.5281/zenodo.20365817},
  url     = {https://doi.org/10.5281/zenodo.20365817}
}