TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration¶
Conference: ICML 2025
arXiv: 2510.05102
Area: Graph Learning
TL;DR¶
TopInG proposes a topologically interpretable graph learning framework based on persistent homology. By learning a "rationale filtration", it identifies stable and persistent rationale subgraphs. It introduces a "topological discrepancy" constraint to reinforce the topological distinction between rationale and irrelevant subgraphs, significantly outperforming existing methods in handling variform rationale subgraphs.
Background & Motivation¶
GNNs have achieved remarkable success in scientific fields, but their lack of interpretability limits their adoption in critical decision-making. Existing inherently interpretable GNNs (such as GSAT, GMT, DIR) face a common issue:
Overly strong implicit assumptions: Existing methods typically assume that the rationale subgraphs of the same class of graphs are nearly invariant in form. In reality, however, variform rationale subgraphs are very common:
- In molecular biology, molecules with the same biological activity may possess different functional groups (aromatic rings, sulfonamide groups, heterocyclic compounds).
- In social networks, the structural reasons for user influence vary (high degree, high betweenness centrality, bridging nodes).
Method¶
Core Idea¶
A graph is viewed as a structure "growing" out of a core rationale subgraph \(G_X^*\): key subgraphs are generated first, followed by the addition of auxiliary structures. Persistent homology is utilized to track the life cycles of topological features during this generation process.
Rationale Filtration Learning¶
A backbone GNN is used as a learnable filtration function \(f_\phi: G \to [0,1]^{|E|}\) to assign an importance score to each edge. These scores induce a graph filtration—a sequence of nested subgraphs:
Edges are incrementally added in descending order of importance (more important edges are added first). The goal is to learn an importance ranking that is consistent with the partition of \(G_X^*\) and \(G_\epsilon^*\).
Topological Discrepancy¶
The topological discrepancy is defined as the 1-Wasserstein distance between the persistent homology distributions:
where \(d_B\) is the bottleneck distance.
Loss & Training¶
The topological distinction between rationale and complementary subgraphs is enhanced by maximizing the topological discrepancy.
Computable Lower Bound¶
Kantorovich duality and Lipschitz continuous, learnable vectorization functions (Rational Hat structuring elements) are leveraged to provide a computable lower bound:
A 2-head attention mechanism is used to select the top-2 maximum values from \(k=8\) Lipschitz representation functions.
Prior Regularization¶
A two-component Gaussian mixture prior is introduced to cluster the edge filtration values:
By fixing \(w=0.5\), \(\mu_1=0.25\), and \(\mu_2=0.75\), unsupervised clustering of \(G_X\) and \(G_\epsilon\) is achieved.
Theoretical Guarantees¶
Theorem 3.4: Assuming that for all \(G\), \(|E_X| < |E_\epsilon|\) and \(G_X^*\) is minimal with respect to \(y_G\), the loss \(\mathcal{L}(\phi)\) is uniquely optimized by \(f_\phi^*(e) = \mathbb{1}\{e \in G_X^*\}\).
Key Advantage: This guarantee does not depend on assumptions about the stability or invariance of the rationale subgraphs, making it unaffected by variform subgraphs.
Experiments¶
Explanatory Performance (AUC)¶
| Method | BA-2Motifs | BA-HouseGrid | SPMotif0.9 | BA-HouseAndGrid | BA-HouseOrGrid | Benzene |
|---|---|---|---|---|---|---|
| GIN+GSAT | 98.85 | 98.55 | 65.25 | 92.92 | 83.56 | 91.57 |
| GIN+GMT-Lin | 97.72 | 85.68 | 69.08 | 76.12 | 74.36 | 83.90 |
| GIN+TopInG | 99.57 | 99.24 | 80.82 | 95.35 | 88.56 | 98.22 |
| CINpp+GSAT | 91.12 | 91.04 | 80.24 | 95.17 | 69.30 | 95.40 |
| CINpp+TopInG | 100.00 | 99.87 | 92.82 | 100.00 | 100.00 | 98.72 |
Key Findings: - The improvements are particularly significant on variform rationale subgraph benchmarks (BA-HouseAndGrid, BA-HouseOrGrid). - CINpp+TopInG achieves 100% AUC on multiple datasets. - The improvement on SPMotif0.9 (which has high spurious correlation) indicates that the method effectively mitigates spurious correlations.
Predictive Performance¶
While maintaining high interpretability, TopInG's predictive accuracy also outperforms or matches baseline methods, overcoming the traditional trade-off between performance and interpretability.
Highlights & Insights¶
- First to introduce persistent homology to inherently interpretable GNNs: It provides a brand-new topological perspective to solve the rationale subgraph identification problem.
- Unique ability to handle variform subgraphs: It demonstrates a significant advantage over existing methods on synthetic datasets.
- Theoretical guarantee independent of invariance assumptions: Rationales can vary freely across different instances.
- Adaptive topological constraints: The prior regularization is insensitive to hyperparameters and requires no tuning.
- Compatible with existing GNN backbones: It can be paired with various architectures such as GIN and CINpp.
Limitations & Future Work¶
- The computation of persistent homology increases training time.
- Only 0-dimensional and 1-dimensional persistent homology (connected components and cycles) are considered, leaving higher-order topological features unexploited.
- The extension from node filtration to edge filtration (star filtration) has limited information.
- Performance on the Mutag dataset is slightly lower than that of MAGE.
- The theoretical guarantee relies on the assumption \(|E_X| < |E_\epsilon|\), which may not always hold.
Rating¶
⭐⭐⭐⭐⭐ (5/5)
This is an outstanding work that excellently combines theory and practice. Introducing topological data analysis to interpretable GNNs is a clever and original idea. The theoretical guarantees are rigorous, and the experiments demonstrate significant advantages across multiple dimensions, especially in tackling the key challenge of variform subgraphs.