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Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity
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Shows higher-order message passing (HOMP) cannot distinguish complexes differing in diameter, orientability, planarity, or homology, and proposes multi-cellular networks (MCN) and scalable MCN (SMCN) that improve topological expressivity and performance on new torus and lifted ZINC benchmarks and real graph datasets.
Problem
Higher-order message-passing (HOMP) architectures for topological data (combinatorial complexes) may be unable to distinguish complexes differing in fundamental topological and metric invariants, and cannot fully exploit graph lifting and pooling.
Approach
Derive a covering-space criterion for HOMP indistinguishability, use it to show blindspots in diameter, orientability, planarity and homology; then add equivariant multi-cellular cochain layers (MCN, IGN-inspired) and a scalable variant (SMCN) that replaces expensive updates with subgraph-GNN-style subcomplex layers over augmented Hasse graphs.
Key results
- On ZINC-12k, SMCN achieves MAE 0.060 ± 0.004, outperforming CIN (0.079) and Subgraphormer+PE (0.063).
- On the Torus dataset, HOMP distinguishes none of the indistinguishable pairs while SMCN distinguishes all of them.
- On lifted ZINC benchmarks, SMCN reaches 92.76% accuracy on cross-diameter and 99.61% on second Betti number, versus 67.87% and 81.76% for a custom HOMP.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use SMCN instead of the strongest reported baseline | graph regression and classification models · ZINC | MAE: 0.063 \pm 0.001 -> 0.060 \pm 0.004 | Helped |
| use SMCN instead of CIN | graph classification models · MOLHIV | ROC-AUC: 80.94 \pm 0.57 -> 81.16 \pm 0.90 | Helped |
| use SMCN instead of the strongest reported baseline | graph regression models · MOLESOL | RMSE: 0.826 \pm 0.010 -> 0.809 \pm 0.037 | Helped |
| use SMCN instead of Custom HOMP | topological property prediction on lifted ZINC graphs · Lifted ZINC cross-diameter | Accuracy: 67.87 \pm 12.26 % -> 92.76 \pm 0.53 % | Helped |
| use SMCN instead of Custom HOMP | topological property prediction on lifted ZINC graphs · Lifted ZINC cross-diameter | MSE: 0.0684 \pm 0.0323 -> 0.011 \pm 0.0008 | Helped |
| use SMCN instead of Custom HOMP | topological property prediction on lifted ZINC graphs · Lifted ZINC second Betti number | Accuracy: 81.76 \pm 10.06 % -> 99.61 \pm 0.12 % | Helped |
| use SMCN instead of Custom HOMP | topological property prediction on lifted ZINC graphs · Lifted ZINC second Betti number | MSE: 0.0391 \pm 0.0208 -> 0.0024 \pm 0.0005 | Helped |
| replace the 3-IGN-style update with PPGN (ablation) | SMCN · Real-world graph benchmarks | performance | No clear effect |
| use an SCL subgraph update rather than PPGN for the replacement of the $\gC^{3\ve_r}$ update (ablation) | SMCN · Real-world graph benchmarks | performance | No clear effect |
Surprising findings
- Standard HOMP architectures cannot distinguish tori with different diameters or tori versus disjoint unions of tori with different homology, despite identical 0-cell counts.
- PPGN-based 3-IGN substitute gave no significant improvement over subgraph networks in the SMCN design.
Limitations
- SMCN runtime scales super-linearly in number of cells, making it intractable for large complexes.
- Unclear whether SMCN fully captures orientability and homology of arbitrary order; only partial results proven.
- Benchmarks are limited in scope of topological properties covered.
Experimental setup
Datasets: ZINC-12k MOLHIV MOLESOL Torus dataset (synthetic) lifted ZINC molecular graphs
Benchmarks: ZINC-12k MOLHIV MOLESOL Torus dataset lifted ZINC cross-diameter and second Betti number
Metrics: MAE ROC-AUC RMSE accuracy MSE
Baselines: GCN GIN CIN CIN++ PPGN SUN Subgraphormer DS-GNN DSS-GNN GNN-SSWL+
Builds on: higher-order message passing invariant graph networks subgraph GNN combinatorial complex neural networks tensor diagrams
Closest papers by meaning
- TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks (ICML 2025 · similarity 0.90)
- Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes (AAAI 2024 · similarity 0.90)
- HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations (NeurIPS 2026 · similarity 0.90)
- Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing (ICLR 2025 · similarity 0.89)
- Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework (NeurIPS 2025 · similarity 0.89)
- Topological Neural Networks go Persistent, Equivariant, and Continuous (ICML 2024 · similarity 0.89)
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