TopoFormer: Topology Meets Attention for Graph Learning
Authors: Md Joshem Uddin
Astrit Tola
Cuneyt Gurcan Akcora
Baris Coskunuzer
Venue: ICLR 2026 Poster (2026)
Keywords: Topological Data Analysis
Transformers
Graph Representation Learning
Graph Classification
Molecular Property Prediction
Primary area: learning on graphs and other geometries & topologies
TL;DR: TopoFormer turns graphs into short sequences of topological tokens, enabling transformers to capture multi-scale structure efficiently. It outperforms strong GNN and TDA baselines on graph learning tasks with theoretical stability guarantees.

Abstract: We introduce *TopoFormer*, a lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is *Topo-Scan*, a novel module that decomposes a graph into a short, ordered sequence of topological tokens by slicing over node or edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by a Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homology pipelines, *Topo-Scan* is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-art performance across graph classification and molecular property prediction benchmarks. Our results show that *TopoFormer* matches or exceeds strong GNN and topology-based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.

Published as a conference paper at ICLR 2026
TOPOFORMER: TOPOLOGY MEETS ATTENTION FOR GRAPH LEARNING
Md Joshem Uddin
Astrit Tola
Department of Mathematical Science
Department of Mathematics
The University of Texas Dallas Richardson, TX 75080, USA edu
Florida State University
Tallahassee, FL 32306, USA atola fsu. edu
mdjoshem. www.eheemmmmiiinneeeeeeeen www.nuuddiinnaaden
Cuneyt Gurean Akcora
Baris Coskunuzer
AI Initiative University of Central Florida Orlando, FL 32816, USA
Department of Mathematical Science
The University of Texas Dallas Richardson, TX 75080, USA coskunuz@utdalles. edu
edd
ABSTRACT
We introduce TOPOFORMER, lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is Topo-Scan, novel module that decom- poses a graph into a short, ordered sequence of topological tokens by slicing over node edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homol- ogy pipelines, Topo-Scan is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-att performance across graph classification and molecular property prediction benchmarks. Our results show that TOPOFORMER matches or exceeds strong GNN and topology-basee based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.
Code: htttp:://gtthubbbbbb http://tttuub.cooo///eeeeeeel /
INTRODUCTION
Graphs are powerful data structures for modeling relational data in biology, chemistry, and social networks. While recent advances in graph learning have produced strong task-specific models, most architectures lack the generalization of foundation models in vision and language (Radford et al., 2021; Bubeck et al., 2023). Achieving such penerol-purpose capability in graphs is difficult due to their irregular, Non-Euclidean structure (Wu et al., 2020), which complicates the design transferable inductive biases.
Topological Data Analysis (TDA) provides a principled approach by encoding global and local structure in way that is stable to perturbations and insensitive to node identity (Hensel et al., 2021; Pham et al., 2025). In principle, Persistent Homology (PH) offers a canonical summary how connectivity and eycles evolve across scales, and has proven useful across domains (Skaf al., 2022; Obayashi et al., 2022; Shultz, 2023). In practice, however, PH pipelines depend on persistence diagrams, which require expensive global reductions and subsequent vectorization step (e.g., images, landscapes, curves) whose design can materially affect downstream performance. On graphs, common
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sublevel/superlevel filtrations also tend to early-saturate, high-valued vertices activate early, quickly filling the complex and suppressing late-emerging features. These computational and modeling frictions have slowed the adoption of PH in graph representation learning, despite the clear promise of topological signals for multi-resolution structure.
To overcome these barriers, develop lightweight yet expressive alternative that bypasses full persistence diagrams while retaining multi-resolution topological information in a form consumable by transformers. We introduce TOPOFORMER, scalable framework that integrates topological descriptors with attention architectures, At its core is Topo-Scan, module that converts a graph into a short, ordered sequence of topological tokens across multiple resolutions. These sequences are directly consumable by attention mechanisms (Vaswani et al., 2017), enabling efficient graph-level representations within the same token- based interface used by large-scale transformer models. We therefore view TOPOFoRMER as a step toward topology-awan graph foundation models, rather than a full foundation model itself, and leave large-scale pretraining on heterogeneous graph corpora to future work. TOPO achieves strong performance graph classification and molecular property prediction under unified evaluation protocols, with theoretical guarantees on the stability of its topologiaal encodings. Our Contributions
follows:
We introduce a scalable method for turning topological structure into attention-ready sequences, en- abling transformers to process graphs without relying node embeddings or heavy preprocessing
We propose a new framework that bridges topological data analysis and deep learning, capturing both local and global graph structure through a unified attention mechanism.
We conduct comprchensive evaluation across diverse graph learning tasks, demonstrating strong performance on both graph classification and molecular property prediction benchmarks.
We provide theoretical guarantees on the robustness of our representations and show that our approach offers predictable and efficient compute, making it practical for large-scale applications.
2 MOTIVATION AND BACKGROUND
This section reviews recent work and highlights the need to integrate advanced topological methods with modern ML to overcome limitations in graph representation learning.
Persistent Homology for Graphs. Persistent Homology (PH) was first defined for filtered simplicial complexes in the early 2000s (Edelsbrunner et al., 2002: Zomorodian & Carlsson, 2005). Early applications centered on point clouds C IR-,,, where Victoris-Rips filtrations generate nested complexes A (X)) C (x c..., allowing topological features to be tracked across scales (Carlsson, 2009). The persistence diagram PD'(A)) = {[b,, d,)) records births and deaths of k d-dimensional features, with longer intervals (d, b,) interpreted more persistent and thus more structurally sigmificant (Dey & Wang, 2022).
PH has since been applied to graphs and images. For graphs, two principal approaches used. Power (distance) filtrations treat nodes as a point cloud with graph distances as pairwise distances, then build Rips filtration (Aktas al., 2019), which is typically computationally heavy. A more practical alternative in graph learning is sublevel filtrations, where scalar node/edge function f induces nested subgraphs that are lifted to simplicial complexes via cliques (upper-star extension standard). A key interpretability difference follows: in power filtrations, bar lengths reflect geometric scale; in sublevel filtrations, they reflect differences in f rather than physical size, so "long bars + important features" need not hold universally. Poorly chosen may yield many short-lived features or early saturation, while task-aligned or learnable filtrations can mitigate these effects (Hofer et al., 2020). Rather than viewing this as an intrinsic weakness, we take it as motivation to design fixed-budget, stable summaries that integrate smoothly with modern ML (See App. C.8 for discussion).
The standard PH pipeline for graphs has three main steps (Coskunuzer & Akçora, 2024): filtration persistence computation, and vectorization. Given a graph G = (V,E), function f V + I with thresholds {ax) J,N induces subgraphs G1 C C GN. where G, contains vertices {u f(v) S a.). Lifting each G to its clique complex G. yields filtration (G.). Persistence diagrams (G, f) = {(b,, d,)} record births and deaths of H₃ (ÿ.) and are typically vectorized via persistence images, landscapes, or Betti curves (Ali al., 2022).
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In recent years, the ML community has increasingly recognized the value of topological encodings for graph-level tasks, with PH-based methods showing strong results across domains (Immonen et al., 2023; Demir et al., 2022; Verma et al., 2024; Loiseaux et al., 2023b; Horn al., 2021; Chen et al., 2024a). Despite this promise, two bottlenecks hinder broader adoption: (1) the computational overhead of persistence computations in large pipelines (Otter al., 2017), and (2) the difficulty of choosing vectorizations that align with downstream objectives (Ali et al., 2022). Our TOPOFORMER framework addresses both by producing a compact sequence of stable, low-cost topological tokens that feed directly into attention layers, thereby bypassing full persistence diagrams and bespoke vectorizations while remaining compatible with efficient graph-specing computations.
Recent methods learn neural approximations of persistence- baasse topological features to reduce the cost of exact PH. RipsNet (de Surrel et al., 2022) estimates Rips persistence diagrams for point clouds directly from raw data, while Yan et al. (2022) approximate graph topological features with GNN. Our approach complementary: instead of approximating persistence diagrams, Topo-Scan bypasses global PH and directly builds short interlevel topologiaa sequences tailored to Transformer encoders.
Transformers. Transformers (Vaswani et al., 2017) underpin transferable models language and vision (Devlin, 2018; Dosovitskiy et al., 2020) by learning from ordered token sequences with long- range dependencies. On graphs, adapting attention is challenging due to variable size, permutation invariance, and irregular connectivity. Our design sidesteps these issues: Topo-Scan yields a short, 1D ordered sequence of topological tokens with a fixed channel width, so positional encodings and attention operate in their native regime, without graph-specific heavy machinery. This makes transformers natural, efficient backend for multi-resolution structural signals. By contrast, recent graph transformer models such as Graphormer (Ying et al., 2021), GPS (Rampášek et al., 2022), and related architectures operate directly on node tokens and inject structure via shortest-path or Laplacian-basee positional encodings and attention biases. In TOPOFORMER, each graph is first compressed into short sequence of topological tokens. so attention runs on a fixed-length, purely topological sequence rather than on all nodes of the original graph.
Molecular Property Prediction. Molecular property prediction (MPP) is central to drug discovery (ADMET). Classical pipelines use engineered fingerprints with RF/SVMs Massaag- et al., 2015); deep learning extends to MLPs on fingerprints. sequence models SMILES (Rong et al., 2020), and GNNs on molecular graphs (Wieder et al., 2020), with recent 3D methods trading accuracy for higher compute and sensitivity rotations (Gasteiger et al., 2021; Li et al., 2022b). Despite progress, DL does not always surpass strong classical baselines on realistic benchmarks (Janela & Bajorath, 2022; Valsecchi et al., 2022), motivating transformer variants (Sultan et al., 2024), geometric models (Liu et al., 2022b), and topological approaches (Demir et al., 2022; I Loiseaux et al., 2023a). Evaluation protocols also vary (e.g., scaffold vs. random splits), affecting reported generalization. Our approach unifies robust topological structure with a scalable attention backend, providing an effective, split-agnostic representation for MPP.
3 TOPO-TRANSFORMEER
TDA captures multi-scale structural patterns while offering robustness to noise, mak- ing attractive for representation learning. pipeline, consisting of filtration construction, persistence diagram computation, and vec- torization, introduces inefficiencies and lacks adaptability, particularly in graph settings. While persistence diagram computation is standardized, vectorization remains ad hoc and significantly impacts model perfor- mance (Ali et al., 2022). Our goal is to develop an efficient and scalable alternative to this workflow for graph representation learning.
However, the standard Persistent Homology
Sqo.11
Our first insight is that the strict nested- ness condition required in PH is not always necessary for graphs. Unlike point clouds, graphs inherently encode structural relation- ships that permit more flexible and direct
S
S
Figure 1: Topo-Scan. Topo-Sean decomposes graph into sequential topological slices node and edge filtrations. The top row shows node-based filtrations; the bottom row, edge-based ones.
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91
92
93
Fx
9s
2
5
TopoScar
G
91..4
G(4,)]
Figure 3: Topo-Scan vs. PH. This toy example highlights the key differences between the Topo-Scan filtration and standard PH filtration, where node values indicate filtration function values. PH, carly-activated nodes quickly saturate the graph, suppressing the emergence of later topological features. As shown, PH yields relatively uninformative barcodes with 3o 21 1, 1) and B1 (0. 0, 0,0,000000 0,0,), while Topo-Scan captures richer topological dynamics, producing 3u (2, 1, 1, 2, 4) and B1 (0. 1,0).
extraction of topological features. Building
on this observation, we bypass persistence diagrams and vectorization by directly extracting topo- logical sequences from structured graph slices. This shift enables efficient and adaptable pattern extraction and forms the foundation of scalable learning framework.
Topo-Scan. Traditional sublevel filtrations on graphs often saturate rapidly, and once most nodes join the subgraph at low thresholds, little new structure emerges and important patterns at larger scales are lost. Topo-Scan overcomes this by first imposing a directional hierarchy via scalar function f: V.V + R. then slicing the graph into a sequence of overlapping subgraphs along increasing values of f. Rather than waiting for single threshold to engulf the entire graph, each slice captures fresh topological information, such as connectivity changes and emerging loops, without early collapse. We compute basic invariants (e.g. Betti numbers) on each slice to form a compact, ordered signature sequence. Feeding these ordered descriptors into transformer lets the model attend to structure at every scale, ensuring no signal is lost to premature saturation (See Fig. 3 for a toy example).
Let f - R be filtration function defined on the vertices, with thresholds og min f(v) 01 < ... < ON = max,ey f()). In most cases, the thresholds are selected either as evenly spaced values based
quintiles. Next, for each dg, define V; {rr E V 0: I f(rr) S ar+m) and G; as the induced subgraph 9, = (V,, &,), where & = (ers E E U,, U, E V.J. The clique complex of Gi denoted G., forms a sequence {G:) called slicing. We call this process Topo-Scan, which decomposes graphs into topological slices, similar to medical scans revealing structural layers. Leveraging a hierarchical structure, it adapts to node and edge filtrations, weighted graphs, and diverse relations, capturing local and global topological patterns for robust, scalable representation learning. It remains robust by tracking short-lived features effectively and is scalable through parallelized slice extraction.
The resolution (N) determines the number of slices, while the thickness (m) speci- fies the range of nodes included in each slice. After constructing ) we compute four outputs for each slice: Bo (G.) (Betti- o, connected components), 31 (G.) (Betti-1. cycles/holes), |V: (node count), and |& (edge count). These outputs form ordered sequences of size N, such as (g)
Berso Tea-Son
(a) IMDB-B Betti-0
(b) IMDB-M 100a-M-Betti--
Figure 2: PH vs. Topo-Scan. Average Betti-O counts under degree filtration with 100 thresholds on IMDB-B - and IMDB- M. Standard PH saturates early, causing a sharp decline near the end of the curve. Topo-Scan avoids early saturation and continues reveal late-emerging features, often surpassing counts at higher thresholds.
((.)) for k: = 0. While (g) the primary topological outputs, {
and { serve as normalization factors (see Figure 1). These sequences are con- catenated into a sequence (vector) [(g) of length 4N where N is the number of slices.
key distinction from PH lies in activation: PH includes all nodes up to a threshold, causing early saturation in dense graphs, while Topo-Scan sliding window to preserve late-emerging features
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and capture fine structure (see Fig. 2 and App. C.8). Slice thickness controls locality, allowing flexibility across datasets. Its localized design ensures robustness to noise and enables parallelization, making it ideal for scalable ML workflows.
Vectorization Choice. Among many possible vectorizations, we deliberately use a very low- dimensional token per slice, (So, $1; [Vl, [&i]). Global vectorizations such persistence images landscapes aggregate information over the entire filtration into a single feature vector, which largely destroys the sequential structure that Topo-Scan is designed expose. In contrast, our Betti-base based tokens preserve how components and cycles evolve across slices; richer per-slice invariants could be plugged in, but we focus on this minimal choice isolate the benefit of the sequential representation.
FORMER. We use the ordered sequences of topological features in Transformers, which excel at capturing sequential structures and complex dependencies through self-attention mechanisms, making them well-suited for tasks requiring order and contextual understanding. While traditional PH processes sequence of simplicial complexes [Ar], , this sequential structure often lost during the persistence diagram and vectorization stages, where outputs are transformed into unordered vectors. Topo-Scan preserves the sequential nature of topological features and aligns them with transformers' ability to model positional relationships.
MI Model. Our transformer architecture (Fig. 4) consists of an embedding layer that processes input sequences, transformer encoder that captures hierarchical dependencies through self-attention mechanisms, and fully connected classification head that maps learned representations to output predictions. To enhance generalization and mitigate overfitting, integrate regularization techniques, such as dropout and weight decay, ensuring robustness across diverse graph learning tasks. Formally, given input sequence x R=XT'xD, where the number of graphs, T the sequence length, and D the token dimensionality, the sequence is embedded via E, with positional encoding P added. This processed sequence is then passed through multi-layer transformer encoder, producing an output representation z, which is flattened and normalized before being classified via fully connected layer.
Expanding this model, we introduce maal-transformer framework with an integrated multi-layer perceptron (MLP) classifier to handle diverse input modalities. The model processes three distinct inputs: X, and X2 through independent transformers Tj and T2, and X3 through an MLP M. Their respective outputs Z1, Z2, and Z3 are combined using a learnable weighted sum, allowing the model to adaptively balance feature contributions:
ccombined
3 22 + (1
3)
where a and 3 are learned during training. The aggregated feature vector then batch-normalized and passed through fully connected layer to produce the final classification output: FC (Z(cobbbinca)) where 9 E RC represents the class probabilities, with C being the number of output classes. Model details given in Appendix C.
3.1 STABILITY OF TOPO-SCAN SEQUENCES
useful graph vectorization should be robust: small changes in the filtration signal should not cause large changes in the output sequence. We formalize this for Topo-Scan on the fixed clique complex & of G (V, E) using upper-star extensions of node functions.
Setup. Let f.g V R be filtration functions, extended to & by the upper rule i(()) maxveg hi(v). Fix shared threshold grid op <... < aN, window width m. stride and windows I. Oits: mm] for / 0,..., T- where T L(N m)/s] + 1. For k {0, 1}, the 1-th Topo-Scan token is the interlevel Betti number
(1) == dim Ha(())))) he (f.g).
Theorem 3.1 (Discrete I stability of Topo-Scan). There exists C = C(,, {ai), m, 8) such that for E 10,
(acs)-Ba(G... S C d M?).
where My denotes the k-dimensional interlevel (level-set) persistence module of the upper-star filtration induced by h G, and d the bottleneck distance between such modules.
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Tepo-Scan Filtration
Filrationn
Graph
Topological Sequence
Topological Sequence
Attention
Concatenation
Encoskr
Geaph Property
Figure 4: TopoFormer Flowchart: Given input graph G, sequential substructures extracted via Topo-Scan. Each substructure encoded into four-dimensional topological signature. These sequences are processed by transformer model, and outputs from multiple filtration functions are fused using attention-base concatenation. A final prediction layer maps the representation to the target graph property.
Corollary 3.2. For upper-star filtrations on fixed complex, interlevel modules satisfy d Mg) |l glloo. Hence f) B., g)ll S C gllooo
Outline. Each token counts classes surviving exactly over I;; under bottleneck matching, only classes within 8 of the interval boundary can change their contribution, so per-window changes are (6)) and summing
windows yields the discrete bound with constant depending the window schedule and the (finite) bar complexity of G. Full details and references are Appendix
Beyond stability, the Topo-Scan sequences 3g (G, h) are closely related to classical PH invariants: they can be viewed as discrete sampling of the rank invariant / Betti curve of the interlevel module M along our window schedule. Thus, Topo-Scan provides a coarse but structured, Transformer-ready discretization of the same homological information underlying barcodes and stable-rank summaries; see Appendix Remark B.3 for further discussion.
4
EXPERIMENTS
4.1
EXPERIMENTAL SETUP
Datasets. We report the TOPOFORMER performance in two graph learning tasks: graph classifi-
cation and molecular property prediction (MPP).
Graph Classification Datasets. We use
graph classification benchmark datasets: (i) molecular graphs from BZR. MUTAG and COX2 (Kriege &
nine
Mutzel, 2012); (ii) biological graphs
PROTEINS (Borgwardt et al., 2005);
and (iii) social graphs, including IMDB-
Table 1: Graph elassification datasets.
Binary, IMDB-Multt REDDIT- SEDDIT-Binary, and REDDIT-Multi-5 (Yanardag et al., 2015). We also include large-scale dataset, OGBG-MOLHIV, from Open Graph Bench- mark (Hu et al.. 2020b). Dataset statistics are provided Table I.
Datasets
#Graphs
[V]
[
Classes
B7R
405
35.75
38.36
COX2
467
41.22
43.45
MUTAG
188
17.93
19.79
PROTEINS
1113
39.06
72.82
IMDB-I B
1000
19.77
96.53
For molecular prop- erty prediction (MPP), we employ seven datasets from MoleculeNet (Wu et al., 2018): BBBP Ablood-brain barrier penetra- tion), Tox21, ToxCast, ClinTox (toxicity
MPP Datasets.
IMDB-M M
1500
13.00
65.94
REDDIT-B
2000
429.63
497.75
REDDIT-SK
4999
508.52
594.87
NGBG-MOLIHE
41127
25.5
27.5

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Table Graph Classification, Accuracy on eight benchmark datasets using 10-fold CV. Baseline results are taken from the respective papers using per column. The last two columns report the average deviation (AvD) from the best best-performing model and the average rank (AvR) across all datascts.
the same setting. We mark the 1st (blue). 2nd (purple), and 3rd (green)
Model
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
REDDIT-SK
AvD]
Avi.
GINN (Erics et al. 2020)
80 80422227
75,80=1.0
71. 20:00:0
49. 10:150
89:902190
16.10
6.0
8.5
(Camibre
2020)
89.80
74.001
71.20
48. 80.0
55.60:
5.2
8.7
(Bodnar atal..22111
84.00).00
15.30.330
73.80
50.90.35:
86.20
51,90,330
FC-V 10 Bray al. 2021)
85.61
a1.0
=7.31
74,54:004
73.84
46. 46.80=233
89.41
52.36
5.7
9.2
SubMix (Yoo al., 2022)
86.34a220
84.68:2700
80.99
67.80:21e
70.30s1,a0
46.47=228
8.4
11.5
G-Mix (Ham et 2022)
$4.15e231
83.832210
81,96 m1
66. 11
69. 40.1.0
46.40=2.
9.1
12.8
RGCL et 2022a)
84.54a147
19.3
87.66
15.03:043
49.31
90.34
56.38
AusoGCL (Yin 20.,
86.27
88.54
75,80:882
72.32
50.60=aac
88.58:140
56.75
4.7
7.0
WWLS (Fang ct al., 202366
88.02
81
31.58
88, 30
75.35
75.08:mma
=1.61
4.5
5.2
PGOT (Qian al. 2024)
87.32
32.98.5531
92.63_2..
73.21
62.90
51. 31.33
6.1
7.5
IMPrChhnn
20124)
88.79 28.79.044
72.78
14.44.0
48.01
91.03.162
54,413319
4.8
7.5
2014)
85.53:140
82.44 14:3
69.06.110
47.93.113
6.9
MP-HSM (Loiseaux et al., 20236)
77.10s330
85.60=130
74.60=210
74.50s2.00
47.90:322
6.9
10.1
TopaCCL (Chen et al.. 2024g)
$7,17
81.45etus
90,09
77.30:
74.67amu2
52.81:921
90.40:055
3.5
4.3
DASP (Yo el al., 2025)
89,40=30
84.80:4.0
91,90=saa
77.20
81.40:500
51,20
57,60=114
1,6
2.8
TOPOFORMER
92.36:411
83.93
94.68=4.
77.64:224
78.90
55.40:479
91.50-1220
57.99
0.5
prediction), SIDER (adverse drug reactions), HIV (replication inhibition), and BACE (3-secretase 1 inhibitors). Dataset statistics are provided in Table 3 (top rows).
Model Setup. We use Topo-Scan generate topological signature sequences. which are fed to Transformer classifiers. Each filtration (20 thresholds, width 2) yields four sequences of length 19 (Betti-0, Betti-1, node count, edge count), giving 76 features per filtration. For graph classification, we use Ollivier- Ricci curvature and Heat Kernel Signature; and for molecular property prediction including the OGBG-MOLHIV dataset, we use atomic weight and Ollivier-Ricci curvature. Indepen- dent Transformers process each filtration, and their outputs are fused by attention before final linear layer.
For MPP, use TOPOFORMER with the standard molecular fingerprints, processed by two-layer MLP and combined with topological features via attention, yielding TOPOFORMER'. We report 10-fold CV accuracy on graph classification, scaffold-split AUCs over three runs for MPP (Fang et al., 2023a), and use the standard split for OGBG-MOLHI,
Hyperparameters. For model optimization, we employed the Adam optimizer with a learning rate of 0.001. We also use standard regularization techniques-dropour (0.5), weight decay (le-4), and batch formalization-commnn architecture was designed with a hidden dimension of 32. Hyperparameters are given in App. C.5.
employed in transformer training. The transformer model
Classical PH requires global boundary- reductions with cubic worst-case cost and poor parallelism (Otter et al., 2017). TOPOFORMER skips persistence diagrams entirely: instead of global reductions. computes 3g and 31 per slice on the clique-complex 2-skeleton. Sn uses union-find on the 1-skeleton, while 31 is derived from sparse -tiingle operators after triangle enumeration (no cycle-rank identity due to clique complexes). This yields O(VV + |F,] + T:) per slice, aggregated as OOEE, (IV + |E + T:)) across k slices and T, filtrations. Since slices are independent, Betti computations are fully parallelizable By bypassing PD computation and vectorization, TOPOFORMER achieves multi-fold runtime and memory gains while retaining task-relevant topological features (Appendix C.3). In Appendix C.6, we further show that TOPOFORMER consistently outperforms classical PH across multiple filtration functions and vectorization schemes in the graph classification task.
Computational Complexity.
Implementation and Runtime. We implemented our approach in Python and conducted ex- periments on a 12th Gen Intel Core i7- 17-12 vPro processor (E-cores up to 3.50 GHz, P-cores up to 4.80 GHz) with 32GB .PDDRS-64000HH RAM. Topo-Scan feature extraction took 269.38 seconds for OGBG-MOLLHIVHHIV and 29.51 seconds for REDDIT-5K; other datasets were faster. The remaining model runtime was negligible. More timeruns and a comparison with PH can be found at Appendix C.3.
4.2 RESULTS
We evaluate our method against 20 state-of-the-ar baselines spanning several categories. These include: GNN-based models such GCN, DGCNN, DiffPool,
Graph Classification Baselines.
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Table SOTA MPP Models. ROC AUC comparison on molecular property prediction with scaffold splitting. We mark the 1st (blue), 2nd (purple), and 3rd (green) per column. The last two columns report the average deviation (AvD) from the best-performing model and the average rank (AvR) across all datasets.
Model
BBBP
Tox21
ToxCast
SIDER
ClinTox
BACE
HIV
AvD
AvR.
Molecules
2,039
7,831
8,577
1,427
1,480
1,513 1
41,913
Task
12
617
27
N- G-GAAM (Liu 2019)
91.2:33
76.1
63.2200
87.5+23
79.1=13
78.7104
8.5
8.4
PT-GNN (Hu et al., 2020s)
70.8=15
78.7
65.7+06
62.7=00
72.6
84.5=07
79.9
12.4
8.7
CMPNN (Song et al., 2020)
92.7=17
80.3
70.8
61.0236
89.8-0a
86.7=02
78.2+22
6.2
MGSSI (Zhang al. 2021)
70.5211
74.0
64. 1197
59 2206
80.7
79.7=08
79.5
13.5
11.7
GEM (Fang et al., 2022)
70.5=20
78.1
68.6+02
63.2+115
90.3+0.7
87.9=10
81.3+
8.9
6.6
GROVER (Rong al., 2020)
86.8+22
82.0#1.
56.8+33
61.2+25
70.3+137
82.8-36
68.2+11.1
13.5
9.9
GraphMVP (L.u al,, 2022a)
72.4
76.5
63,1
63.9a11
19.1
81.2=09
77.0
12.7
10.4
MolCLR (Wang et al., 2022)
72.6
77.2:06
65.9+21
61.3-16
89.8+2.7
88.5-22
77.4+96
9.9
7.8
MoICLR-2 (Wang et 2022)
72.4=07
78.4
69.1
59.7=34
88.0=40
85.0=24
77.8+5..5
10.2
8.6
KANO (Fang al., 2023a)
96.0:14
83.7:11
73.2:16
65.2:99
94,4
93.1=21
85.1
2.0
MV- (Lun et al., 2024)
73.6202
80.3
70.0g04
67.3:408
95.6114
88.2=0A
81.4:4.3
6.5
3.6
MolFuse (Zheng et al., 2024)
74.3
77.6a0.
64.1a03
69.5=10
95.5+333
87.2=113
78.6+0.9
7.9
6.2
MORE (Son al., 2025)
71.9209
75.6s05
64.61u6
60.9:06
81.0a07
82.8=13
77.0
12.6
11.1
TOPOFORMER*
89.5
82.7.99
75.3ins
63.1
96.510
95.9_03
81.2a0s
2.5
2.8
ECC, GIN, and GraphSAGE (with the best results reported by Errica et al. (2020)); topological methods including PersLay, DMP, FC-V, WWLS, MP-HSM, and EMP: GNNs with data augmentation such as SubMix, G-Mix, and EPIC; contrastive learning methods including RGCL, AutoGCL, and TopoGCL; and prototype-based methods such as PGOT. We further include the recent graph kernel method DASP (Ye et al., 2025). complete list of baselines is provided in Table 2.
Graph Classification Results. In graph classification, TOPOFORMER attains the best second- best accuracy on 7 out of 8 benchmarks (Table 2). Aggregating across datasets, it achieves an average deviation (AvD) of 0.5 from the best model and average rank (AvR) of 1.5, demonstrating consistent top-tier performance. Notably, TOPOFORMER establishes new state-of-the-ar on BZR, BUTAA..POOTINNS IMDB-M, REDDIT-B, and REDDIT-5K, while remaining highly competitive elsewhere. It also surpasses common pooling- based methods on these datasets (see Table 12). On the large-scale
benchmark (Table 4), TOPOFORMER" reaches an AUC within 2 points of the strong Graphormer baseline, underscoring both its scalability and the strength of topological signals as an inductive bias in graph learning. For this table, we restrict baselines to peer-reviewed published methods reported in the literature, rather than including unpublished leaderboard entries in (Hu et al., 2020b).
MPP Baselines. We compare against strong supervised, self-supervised, and contrastive meth- ods for molecular property prediction (MPP). Su- pervised: CMPNN (message passing on molec- ular graphs). Predictive relf-supervision: N- GRAM, PTTGNN, GROVER, MGSSL, GEM. Con- trastivelangmentationn and 3D: GraphMVP (with
Table 4: ROC AUC results for OGBG-MOLHIV dataset.
Model
ROC AUC
GIN-VN (Xu al., 2018)
77.80=1.81
HGK-WL (Togninalli al., 2019)
79.05
WWL (Borgwardt et al., 2020)
75.58
3D), MoICLR, Mo1CLR-2, Knowledge-awarn / prompts: KANO.
PNA (Corso al., 2020)
79.05=1.32
DGN (Beaini al., 2021)
79.70=u99
GraphSNN (Wijesinghe al., 2021)
79.72.1180
GCN-GNorm (Cai et al., 2021)
78.83.1100
Recent multi wiwlfiiview/fussinn models: MV-Mol (multi- view molecular representations), MORE (modality- aware molecular representation learning), and Mol- Fuse (fusion of heterogeneous molecular signals). See Table 3 for full references.
Graphormer (Ying et al., 2021)
$0.51
Cy2C-GCN (Choi et al., 2022)
78.02=0000
GAWL (Nikolentzos et al., 2023)
78.3410099
LLM-GIN (Zhong et al., 2024)
79.22+NA
GMoF-GIN (Wang et al., 2023)
76.90=0000
TopER (Tola et al., 2025)
80.21 +
On molecular property predic- tion, TOPOFORMER shows strong adaptability when paired with Extended Connectivity Fingerprints. Against state-of-the-art supervised, contrastive, and fusion baselines, TOPOFORMER* achieves the best ROC AUC on ToxCast, ClinTox, and BACE, and is the runner-up on Tox2/ (Table 3). remains competitive HIV, trailing the leader by only a small margin. Aggregating across all seven benchmarks, TOPOFORMER attains the second-lowest average deviation from the column best (AvD 2.5) and the second-lowest average rank (AvR 2.8), confirming consistent top-tier performance
MPP Results.
TOPOFORMER
78.19 +0.19
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Published as a conference paper at ICLR 2026
Table 5: TOPOFORMER vs. PH: Accuracy results for three topological models using degree centrality, Ollivier-Riee and HKS filtrations. The PH-MLP model utilizes Betti vectors derived from regular sublevel filtrations combined with an MLP, while PH-TR applies transformers to the same vectors. The TOPOFORMER uses Betti sequences generated via the Topo-Scan on the same filtration function and applies transformers.
Filtration
Model
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
PH-MLP
82.71
36.442530
84.062465
68.37-3397
65.70=4.0
45.07=259
89.50=257
Degree
PH-TR
86,432435
18.15
86, 11
77.54-244
75.00:211
50.67=347
92.30:
TOPOFORMER
91.10=514
80.27
92.54:5.12
77.45=4.02
74.20
50,33=1.52
89.75
PH-MLP
85.45:336
78.16:55.
84.06=521
65.50=4426
68.00+3.55
44.87=365
85.65
O.Ricci
PH-TR
88 28.62a540
78.1645573
87.61
77.27-3.08
72.20
48.00=433
90.65
TOPOFORMER
90.38
80.72:s.44
92.54:447
77.90-3.17
74.70
51.53=3349
91.90=273
PH-MLP
44.96:4442
78.19 a8.19
84.09
70.80=4.m
71.10
47,93=320
18.10
HKS
PH-TR
89.60=584
79.89
94. 12:422
77.18=3.15
76. +3390
53.60=331
87.25
TOPOFORMER
90.62:491
83,95
95.32::5.58
77.35-2.86
77.90
54.07=2.54
90.05 22.41
alongside recent SOTA models such as KANO, MV-Mol, and MolFuse. We also benchmarked against hybrid classical (HC) models (Appendix A.3), where TOPOFORMER achieves the best result on four out of seven datasets and highly competitive results on others (Table 8). These findings highlight that transforming topology into compact, attention-ready tokens yields a robust and adaptable molecular predictor.
We further report Hybrid Classical baselines combining fingerprints, SMILES, and graph features with standard learners in Table 8. See Appendix A.3 for details of these models.
4.3 ABLATION STUDIES
We conduct four ablation studies, as follows.
TOPOFORMER vs. PH (Table 5): We compare TOPOFORMER with two persistent homology models using the same filtration functions and thresholds. PH-MLP uses sublevel filtrations with Betti vectorization followed by an MLP, while PH-TR replaces the MLP with Transformer, treating Betti vectors as sequences. TOPOFORMER instead uses our proposed Topo-Scan to directly extract topological sequences. PH-TR outperforms PH-MLP, showing that sequential encodings preserve richer information than static features. TOPOFORMER further improves on PH-TR, indicating that Topo-Scan captures more expressive structure than standard PH filtrations.
We evaluate TOPOFORMER and Sriendee-Coonectvviig Fingerprints (ECFPs) both independently and in combination, including integration with PubChem descriptors. While topological and fingerprint models perform moderately on their own, their combination consistently outperforms individual baselines, suggesting that topological features complement domain-specific descriptors.
Effect of molecular fingerprints (Table 7):
Sensitivity to width parameter (Table 6): We analyze how the sliding window size influences the performance of Tcan-Scan. See Appendix C.5 for further details.
Table 6: Width Parameter. Performance comparison for different window width parameters across datasets.
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
m 2
89.89 1374
78.361493
92.02.724
77.28
74.20
51.53:334
86.60:299
Degree Centrality
3
88.64
78.38
90.41
76.9243mt
33.20
51.132308
86.90:2a1
88.86.1433 m
78.1616077
92.57.1565
76.91
74.101398
49.67.1536
85.851288
90.60
78.60
89.91
77.26
79.10
54.53
91.40
Ricci
m
3
89.14 44.1444
78.154573
91.02::6445
77.72:336
78.80
53.73-44006
89.95:224
4
88.39.26.4
78. 17:550
89.85 40
77.35=4405
78. 10
53.87+537
89.65
2
90.62:491
83.95
95.32
77.35
77.90
54,07:254
90.05
HKS
91.09::4.88
85.01:4844
95.23
78.17:4..4
76.90
53.60+3300
88.80=
m 4
90.63-14.09
83.75
95.12:4605
78.07=22.4
77.00
53,40
8..00
Single vs. multiple filtration functions (Table 13): We test several node based and edge based functions to study how filtration choice affects performance. We observe that single-filtration

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Published as a conference paper at ICLR 2026
TopoFormer (for example, using only HKS or only Ollivier-Ricei already achieves strong results, while combining filtrations yields modest but consistent improvements on some datasets. This indicates that multiple filtrations are flexible way to incorporate complementary structural signals rather than a requirement for good performance.
Discussion. TOPOFORMER delivers consistently strong performance across broad range of graph classification benchmarks, outperforming state-of-the-ar baselines and achieving the best overall accuracy on datasets. These results demonstrate the model's ability extract essential structural information through topological patterns while producingffxeed-sizzz fixed-size sequential representa- tions. Such representations are particularly well-suited for Graph Foundation Models, which require consistent and transferable embeddings across graphs of varying sizes and domains. Table 2 further reveals that among the six topological baselines (PersLay, DMP, FC-V, EMP, MP-HSM, TopoGiCI.), TOPOFORMER MER achieves the best performance, despite being architecturally simpler and more compu- tationally lightweight. This supports our design philosophy that robust topological summaries, when properly structured, can outperform more complex pipelines.
Crucially, TOPOFORMER departs from the standard GNN paradigm of first learning node embeddings followed by global pooling. While effective, this node-centric strategy treats graphs as unstructured point clouds in latent space, requiring repeated updates as embeddings evolve, often at the cost of coherence and efficiency (Mesquita et al., 2020; Liu et al., 2023). In contrast, topological models treat graphs as structured wholes and directly encode global patterns. By bypassing intermediate node embeddings, TOPOFORMER provides streamlined and principled approach for learning stable, transferable graph-level representations.
Limitations and future work. Our focus in this work is on a streamlined, graph-level instantiation of TOPOFORMER, which also suggests several natural extensions. We restrict attention to low- dimensional homology (Ho, H1) on a fixed clique complex with small set of standard filtrations (degree, curvature, HKS); incorporating richer per-slice invariants or learnable filtrations could further boost expressivity while keeping the same Topo-Scan Transformer backbone. Likewise, we concentrate on widely used graph-classfficttion and molecular benchmarks, leaving node-/edge-level tasks and more heterogeneous settings (e.g., temporal or citation graphs) to future work. Finally, Topo-Scan is designed persistent homology, and we see developing additional theory and applications for such summaries as an interesting direction for the TDA community.
lightweight, scan-style summary that complements rather than replaces full
5 CONCLUSION
Fixed-size, transferable representations remain a central challenge in graph learning. We introduce ToPOFORMER, a scalable framework that encodes multi-scale topological structure into attention- ready sequences. By replacing full persistence diagrams with lightweight, slice-wise invariants via Topo-Scan, our method integrates seamlessly with transformer architectures while offering theoretical stability guarantees. TOPOFORMER achieves -iie-offthheart results across graph classification and molecular property prediction tasks, with predictable compute and strong generalization. Looking ahead, aim to extend this framework toward graph foundation models by combining topological and spectral signals through large-scale self-supervised pretraining, and by adapting to dynamic and heterogeneous graphs via learnable filtrations.
ACKNOWLEDGMENTS
This work was partially supported by Canadian NSERC Discovery Grant RGPIN-2020-056655 Data Science Blockchains, National Science Foundation under grants DMS- DMS-2220613, and DMS- 2229417. The authors acknowledge the Texas Advanced Computing Center (TACC) at UT Austin for providing computational resources that have contributed the research results reported within this paper.
REFERENCES
Henry Adams, Tegan Emerson, Michael Kirby. Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A
10

---

Published as a conference paper at ICLR 2026
stable vector representation of persistent homology. Journal of Machine Learning Research, 18, 2017.
Mehmet E Aktas, Esra Akbas, and Ahmed El Fatmaoui. Persistence homology of networks: methods and applications. Applied Network Science, 4(1):1-28, 2019.
Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo, and Manuel Periano-Trriuergn A survey of vectorization methods in topological data analysis. arXiv preprint 22xi::2212.09703, 2022.
Håvard Bakke Bjerkevik. On the stability of interval decomposable persistence modules. Discrete Computational Geometry, 66(1):92-121, 2021.
UIrich Bauer and Michael Lesnick. Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the thirtieth annual symposium on Computational geometry, pp- 355-364, 2014.
Dominique Beaini, Saro Passaro, Vincent Létourneau, Will Hamilton, Gabriele Corso, and Pietro Liò. Directional graph networks. In International Conference on Machine Learning, pp. 748-758. PMLR. 2021.
Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Spectral clustering with graph neural networks for graph pooling. In International conference machine learning, pp. 874-883. PMLR, 2020.
Cristian Bodnar, Cätälina Cangea, and Pietro Liò. Deep graph mapper: Seeing graphs through the neural lens. Frontiers in big Data, 4:680535, 2021.
Karsten Borgwardt, Elisabetta Ghisu, Felipe Llinares-López, Leslie O' D'Bray, and Bastian Rieck. Graph kernels: State-of-the-ar and future challenges. Foundations and Trends in Machine Learning, 13 (5-6):531-712, 2020.
Karsten Borgwardt, Cheng Soon Ong, Stefan Schönauer, SVN Vishwanathan, Alex Smola, and Hans-Peter Kriegel. Protein function prediction via graph kernlss. Bioinformatics, 21 21(uppl__)) i47-i56, 2005.
Magnus Botnan and Michael Lesnick. Algebraic stability of zigzag persistence modules. Algebraic & geometric topology, 18(6):3133-3200 2018.
Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, et al. Sparks of artificial general intelligence: Early experiments with gpt-4. arXiv preprint 20X.v:2303.12712. 2023.
Chen Cai and Yusu Wang. Understanding the power of persistence pairing via permutation test. arXiv preprint arXiv:2001.06058 2020.
Tianle Cai, Shengjie Luo, Keyulu Xu, Di He, Tie-yan Liu, and Liwei Wang. Graphnorm: A principled approach to accelerating graph neural network training. International Conference on Machine Learning, pp. 1204 1215. PMLR, 2021.
Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308. 2009.
Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In International Conference on Artificial Intelligence and Statistics, pp. 2786-2796. PMLR, 2020.
Adrià Cereto-Massaguee María José Ojeda, Cristina Valls, Miquel Mulero, Santiago Garcia-Vallvé, and Gerard Pujadas. Molecular fingerprint similarity search in virtual screening. Methods, 71: 58-63, 2015.
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes and silhouettes. In Proceedings of the thirtieth annual symposium Computational geometry, pp. 474 2014.
11

---

Published as a conference paper at ICLR 2026
Yuzhou Chen, Jose Frias, and Yulia R Gel. Topogel: Topological graph contrastive learning. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 11453-11461, 2024a.
Yuzhou Chen, Ignacio Segovia- -rooii--D Cuneyt Gurcan Akcora, Zhiwei Zhen, Murat Kantar- cioglu, Yulia Gel, and Baris Coskunuzer. Emp: Effective multidimensional persistence for graph representation learning. In Learning on Graphs Conference, pp. 24-1. PMLR, 2024b.
Kyunghyun Cho, Bart van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 1724-175 1734, 2014.
Yun Young Choi, Sun Woo Park, Youngho Woo, and U Jin Choi. Cycle to clique (cy2c) graph neural network: A sight to see beyond neighborhood aggregation. In The Eleventh International Conference Learning Representations, 2022.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007. doi: 1110077/00444-00--2776666
Gabriele Corso, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Veličković. Principal neighbourhood aggregation for graph nets. Advances in neural information processing systems, 33: 13260-13271, 2020.
Baris Coskumuzer and Cüneyt Gürcan Akçora. Topological methods in machine learning: A tutorial for practitioners. arXiv preprint arXiv:2409.02901 2024.
Thibault de Surrel, Felix Hensel, Mathieu Carrière. Théo Lacombe, Yuichi Ike, Hiroaki Kurihara, Marc Glisse, and Frédéric Chazal. Ripsnet: a general architecture for fast and robust estimation of the persistent homology of point clouds. In Topological, algebraic and geometric learning workshops 2022, pp. 96-106. PMLR, 2022.
Andac Demir, Baris Coskunuzer, Yulia Gel, Ignacio Daaviiaa-Dminuuez, Yuzhou Chen, and Bulent Kiziltan. Todd: Topological compound fingerprinting in computer-aided drug discovery. Advances in Neural Information Processing Systems, 35:27978-27993, 2022.
Jacob Devlin. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint urXiv:1810.04803 2018.
Tamal Krishna Dey and Yusu Wang. Computational Topology for Data Analysis. Cambridge University Press, 2022.
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, and Sylvain Gelly. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929 2020.
Edelsbrunner, Letscher, and Zomorodian. Topological persistence and simplification. Discrete & computational geometry, 28(4)5511-333, 2002.
Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. fair comparison of graph neural networks for graph classification. In ICLR, 2020.
Xiaomin Fang, Lihang Liu, Jieqiong Lei, Donglong He, Shanzhuo Zhang, Jingbo Zhou, Fan Wang, Hua Wu, and Haifeng Wang. Geaaeery--enhanced molecular representation learning for property prediction. Nature Machine Intelligence. 4(2): 2022.
Yin Fang, Qiang Zhang, Ningyu Zhang, Zhuo Chen, Xiang Zhuang, Xin Shao, Xiaohui Fan, and Huajun Chen. Knowledge graph-enhanced molecular contrastive learning with functional prompt. Nature Machine Intelligence, 5(5):542-553) 2023a.
Zhongxi Fang, Jianming Huang, Xun Su, and Hiroyuki Kasai. Wasserstein graph distance based on ll approxxmmated tree edit distance between weisfeiler- subtrees. Proceedings of the AAAI Conference Artificial Intelligence, volume 37, pp. 7539-7549, 2023b.
12

---

Published as a conference paper at ICLR 2026
Hongyang Gao and Shuiwang Ji. Graph u-nets. In international conference machine learning, pp. 1083-2092, PMLR, 2019.
Johannes Gasteiger, Florian Becker, and Stephan Günnemann. Gemnet: neural networks for molecules. Advances in neural information processing systems, 34:6790-688 6802, 2021.
Universal directional graph
Xiaotian Han, Zhimeng Jiang, Ninghao Liu, and Xia Hu. G-mixup: Graph data augmentation for graph classification. In International conference on machine learning, pp. 8230-8248. PMLR, 2022.
Felix Hensel, Michael Moor, and Bastian Rieck. survey of topological machine learning methods. Frontiers in Artificial Intelligence, 4:52, 2021.
Jaeseung Heo, Seungbeom Lee, Sungsoo Ahn, and Dongwoo Kim. Epic: Graph augmentation with edit path interpolation via learnable cost. In IJCAI, 2024.
Christoph Hofer, Florian Graf, Bastian Rieck, Mare Niethammer, and Roland Kwitt. Graph filtration learning. In International Conference Machine Learning, pp. 4323- PMLR, 2020.
Shion Honda, Shoi Shi, and Hiroki R Ueda. Smiles transformer: Pre-trained molecular fingerprint for low data drug discovery. arXiv preprint arXiv: /9/////773, 2019.
Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau, Bastian Rieck, and Karsten Borgwardt. Topological graph neural networks. In ICLR, 2021.
W Hu, Liu, Gomes, Zitnik, Liang, V Pande, and J Leskovee. Strategies for pre-training graph neural networks. In ICLR, 2020a.
Weihua Hu, Matthias Fey, Marinka Zitmik, Yuxiao Dong. Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning graphs, Advances in neural information processing systems, 33:22118- 18-22133, 2020b.
Johanna Immonen, Amauri Souza, and Vikas Garg. Going beyond persistent homology using persistent homology. Advances in neural information processing systems, 36:63150 150-63133, 2023.
Tiago Janela and Jürgen Bajorath. Simple nearest-neighhour analysis meets the accuracy of compound potency predictions using complex machine learning models. Nature Machine Intelligence, 4(12): 1246-1255, 2022.
Dejun Jiang, Zhenxing Wu, Chang- Yu Hsich, Guangyong Chen, Ben Liao, Zhe Wang, Chao Shen, Dongsheng Cao, Jian Wu, and Tingjun Hou. Could graph neural networks learn better molecular representation for drug discovery? comparison study of descriptor-based and graph-based models. Journal of cheminformatics, 13:1-25 2021.
Talia B Kimber, Maxime Gagnebin, and Andrea Volkamer. Maxsmi: maximizing molecular property prediction performance with confidence estimation using smiles augmentation and deep learning. Artificial Intelligence in the Life Sciences, 1:100014, 2021.
Nils Kriege and Petra Mutzel. Subgraph matching kernels for attributed graphs. In Proceedings of the 29th International Coference on International Conference Machine Learning, pp. 291-298, 2012.
Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In International conference on machine learning. pp. 3734-3743, pmlr, 2019.
Sihang Li, Xiang Wang, An Zhang, Yingxin Wu, Xiangnan He, and Tat-Seng Chua. Let invariant rationale discovery inspire graph contrastive learning. In International conference on machine learning, pp. 13052-13065. PMLR, 2022a.
Zhen Li, Mingjian Jiang, Shuang Wang, and Shugang Zhang. Deep learning methods for molecular representation and property prediction. Drug Discovery Today, 2733) 2022b.
13

---

Published as a conference paper at ICLR 2026
Chuang Liu, Yibing Zhan, Jia Wu, Chang Li, Bo Du, Wenbin Hu, Tongliang Liu, and Dacheng Tao. Graph pooling for graph neural networks: progress, challenges, and opportunities. In Proceedings of the Thirty-Second International Joint Conference Artificial Intelligence, pp. 1712267222 2023.
Shengchao Liu, Mchmet F Demirel, and Yingyu Liang. N-gram graph: Simple unsupervised repre- sentation for graphs, with applications to molecules. Advances in neural information processing systems, 32, 2019.
Shengchao Liu, Hanchen Wang, Weiyang Liu, Joan Lasenby, Hongyu Guo, and Jian Tang. Pre- training molecular graph representation with 3d geometry. In ICLR Workshop on Geometrical and Topological Representation Learning. 2022a.
Yi Liu, Limei Wang, Meng Liu, Yuchao Lin, Xuan Zhang, Bora Oztekin, and Shuiwang Ji. Spherical message passing for 3d molecular graphs. In ICLR, 2022b.
David Loiseaux, Luis Scoccola, Mathieu Carrière, Magnus Bakke Botnan, and Steve Oudot. Stable vectorization of multiparameter persistent homology using signed barcodes as measures. NeurIPS, 2023a.
David Loiseaux, Luis Scoccola, Mathieu Carrière, Magnus Bakke Botnan, and Steve Oudot. Stable vectorization of multiparameter persistent homology using signed barcodes measures. Advances in neural information processing systems, 36:68316 663422, 2023b.
Yizhen Luo, Kai Yang, Massimo Hong, Xing Yi Liu, Zikun Nie, Hao Zhou, and Zaiqing Nie. Learning multi-view molccular representations with structured and unstructured knowledge. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 2082-2093, 2024.
Yao Ma, Suhang Wang, Charu C Aggarwal, and Jiliang Tang. Graph convolutional networks with eigenpooling. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 723-731 2019.
Diego Mesquita, Amauri Souza, and Samuel Kaski. Rethinking pooling in graph neural networks. Advances in Neural Informaionn Processing Systems, 33:2220-223 2231,, 2020.
Giannis Nikolentzos et al. Graph alignment kemels using weisfeiler and leman hierarchies. In International Conference Artificial Intelligence and Statistics, pp. 2019-2034. PMLR, 2023.
Ippei Obayashi, Takenobu Nakamura, and Yasuaki Hiraoka. Persistent homology analysis for materials research and persistent homology software: Homcloud. journal of the physical society of japan, 91(9))091013 101133,, 2022.
Leslie O' Bray et al. Filtration curves for graph representation. In Proceedings of the 27th ACM SIGKDD Conference Knowledge Discovery & Data Mining, pp. 1267-1275, 2021.
Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod, and Heather A Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6:1-38, 2017.
Phu Pham, Quing-Thinh Bui, Ngoc Thanh Nguyen, Robert Kozma, Philip S Yu, and Bay Vo. Topological data analysis in graph neural networks: Surveys and perspectives. IEEE Transactions Neural Networks and Learning Systems, 2025.
Chen Qian, Huayi Tang, Hong Liang, and Yong Liu. Reimagining graph classification from a prototype view with optimal transport: Algorithm and theorem. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. pp. 2454-2454, 2024.
Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. PmLR, 2021.
14

---

Published as a conference paper at ICLR 2026
Ladislav Rampášek, Michael Galkin, Vijay Prakash Dwivedi, Anh Tuan Luu, Guy Wolf, and Do- minique Beaini. Recipe for general, powerful, scalable graph transformer. Advances in Neural Information Processing Systems, 35: 35:14501-145155 2022.
David Rogers and Mathew Hahn. Extended-connectivig fingerprints. Journal of Chemical Informa- lion and Modeling, 50(5):742-754, 2010.
Yu Rong, Yatao Bian, Tingyang Xu, Weiyang Xie, Ying Wei, Wenbing Huang, and Junzhou Huang. Self-supervised graph transformer on large-scale molecular data. Advances in neural information processing systems, 33:12559-12571, 2020.
Christopher Shultz. Applications of topological data analysis in economics. Available at SSRN 4378151, 2023.
Yara Skaf et al. Topological data analysis in biomedicine: A review. Journal of Biomedical Informatics, 130:104082, 2022.
Yeongycong Son, Dasom Noh, Gyoungyoung Heo, Gyoung Jin Park, and Sunyoung Kwon. More: Molecule pretraining with multi-level pretext task. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pp. 20531-20539 2025.
Ying Song, Shuangjia Zheng, Zhangming Niu, Zhang-Hur Fu, Yutong Lu, and Yuedong Yang. Communicative representation learning on attributed molecular graphs. In IJCAI, volume 2020, pp. 2831-2838, 2020.
Afnan Sultan, Jochen Sieg, Miriam Mathea, and Andrea Volkamer. Transformers for molecular property prediction: Lessons learned from the past five years. Journal of Chemical Information and Modeling, 64(16))6559-6280, 2024.
Matteo Togninalli, Elisabetta Ghisu, Felipe Llinares-López. Bastian Rieck, and Karsten Borgwardt. Wasserstein weisfeiler-lehman graph kernels. Advances in neural information processing systems, 32, 2019.
Astrit Tola, Funmilola Mary Taiwo, Cuneyt Gurean Akcora, and Baris Coskumuzer. Toper: Topo- logical embeddings in graph representation learning. Advances in neural information processing systems, 2025.
Cecile Valsecchi, Magda Collarile, Francesca Grisoni, Roberto Todeschini, Davide Ballabio, and Viviana Consonni. Predicting molecular activity on nuclear receptors by multitask neural networks. Journal of Chemometrics, 36(2):e3325, 2022.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. Advances in neural information processing systems, 30, 2017.
Yogesh Verma, Amauri Souza, and Vikas Garg. Topological neural networks go persistent, equivariant, and continuous. In International Conference Machine Learning, pp. 49388-494499 PMLR, 2024.
Haotao Wang, Ziyu Jiang, Yuning You, Yan Han, Gaowen Liu, Jayanth Srinivasa, Ramana Kompella, Zhangyang Wang, al. Graph mixture of experts: Learning on large-scale graphs with explicit diversity modeling. Advances in neural information processing systems, 36:50825-50837, 2023,
Yuyang Wang, Jianren Wang. Zhonglin Cao, and Amir Barati Farimani. Molecular contrastive learning of representations via graph neural networks. Nature Machine Intelligence, 4(3):299-877 2022.
Yu Guang Wang, Ming Li. Zheng Ma, Guido Montúfar, Xiaosheng Zhuang, and Yanan Fan. Haar graph pooling. In ICML, 2020.
Oliver Wieder, Stefan Kohlbacher, Mélaine Kuenemann, Arthur Garon, Pierre Ducrot, Thomas Seidel, and Thierry Langer. A compact review of molecular property prediction with graph neural networks. Drug Discovery Today: Technologies, 1-21-12, 2020.
15

---

Published as a conference paper at ICLR 2026
Daniel S Wigh, Jonathan M Goodman, and Alexei A Lapkin. A review of molecular representation in the age of machine learning. Wiley Interdisciplinary Reviews: Computational Molecular Science, 12(5):e 1603, 2022.
Asiri Wijesinghe et al. A new perspective on" how graph neural networks go beyond weisfeiler- lehman?". In ICLR, 2021.
Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh Pappu, Karl Leswing, and Vijay Pande. Moleculenet: benchmark for molecular machine learning. Chemical science, 9()):133-300 2018.
Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A comprehensive survey on graph neural networks. IEEE transactions on neural networks and learning systems, 32(1):4 24, 2020.
Jun Xia, Lecheng Zhang, Xiao Zhu, Yue Liu, Zhangyang Gao, Bozhen Hu, Cheng Tan, Jiangbin Zheng, Síyuan Li, and Stan Z Li. Understanding the limitations of deep models for molecular property prediction: Insights and solutions. Advances in neural information processing systems, 36:64774-64792, 2023.
Zhaoping Xiong, Dingyan Wang, Xiaohong Liu, Feisheng Zhong, Xiaozhe Wan, Xutong Li, Zhaojun Li, Xiaomin Luo, Kaixian Chen, Hualiang Jiang, et al. Pushing the boundaries of molecular representation for drug discovery with the graph attention mechanism. Journal of medicinal chemistry, 63(16):8749 -7660, 2019.
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? ICLR, 2018
Zuoyu Yan, Tengfei Ma, Liangcai Gao, Zhi Tang. Yusu Wang, and Chao Chen. Neural approximation of graph topological features. Advances in neural information processing systems, 35:33357-3333 2022.
Pinar Yanardag al. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1365-1374, 2015.
Jingbo Yang, Yiyang Cai, Kairui Zhao, Hongbo Xie, and Xiujie Chen. Concepts and applications of chemical fingerprint for hit and lead screening. Drug discovery today, 27(11):103356 2022.
Chun Wei Yap. dadel-des:riptor: An open source software to calculate molecular descriptors and fingerprints. Journal of Computational Chemistry, (477))1466-1774,,
Wei Ye, Shuhao Tang, Hao Tian, and Qijun Chen. Beyond histogram comparison: Distribution-warre simple-path graph kernlls. IEEE Transactions on Arlificial Intelligence, 2025.
Yihang Yin, Qingzhong Wang, Siyu Huang, Haoyi Xiong, and Xiang Zhang. Autogel: Automated graph contrastive learning via learnable view generators. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 8892-8900, 2022.
Chengxuan Ying, Tianle Cai, Shengjie Luo, Shuxin Zheng, Guolin Ke, Di He, Yanming Shen, and Tie Yan Liu. Do transformers really perform badly for graph representation? Advances Neural Information Processing Systems, 14:28877--28888 2021.
Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, William Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems, volume 31, pp. 4800-4810. Curran Associates, Inc., 2018.
Jaemin Yoo al. Model-agnostic augmentation for accurate graph classification. In Proceedings of the ACM Web Conference 2022, pp. 1281-1291, 2022.
Zaixi Zhang, Qi Liu, Hao Wang, Chengqiang Lu, and Chee- Kong Lee. Motif-based graph self- supervised learning for molecular property prediction. Advances in Neural Information Processing Systems, 34:15870-15882. 2021.
16

---

Published as a conference paper at ICLR 2026
Yan Zheng, Song Wu, Junyu Lin, Yazhou Ren, Jing He, Xiaorong Pu, and Lifang He. Cross-view contrastive fusion for enhanced molecular property prediction. In Proccedings of the Thirty-Thin International Joint Conference Artificial Intelligence, volume 2, 2024.
Zhiqiang Zhong, Kuangyu Zhou, and Davide Mottin. Benchmarking large language models for molecule prediction tasks. arXiv:2403.0507,. 2024.
Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249 274, 2005.
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Appendix
A TOPOFORMER": TOPOFORMER FOR MPP
A.1 MOLECULAR FINGERPRINTS
Molecular fingerprints are widely used in computational chemistry and machine learning to represent molecular structures fixed-length numerical vectors (Cereto-Massagué et al., 2015). They encode features such as atomic connectivity and substructural patterns, enabling efficient similarity search and predictive modeling. Popular methods include ECFP (Extended Connectivity Fingerprints) and PubChemFP, both extensively applied in drug discovery, virtual screening, and bioinformatics (Yang al., 2022).
ECFP Fingerprints. Extended Connectivity Fingerprints (ECFP) capture structural features by iteratively hashing local atomic environments up to a specified radius (Rogers Hahn, 2010). Unlike traditional hashed fingerprints, ECFP preserves substructural detail, making it effective for similarity search, QSAR modeling, and property prediction. is invariant to atom ordering while retaining connectivity, enabling fine-grained molecular feature analysis. For recent overview of ECFP fingerprints and their role in modern biochemical ML pipelines, see (Wigh et al., 2022).
A.2 TOPOFORMER' MODEL
For Molecular Property Prediction Task, employ hybrid model, TOPOFORMER'S combining ECFP Fingerprints and our TOPOFORMER Model. This hybrid model shows the versatility of our TOPOFORMER model its effective integration with complementary information (Table 7). We give the flowchart of our hybrid model in Figure 5. In our hybrid model, we used the same experimental setup for the TOPOFORMER component. For the MLP component, we employed a two-layer MLP with hidden dimension of 200, ensuring that its output dimension matches the output dimension of TOPOFORMER model. The model was optimized using the Adam optimizer with a learning rate of 0.01 and weight decay of le-4. Both the MLP and TOPOFORMER components were trained in an end-to-end manner, allowing the model to leverage both topological signatures and complementary graph information, ultimately lcading to improved performance.
Table 7: Performance comparison of standalone models (ToPoFoRMER and FP-MLP) and the hybrid model (ToPoFoRMER in random splitting (8:1:1).
PH-TR
TOPOFORMER
FP-MLP
PH+ECFP+TR
TOPOFORMER*
BACE
72.41=315
83.29
90.29
90.60:2.0
91.60:
HIV
69.29=165
83.26
83.97=151
85.10:0.4
BBBP
83.37
94.54.1.0
89 29.6882346
93.4 33.47
95.90
Cli
75.8926.0
83.422a235
76. 76.34
82. 22.04
86.20:330
SIDER
62.912149
62.10.::4
65. 359
66.99-1.8
66. 2/
Tox21 68.242160
80.87:0.1
1..8
79. 19
$1.50
ToxCast
64.742229
73.37
74.69
75.73
78.40
A.3 HYBRID CLASSICAL MPP BASELINES
We refer to the classical models combined with modern ML models Hybrid Classical (HC) Models. The first family of HC baseline models consists of Fingerprinting models (Jiang et al., 2021), which use vectorized molecular fingerprints as input to traditional machine learning models, including SVM, XGB, RF, and MLP. The input fingerprints are concatenation of 881 381-dimensional PubChem fingerprints (PubChemFP), JBbCChemFPP.337-dddmmmnnnno 307-diimnnio substructure fingerprints (SubFP), and 206-dimensional MOE 1-D and 2-D descriptors ( (Yap, 2011). The second family of baseline models comprises SMILES models, which treat SMILES strings as sequential input to ID CNN (Kimber et al., 2021), ), a 3-layer bidirectional GRU (Cho et al., 2014), and pre-trained SMILES transformer (TRSF) (Honda et al., 2019). The third family is GNN models which use 2D graph-based representations of compounds, where atom and bond features and AFP models (Xiong et al., 2019). Another baseline is the SPN model, using SphereNet (Liu al., 2022b), which employs 3D graphs of compounds as input.
encoded using one-hot schemes and fed into GCN, MPNN, GAT,
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Tepa-Scza
Transformer
Tepo-Scoa uvirg C
Transformer
Propett
Propatt
Profiction
MLP
Fingerprint
Attrakio Rose Concetraaio
Figure 5: TOPOFORMER". To successfully integrate complementary graph information, such as ECFP, with our TOPOFORMER model, we employ MLP. The MLP output is combined with the TOPOFoRMER model using attention mechanism, and the combined representation is then passed through a graph prediction network to perform the final prediction task.
Table 8: Hybrid Classical MPP Models. ROC AUC results of models for molecular property prediction tasks with random splitting (8:1:1). The baseline results are reported from (Xia et al., 2023). The best and the second best performances are given in bold, and underlined, respectively.
Fingerprinting Models
SMILES Models
GNN Models
Ours
Dataset
SVM
XGB
RF
CNN
RNN
TRSF
MLP
GCN
MPNN
GAT
AFP
SPN
TOPOFORMER"
BBBP
91.3
92.6
92.3
89.7
76.0
69.3
89.7
91.8
91.5
87.2
90.2
90.5
96.6
Tox21
82.0
83.1
81.2
73.7
76.8
79.9
84.6
82.1
82.7
82.5
81.5
ToxCast
72.5
78.5
77.8
73.5
67.8
78.0
78.1
76.7
78.8
77.2
76.8
77.2
78.4
SIDER
62.6
63.8
64.4
59.1
51.5
64.1
61.7
62.3
60.3
62.0
61.3
61.3
66.8
Cox
87.9
93.0
88.8
68.5
96.3
93.0
88.9
86.8
89.8
87.9
91.2
86.2
BACE
88.6
89.6
89.0
81.5
55.9
83.5
88.7
88.0
84.6
88.6
87.9
88.2
91.6
HIV
81.7
83.2
82.0
82.6
73.3
74.8
79.1
83.4
81.2
81.8
81.8
85.1
B PROOFS OF STABILITY THEOREMS
work on the fixed clique complex G of G (V, E). For a node function + R, we use upper-star extension li(o)= maX.ea h(v) and the associated sublevel filtration on G. Throughout, (0, } is the homological dimension used our tokens.
Preliminaries. For S b. define the interlevel (level-set) subcomplex G [..j == fa G minveo h(v) and max.co h(v) < 6}. The associated pointwise finte-diieeniionll interlevel persistence module the functor M (a,b) + H&(G h (a,b)). Given a shared grid ag <... aN window width m, and stride 8, the Topo-Scan token at window is
(t))) dim M (aus) ats+m):
O,..., T-1, T
We write (M MG) for the bottleneck distance between the interval decompositions (barcodes) of the interlevel modules Mf and Mg.
Two stability lemmas.
Lemma B.1 (Interlevel stability). (Botnan & Lesnick, 2018, Thm 1.1 & 1.2) For 2 O, the interlevel modules of the upper-star filtrations induced by f. V + IR on the fixed clique complex G satisfy
do(M/ Mg) < | -glbo-
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Lemma B.2 (Lipschitzness of interval rank). (Bauer & Lesnick, 2014; Bakke Bjerkevik. 2021) Let M,N be interval- deppraaddcccopoabblle p.f.d. modules with dg (M, N) S 6. For any interval I = [aa b).
dim M (a, b) - dim (a, b) & BMr (1, 0) (x(),
where (Tr 8) counts bars in Bar(M) whose endpoints lie within 8 of the boundary {a, b) (and similarly for N).
Theorem 3.1. With the setup above, there exists C C(G, (0s); m, such that
T-1
Σ (1) 32()) s C d
Proof of Theorem 3.1. Fix & and write 4, = Jats atstmm J. By Lemma B.2, there exists a finite constant Co 1) such that dim M/ (Ix) dim Mg (I)) I Co da (M/ Mg). Summing over gives
T-1 (t() Bg(
E Colâ, 1i) db(M,, M2) == Cds(ML.
On a fixed finite complex and fixed grid, the Co(G, It) are finite and can be uniformly bounded, yielding C' = T To.
Corollary 3.2. For star on a fixed complex, dg dg(M) ME) s ol (Lemma B.
hence (G, f) BG S IIf gll oo.
Proof of Theorem 3.1. By Theorem 3.1, we have
Cd Mg).
By Lemma B.1 (interlevel/level-set stability on the fixed clique complex), d Mg) s If gllec. Combining the two inequalities yields the claim.
Connection classical sublevel stability. The inequality dg glooo is classical for sublevel filtrations on fixed space (Cohen-Steings et al., 2007). Our Lemma B.1 is the level-set (interlevel) analogue on the fixed clique complex, following algebraic stability for zigzag/level-set modules (e.g., Botnan & Lesnick, 201 8). We use this interlevel version to handle windowed intervals (a, bj appearing in Topo-Scan
Shared thresholds.
The theorem assumes shared grid {a.}. If thresholds are chosen separately (e.g.. per-function quantiles), a monotone reparameterization of the filtration axis induces an additional term proportional to the grid displacement, which be absorbed into C.
Remark B.3 (Relation to PH invariants and stable ranks). Our stability theorem focuses the & robustness of the discrete Topo-Sean sequences Be (G, h). but these sequences implicitly encode familiar PH objects. For a fixed filtration function h, the map + (t) can be viewed as a sampled version of the rank invariant (a, b) + rank Ha ((r), a..)) associated with the interlevel module M In this sense, Topo-Scan produces coarse, structured discretization of the same information that barcodes and stable vectorizations of persistence diagrams, such as persistence landscapes, silhouettes and persistence images (Chazal et al., 2014; Adams et al., 2017), summarize in continuous form. Similarly. Graph Filtration Learning (Hofer et al., 2020) can be seen as learning the filtration function h, while our work fixes h and instead changes the representation from global barcodes local interlevel sequences. A full expressivity comparison and formal information-los bounds relative to complete barcodes interesting directions for future work.
C MORE ON TOPOFORMER
C.1 BASE MODEL: TRANSFORMER
Our TOPOFORMER model is designed for classification tasks using sequential inputs, harnessing transformers for efficient feature extraction. The architecture includes embedding layer, a trans- former encoder, and a fully connected (FC) classification head, with regularization techniques applied to mitigate overfitting.
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Let (Ø1, .2...., .T) I NxTxP represent the input sequence, where N is number of graphs, is the sequence length, and D is the dimensionality each imput token. The input sequence first passed through embedding layer E R" + IR where H denotes the embedding dimension. In addition, a positional encoding matrix P E RIxT*H is added to the embeddings to encode the positional information of the sequence, resulting in sequence of embedded vectors e E( + Pa
for = 1, T, where P; is the positional encoding for position t.
The sequence of embeddings then passed through multi-layer transformer encoder, where the encoder operates on the embedded sequence Exx) - P & R with B representing the batch size. The transformer encoder generates a new sequence of output representations (21, 22..... 2T) E I. TxxBxx After processing through the encoder, the output sequence is permuted and reshaped to a flattened vector of size x (T H), ensuring compatibility with subsequent fully connected layers.
The flattened representation Zia E BBx(T-H) is then passed through a batch normalization layer, BN (Z which normalizes the activations across the batch stabilize the training pro- cess. A dropout layer D(-) is then applied to the normalized output to regularize the model and mitigate overfitting. The final classification output is obtained through a fully connected layer FC: Bx((H)) IR
C.2 DUAL TRANSFORMER WITH MULTI-LAYER PERCEPTRON CLASSIFIER
This model combines multiple sources of input data through a hybrid architecture that integrates two independent base models and a multi-layer perceptron (MLP). This model is designed to handle diverse input modalities by leveraging the strengths of both transformers and MLPs for feature extraction and classification.
Let X₁ E NRxxxxx,, X₂ NxTxx2, and X3 E NxL represent the three distinct input graph encoding, where T; denotes the sequence length, D, the dimensionality of the inputs for each modality and L the dimension of fingerprints. Each input is processed through its respective component: the first sequence X, is passed through transformer' TI, the second sequence X₂ through transformer Tz, and the third sequence X3 through an MLP M.
The output of the first transformer TI, denoted 21 RTixxxx, is obtained by passing X, through the transformer encoder. Similarly, the output of the second transformer' T2, denoted 22 E R22xBxH is obtained by processing X2. Finally, the output of the MIP M is denoted 23 Bx
The outputs 21, 22, and 23 are then combined through a learnable weighted sum. Specifically, the combined feature vector Zoobbined is computed as:
Zcombined Q 21 3 22
32 Z3
where a and [ are learnable parameters that control the contribution of each modality the final representation. This weighted combination allows the model to adaptively learn the most relevant contribution of each input sequence.
The combined feature vector Zeombined is then passed through batch normalization layer B to normalize the activations, improving training stability. A final fully connected layer FC produces the classification output: ý = FC((combined)) where R represents the predicted class probabilities, with C' being the number of possible output classes.
C.3 RUNTIME ANALYSIS
To assess the computational efficiency of our method, we report the total runtime across two key stages: topological signature extraction using Topo-Scan (via Degree Centrality and Ollivier-Rica curvature), and (ii) model training using the transformer-basee classifier. Table 9 presents a detailed breakdown of runtimes (in minutes) for five benchmark datasets.
As expected, Degree Centrality is extremely fast to compute and contributes negligible overhead. Ollivier-Ricel curvature, while more computationally intensive, remains tractable even for large graphs, as evidenced by reasonable runtimes on datasets such as REDDIT-5K and OGBG-MOLHIV Transformer training times scale smoothly with dataset size and remain within practical limits.
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Overall, method maintains scalability while offering strong performance, demon- strating the feasibility of integrating topo- logical signatures into deep graph models scale.
Table 9: Runtimes. Total runtime (in minutes) per dataset. The second and third columns report the time to compute scalar filtration values and Topo-Scan vectorizations for de- gree centrality and O.Ricci curvature, respectively, and the final column shows the Transformer training time.
TopoSean vs. PH. We report in Table 10 the runtime for Topo-Scan and standard PH on four benchmark datasets with degree centrality filtration (already computed), us- ing the same backend (pyflagser) for both pipelines. For Topo-Scan, we invoke the un- weighted flagser routine, since our method only requires Betti numbers on unweighted clique complexes. For PH, we use the weighted flagser routine, which constructs full filtration and computes persistence diagrams.
Dataset
Degree C.
O. Ricci
Transformer
IMDB-B IMDB-M REDDIT-B REDDIT-SK
0.51
5.04
3.05
0,49
7.62
4.57
5,30
23,70
6.10
36.74
109.98
15.65
OGBG-MOLHIV
14.70
339.06
21.67
The clustering coefficient column serves as a proxy for graph density and hence clique complexity. On highly clustered graphs such as IMDB-B and IMDB-M B- (coefficients = 0.95-0.97), PH is roughly 13-14x slower than Topo-Scan, reflecting the com- binatorial explosion of cliques and the cost of global boundary-matrix reductions, whereas on the sparser REDDIT datasets the gap smaller but still consis- tent (about 2x on REDDIT-B and 1.5x on REDDIT- 5K). These results empirically confirm that Topo- Scan achieves multi-fold runtime savings over stan- dard PH pipelines on dense graphs while remaining uniformly more efficient across all tested datasets.
Table 10:
Runtime for PH vs. Topo-Scan. Run- time (in seconds) per dataset for computing topo- logical features using the same backend (pyflagser).
Dataset
Clus. Coeff.
Topo-Sean
PH
IMDB-B
0.947
9.67
135.48
IMDB-M
0,969
17.81
234.30
REDDIT-B
0.048
12.29
24.25
REDDIT-SK
0.027
29.51
43.92
Comparison with other methods. We also compare the runtimes of two PH-based baselines, PersLay (with degree centrality input) and TopoGCL (using only the topol- ogy derived component), against Topo-Scan on IMDB-I B and REDDIT-B (Table 11). All times are reported in seconds. For Topo- Scan, we include both the scalar filtration computation and Topo-Scan feature extrac-
Table 11: Runtimes (in seconds) for PH-based baselines (Per- sLay, TopoGCL) and Topo-Scan on IMDB-BI B- and REDDIT-B.
Method
IMDB-B
REDDIT-B
Notes
Perslaa TopoGCL
97.02
454.78
PH hased layer degrue input
435.49
4010.08
Only topological component
Topo-Scan
15.16
290.00
Filtration Topo-Scan
tion. On IMDB-B, Topo-Scan is about 6 times faster than PersLay and around 29 times faster than the topological part of TopoGCL; on REDDIT-B, it remains faster than PersLay and roughly 14 times faster than TopoGCL. These results further support the practical efficiency of Topo-Scan compared with PH-based pipelines.
C.4 COMPARISON WITH POOLING METHODS
Table 12 compares TOPOFORMER with six representative graph pooling methods designed to adapt GNNs to graph-level tasks. DiffPool (Ying et al., 2018) learns soft assignment matrix that hierarchically clusters nodes in a differentiable, end-to-end manner. Top-KPooling with Graph U-Nets (Top-K) (Gao & Ji, 2019) ranks nodes using learnable projection score and retains the top-k fraction to coarsen the graph. EigenPool (EigenGCN) (Ma et al., 2019) projects node features onto the leading eigenvectors of the graph Laplacian to preserve global spectral properties. SAGPool (Lee et al.. 2019) computes attention scores through a GNN layer. pruning low -owiipprt nodes and re- ee-wiring the remaining graph. MinCutPool (Bianchi et al., 2020) casts pooling as a relaxed spectral clustering problem by optimizing a minimum-cut objective to form node clusters. HaarPool (Wang et al., 2020) applies a Haar wavelet transform to graph signals and performs pooling by selecting key wavelet coefficients. Our model TOPOFORMER takes a different approach by integrating multiscale topological filtrations with a transformer based attention mechanism, enabling the pooling of substructures across scales and yielding robust higher-order graph representations. As shown in Table 12, TOPOFORMER consistently outperforms all baselines, achieving the best accuracy on six out of seven datasets and ranking second on the remaining one.
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Table 12: Comparison with Pooling Methods. Accuracy results of six baseline pooling methods and TOPOERRMER on seven graph classification benchmark datasets.
Model
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
Top-K
70.40.1.2
40.30.4.21
67.61.3.36
13.50
73.17 44.84
48, 3.11
79.40.17.4
MinCutPool
82. 22.6445.00 05
80.07=3.85
79. 19.171
76.52=2.58
70.77=4.89
49.00=2.88
87.20
DifiPool
83.93+4.41
79.66=2.44
79. 29=1022
73.63=3.60
08.60=3.10
45.70-3.40
79.00=1.10
EigenGCCN
83.05#6.00 00
80.16=5.80
79.50=0.66
74.10=3.10
70.40=3:30
47.20=3.00
N/A
SAGPool
82.95:14.91
79.45
76.78 28:12222
71.86
74.87
49.33=4.50
84. 74.44.4
HaarPool
83.95 68
82.61 = 6.0
90.00=3.60
73.23=2.51
73.29=3.40
49.98=5.70
N/A
TOPOFORMER
92.36=4.11
83.93=4408
94.68
77.64+3.61
78.90+3.31
55.40+4.78
91.50+1.80
C.5 TOPO-SCAN HYPERPARAMETERS
In the Topo-Scan algorithm, two key hyperparameters play crucial role: the width parameter, which controls the thickness of slices, and the filtration function, which defines the hierarchical importance of nodes or edges. To determine the optimal hyperparameter settings, we conducted extensive experiments to validate their impact on model performance.
Width Parameter Selection. To determine the optimal width parameter m, conducted experi- ments using degree centrality and Ollivier- curvature as filtration functions for the Topo-Seanner on graph classification datasets. We evaluated Scanner feature vectors and using them as inputs to a transformer model. The results presented in Table 6 indicate that, for most of the datasets, the Topo-Scanner features achieve the best performance when m = 2 for both filtration functions. Based on this experimental analysis, we select m 2 as the optimal parameter for model.
2,3, 3, and 4, extracting the corresponding Topo-
Different filtration functions impose distinct hierarchical orderings on nodes (or edges), enabling our model to capture diverse topological patterns in the induced sequences. This allows the Topo-Scan process to effectively integrate domain-specific information. To fully leverage multiple filtrations, TOPOFORMER applies separate transformers for each filtration function and combines their outputs using a learnable attention mechanism. This mechanism dynamically assigns higher weights to the most relevant topological signatures, ensuring optimal feature selection and enhanced performance. As shown in Table 13, ToPoFor MER employing multiple functions consistently outperforms models using a single filtration function, demonstrating the advantages of multiple filtrations.
Multiple Filtrations.
This approach enhances model robustness and stability by incorporating diverse topological perspec- tives.
Table 13: Filtration Functions. Performance comparison of single filtration and multiple filtrations with TOPOFORMER across different datasets. The best values in each column are highlighted in bold.
Filtrations
MUTAG
PROTEINS
BZR
COX2
IMDB-B
IMDB-M
REDDIT-B
Degree only
92.02
77.28
89.8913.74
78.36114.3
74,20
51.53
86.601297
O. Ricci only
89.91
77.26
90.60=3.00
78.60 78.609:4779
79.10
54.53
91.40
HKS Only
95.32:5..58
77.35=2266
90.62::491
83.95
76.90
54.07=2.44
90.05
Deg.+O.Rica
93.01
78.35
91.12:14.88
81.80-15.4
78. 18.80
53.87
90.65+22.2
HKS+O.Ricci
4.33
7..64
92.36++411
83.93
78.90+331
55.40=4478
91.50++199
HKS+Degree
95.26=3.88
78.08
91.09+5.53
13.71
77.30+241
52.60=2.25
89,90+235
C.6 TOPOFORMER VS. PH WITH DIFFERENT VECTORIZATIONS
TOPOFORMER consistently outperforms Persistent Homology methods in both accuracy and compu- tational efficiency. As shown Table 14, we compare against the best PH results reported in (Cai & Wang, 2020), which evaluates 16 combinations of four filtration functions (degree, O.Ricci, Fiedler, closeness centrality) and four vectorization techniques (Sliced Wasserstein, Pervec, Filvec, SW-p) per dataset. TOPOFORMER achieves higher accuracy all six benchmarks.
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Table 14: Accuracy results for TOPOFORMER (HKS) vs. Persistent Homology in graph classification tasks. In PH row, we report the best performance of 16 combinations with four filtration functions combined with four vectorizations.
BZR
COX2
PROTEINS
IMDB-B
IMDR-M
PH (Best of 16 comb) TOPOFORMER
08.4
82.0
74.0
69.5
46.5
90.6=449
82.0
77.4=29
77.9434
54.1+25
Table 15:
TOPOFORMER vs. PH Performance Comparison: Accuracy three topological models under seven filtration:: Degree, Ollivier-Rieci, HKS, Betweenness centrality, Closeness centrality, Eigenvector centrality, and Forman-Rinn Ricci curvature. The last column reports the average accuracy improvements our models PH-TR and TOPOFORMER over the classical TDA pipeline PH-MLP for the same filtration function.
Filtration
Model
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
Av.Imp.
PH-MLP
82.71=651
76.44=5.39
84,06+4455
68. 68.37=397
65, 70
45.07=259
89 29.50-247
Degree
PH-TR
86,43-431
15
77.54:241
75.00:211
50.67:387
92.30-1177
4.91
TOPOFORMER
91,10_x44
80.27
12154
77.45
74.20
50.33
89.75_
6.26
PH-MLP
85.45=336
18.16=5.0
84.06-521
63. 05.50=436
68
44.87=3x5
85 85.65-243
O.Ricel
PH-TR
88 52:
78. T6
2:5m
77.27
72. 72.20
48.00
90.65_106
4.40
TOPOFORMER
90.38
80.72..6.4
92.541447
77.90
74.70
51.53
91.90__775
6.85
PH-MLP
84, 96:4444
78. 19:494444
84,0% 72
70. 80:40.0
71. 10,10
47.93=1.20
88. 1.657
HKS
PH-TR
89.60 541
79.89_240
12
77.18
76.80.117
53.60
87.25_...9
4.75
TOPOFORMER
90.62 4.9
83.95_2.
95.32
77.90
54.07 1::4
90.05 2411
6.30
PH-MLP
84,95=415
80
89, 89.944799
7161+185
68. 10
43
79,
Betweenness
PH-TR
85 85.43=4.13
81.60=6.0
50.52+5
74.13=205
69 90=244
45.40=1.9
84 84.05-131
1.79
TORMER
87.41=487
80,74
90.99
76.73=247
73.90
51.47=1.6
86.55-111
4.19
PH HLP
84.21
79.07
88 24
74.39:x:9
65. 65,30:4m1
47.47
66.85
Closeness
PH-TR
87,43-433
79.65=496
29
75.93=3332
69.70s450
50.47=3.72
77.20-331
TOPOFORMER
85. 13-816
81.17:532
90.88:2445
73.64:446
73:20=2220
51.07=32
86,40=133
5.61
PH-MLP
83.97=14e
80.56
89.39
67.20
66. 66.70:221
17.40:1m
79.40=321
Eigenvector
PH-TR
87 37.41=421
19.88=6.0
91.57+5..0
70.53=2450
72.40a444
50.13=7744
89 19.25=1440
3.79
TOROFRRMER
90.59
82.87=333
61
77.35
76.10:363
51.00:314
91.85-1145
6.59
PH-MLP
82.46
80.13::5.
87.81
73.95
66.60)a37
45.53
73.70=378
F. Ried
PH-TR
16-186
81,97=662
91,99+4.0
76.09=404
70.80+447
50.67=1.9
77.20-221
3,53
TOPOFORMER
88.41=684
81.02:26.6
92.08::5m7
77.81
79.40:am
54.47=331
88.95-1.M
7.42
C.7 TOPOFORMER VS PH PERFORMANCE
Table 15 extends our ablation (Table 5) from three to seven filtration functions and compares three topological pipelines under the same filtration function: the classical PH-MLP baseline (sublevel PH + Betti vector + MLP), our PH-TR variant (same Betti vectors but processed as sequences by a Transformer), and TOPOFORMER (Topo-Scan sequences with sliding-window window interlevel filtrations). Across all seven filtrations, replacing the MLP with a Transformer already yields consistent gains: PH- TR improves over PH-MLL by roughly 2-4 accuracy points on average (see the "Av.Imp." column), confirming that treating Betti curves as ordered sequences is beneficial even without changing the underlying filtration.
TOPOFOR MER further improves on PH-TR for almost every filtration, typically adding another 1-3 points most datasets and yielding average gains of 6-7 points over PH-MLP. The effect is especially pronounced on more challenging benchmarks such as IMDB-M and REDDIT-B, where sliding- window interlevel slices capture richer late-emerging structure than standard sublevel PH. Importantly, this pattern holds not only for the three filtrations used in the main text (degree, Ollivier-Rice HKS) but also for the four additional ones (betweenness, closeness, eigenvector centrality, Forean-Ricei curvature), indicating that the benefit of Topo-Scan is robust to the choice of scalar function. Together, these results support our central claim: the main performance gains come from the Topo-Scan sequential representation (and its integration with Transformers), rather than from a particular hand- picked filtration function.
C.8 EARLY SATURATION IN PH FILTRATIONS AND TOPO-SCAN
Goal. We compare classical PH (sub/superlevel on a fixed clique-complee 2 skeletton)) with Topo- Scan show how PH frequently early-saturates graphs, i.e., after relatively small portion of the
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BCR Meto (wrmekend
MUTAI
PROroIs bet
0.3
09
(a) BZR Betti-O
(b) MUTAG Betti-0
(c) PROTEINS Betti-0
Figure 6: PH vs. Topo-Sean. Normalized average Betti-0 values 20 thresholds of the degree-centralij filtration on (a) BZR, (b) MUTAG, (c) PROTEINS. Under the classical PH pipeline, feature counts decline monotonically with increasing threshold, whereas Topo-Scan maintains elevated values at higher thresholds, revealing late-energing topological features that PH alone misses.
threshold range. new features cease to appear, whereas Topo-Scan continues to surface structure by sliding windows over the same signal.
Protocol. For each dataset and filtration function f (e.g., degree or Selivier-Ricci] we fix common grid of T thresholds and evaluate both methods on the same clique-complex 2 skeleton (upper-star from nodes). PH: sublevel filtration evaluated at the same grid points; Betti counts are read at each threshold. Topo-Scan: window width m and stride s define 7 overlapping slices whose vertex sets correspond consecutive value ranges in the same grid. Betti counts computed per slice. To make cross- dassset plots visually comparable, we report (i) normalized Betti-O curves when scales differ markedly (Fig. 6) and (ii) unnormalized Betti-0 when PH and Topo-Scan share similar ranges (Fig. 2). Betti-1 frequency barplots are shown to illustrate higher-order behavior (Fig. 7).
How to read the figures. A positive Betti-O value position means additional connected components are present that slice/threshold; persistent nonzero values toward the right side the horizontal axis indicate late-emerging structure. For B darker bars at higher thresholds indicate more cycles appearing later in the filtration. Because Topo-Scan slices are value-localized ranges rather than one-sided sublevels, they retain visibility into regions that are otherwise drowned out once early high- or low-valued nodes saturate the PH complex.
Results on small biochemical graphs (BZR, MUTAG, PROTEINS). Figure plots normalized Betti-0 curves over 20 degree thresholds. Across all three datasets, PH curves drop quickly and remain low: after early rise, new components rarely appear as the complex fills up. In contrast, Topo-Scan maintains elevated values deeper into the axis, indicating that the sliding window moves, continues to expose distinct local subgraphs in later value ranges. This pattern is precisely the late-structure retention we aim to capture.
Results on social graphs (IMDB-B, IMDB-M). Figure 2 shows unnormalized Betti-O with 100 thresholds (comparable scales). Here, PH exhibits sharp taper near the end: once the core of the
PROTEINS PH Bett-1
(b) PROTEINS Tapo-Sean Bent-1
Figure 7: PH vs. Topo-Scan. Bar plots of Betti-1 counts at each O. O.Ricci filtration threshold on the PROTEINS dataset, with bar color intensity encoding the frequency of cach integer value at that threshold, In (a) classical PH features rapidly taper off and plateau carly, whereas in (b) Topo-Scan shows increasingly darker bars higher thresholds, evidence of continued cycle emergence beyond PH's saturation.
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graph enters the complex, subsequent thresholds add little. Topo-Scan avoids this collapse: activity persists and often exceeds PH in the tail, reflecting components that are still exposed by the windowed slices even when global sublevels have already merged them away.
Higher-order signal (Betti-1 on PROTEINS, O. Ricci). Figure provides barplots where color intensity encodes the frequency of each integer Betti-1 value per threshold. Under PH (left), bars fade and plateau early, showing few cycles after the initial growth phase. Under Topo-Scan (right), darker bars persist across later thresholds, demonstrating continued cycle emergence that PH no longer reveals once the complex has saturated.
Why does this happen? In sub/superlevel PH, extreme-valued vertices enter early, the induced complex quickly fills in, so later additions create little new topology, especially on graphs where dense regions are correlated with the signal. Topo-Sean, by scanning ranges of values with overlap. repeatedly re-centers attention on late parts of the signal, preventing early regions from dominating the entire sequence. Importantly, this not claim that sublevel is intrinsically flawed; task-aligned or learned filtrations can mitigate early saturation. Our point is empirical and architectural: a fixed-budget sliding-windon view preserves late signal by design.
Controls and caveats. (i) We use the same signal, grid, and complex for both methods to avoid confounding factors. (ii) Normalization applied only for visualization when scales differ; conclu- sions do not depend normalization. (iii) Sublevel and superlevel yield the same multiset of slices in reverse order; Topo-Scan's s behavior is insensitive to that choice. (iv) Window hyperparameters (m, s) trade locality for coverage; keep them fixed across datasets in these plots for clarity.
Takeaway. Across biochemical and social benchmarks, PH curves commonly flatten early, while Topo-Scan remains active in the tail (Betti-0 and Betti-1), revealing late- lateeenerging components and cycles. This supports our central design choice: turning topology into short, ordered, range-localized tokens helps retain information that standard PH pipelines often lose the complex saturates.
26


Reviews:
[{"review_id": "4WJM4UJlau", "reviewer": "Reviewer_zs8A", "summary": "This work introduces TOPOFORMER, a graph representation framework that integrates topological structure into Transformer architectures. The core component, Topo-Scan, generates compact sequential topological tokens by sliding over node/edge filtrations and extracting slice-wise invariants (Betti numbers, node/edge counts). This avoids traditional persistent homology pipelines and costly persistence diagram computations. The authors provide stability guarantees, demonstrate scalability, and report state-of-the-art results on graph classification and molecular property prediction benchmarks.", "questions": "1. Could Topo-Scan be extended to learn filtration functions end-to-end? How would stability guarantees extend to such a case?\n\n2. For large-scale graphs (e.g., OGB-MOLHIV, REDDIT), what are memory/time trade-offs compared to graph Transformers with pooling?\n\n3. Beyond Betti numbers, would incorporating additional topological descriptors (e.g., persistence landscapes) meaningfully improve results?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a framework to inject TDA signals into Transformers without persistence diagrams, which articulates limitations of classical PH pipelines (saturation, vectorization design burden) and motivates a practical alternative.\n\n2. The paper provides stability guarantees for the proposed topological sequences, linking back to interlevel persistence.\n\n3. The proposed method avoids heavy PH computation and vectorization. It emphasizes parallelizable slice computation with predictable runtime.", "weaknesses": "1. Although the paper emphasizes advantages over classical persistent homology pipelines, the empirical comparison against methods that incorporate learnable filtrations remains limited (e.g., Horn et al., 2021; Immonen et al., 2024). \n\n2. While the proposed Topo-Scan framework leverages fixed scalar functions to generate topological slices, the methodology may still exhibit sensitivity to the choice of these filtration signals. The current study provides limited analysis of this dependency, and a deeper investigation—particularly involving learnable or data-driven filtration functions—could strengthen the robustness and interpretability claims.\n\n3. The approach demonstrates competitive performance across multiple graph classification and molecular property prediction benchmarks. However, on large-scale datasets such as OGB-MOLHIV, the framework does not achieve the top reported results, suggesting potential room for improvement in scaling to very large graphs or addressing complex molecular tasks relative to specialized architectures.\n\n4. A few relevant contributions, e.g., [1,2], to learning persistent-homology-based representations are not included in the reference.\n\n[1] de Surrel et al. \"RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds.\" ToGL 2022.\n\n[2] Yan et al. \"Neural approximation of graph topological features.\" NeurIPS 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JF9SVT2gkK", "reviewer": "Reviewer_dgUd", "summary": "This paper introduces TOPOFORMER, a framework for graph-level representation learning. The core contribution is \"Topo-Scan,\" a module that decomposes a graph into an ordered sequence of topological tokens .\n\nThe authors claim that standard Persistent Homology (PH) suffers from an \"early saturation\" problem on graphs. They propose Topo-Scan as a lightweight solution. Instead of the standard cumulative filtration, Topo-Scan uses a simple \"sliding window\" to \"slice\" the graph . For each slice, it computes basic invariants (Betti-0, Betti-1) . This process creates an ordered sequence fed into a standard Transformer.\nThe authors claim this simple method solves the \"early saturation\" problem and achieves SOTA results on graph classification and molecular property prediction", "questions": "1.\tThe SOTA claim in Table 4 relies on TOPOFORMER* (which includes ECFP features), while many baselines do not . This is an unfair comparison. To justify the method, can the authors provide results for a \"PH-TR + ECFP\" baseline?\n\n2.\tThe paper's core efficiency claim is that Topo-Scan achieves \"multi-fold runtime... gains\" by avoiding PH's \"global boundary-matrix reductions\" . Can the authors provide experimental runtime data for this specific claim?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "The paper is well-organized and easy-to-follow. It also provides a lot of experimental results to understand the empirical behavior of this method. The Figures and tables are carefully presented. Additionally, the paper is well-motivated, clearly identifying the \"early saturation\" problem , and its core originality comes from combining topological slicing with a standard Transformer architecture .", "weaknesses": "The paper's central claims are undermined by a significant gap between its motivation and its empirical results. The core contribution (Topo-Scan) is a conceptually simple modification, but the experiments fail to provide statistically significant evidence that this modification offers a meaningful advantage over the standard methods it claims to improve upon.\n\n1.\tThe core idea is trivial and lacks justification: The paper's main technical contribution, Topo-Scan, replaces the standard cumulative sublevel filtration $V_i = \\{v: f(v) < a_{i}\\}$ with a slicing window $V_i = \\{v: a_{i} <f(v) < a_{i+m}\\}$. This is a conceptually simple modification.\n\n2.\tThough the authors justify the contribution of solving early saturation by showing the empirical Betti-0 comparison with increasing thresholds, the trends only differ at the very late stage (the last 10 steps on IMDB-B and only from the ~72nd to 82nd step on IMDB-M). The relation of this phenomenon with the eventual performance on property prediction tasks is unclear. Table 5 is the key experiment meant to provide justification, comparing TOPOFORMER (Topo-Scan + Transformer) directly against \"PH-TR\" (standard PH + Transformer). However, the paper's own data shows no statistically significant improvement.\nFor example, on IMDB-B, where Figure 3 implies that early saturation is addressed by Topo-Scan, the performance with HKS filtration function for PH-TR is (76.8, 3.97) and for TOPOFORMER is (77.9, 5.72). The second's CI almost overlaps with the first one, which gives no evidence to support that Topo-Scan is better than standard PH.\n\n3.\tThe comparison on molecular property prediction is misleading: The results on Table 4 are also based on an unfair comparison. The model used is “TOPOFORMER*”, which is a hybrid model fusing the ECFP fingerprints. However, not all the cited methods used this additional feature. Such comparison exaggerates the advantages. It would be natural to ask whether PH-TR + ECFP gives similar results, but this is not provided either.\n\n4.\tThe paper claims to be \"lightweight\" and \"efficient\". However, Appendix C.3 (Table 10) shows that the preprocessing for the SOTA results (using Ollivier-Ricci) takes 339 minutes (>5.5 hours) on OGBG-MOLHIV. It is still a significant cost, and limits the possibility of applying this to larger scale datasets", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WAZ8rDS40w", "reviewer": "Reviewer_a6GH", "summary": "This paper introduces TopoFormer, a framework for injecting topological information into graph Transformers by turning a graph plus a filtration into a short, fixed-length sequence of “topology tokens.” Instead of running standard persistent homology and then vectorizing diagrams, the authors propose Topo-Scan: they slide overlapping windows over a node/edge filtration and, for each window, record four inexpensive quantities. Multiple filtrations (degree, HKS, Ollivier–Ricci) produce multiple sequences, which are encoded independently and fused to create a final graph representation. A tailored stability result shows these interlevel Betti sequences change in a controlled way under small perturbations of the filtration, which helps justify the construction.\n\nEmpirically, the method is evaluated on 9 graph classification datasets and 7 molecular property prediction tasks. It achieves first or second place on most small/medium benchmarks and remains competitive on the larger OGBG-MOLHIV (refer to questions). The ablations are well designed: they hold filtrations fixed and swap in PH+MLP, PH+Transformer, and the proposed Topo-Scan, which cleanly attributes gains to the sequence-based topological view rather than to feature choice. The authors also report concrete extraction times and argue their approach avoids the “early saturation” commonly seen in PH.", "questions": "At the end of 4.3 you mention that this architecture is particularly well suited for Graph Foundation Models. What makes you claim this? I do not see anything empirically to back this up. Simply working well on multiple scales (which is also only demonstrated in a limited way) is not sufficient reason to make such a claim. \n\nHow did you decide on the list of models in Table 3? It seems to leave out many top performing models of the ogb leaderboard (link above).\n\nAre the results in Table 2, and Table 3 all for the exact same filtration setup (OR curve + HKS for Table 2, atomic weight + OC curvature for Table 3)?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I find the core idea clean and promising. The proposed methods to address real challenges in applying TDA to graphs (global PH + saturation) and nicely enough it does so by simplifying in a well motivated fashion. The resulting sequences naturally fit Transformer architectures and I think this is a clean and reusable concept.\n\nThe comparisons to PH+MLP and PH+Transformer under the same filtrations isolate the benefit of the proposed representation rather than conflating it with different signal sources. This supports the central claim.\n\nThe empirical results are strong even though some common benchmarks are surprisingly missing. The model is tested on many graph classification datasets and several molecular property prediction tasks, and it is consistently at or near the top on the small/medium ones, showing the idea is not tuned to a single benchmark.", "weaknesses": "**Architecture**\n\nThe part of the paper that introduces the “new” architecture for using multiple filtrations and then combining them at the end is poorly motivated. It is also not clear if you consider this just something you tried or an important part of the contribution of this submission.\n\nTable 12 does not show any convincing evidence that using multiple filtrations in this fashion has any real benefit over using only one filtration. A discussion of this would be important and appropriate where the ablation is mentioned in the main body.\n\nMoreover, the description of the architecture suffers from some over the top phasing. For instance “To enhance generalization and mitigate overfitting, we integrate regularization techniques,\nsuch as dropout and weight decay, ensuring robustness across diverse graph learning tasks.” \nFor one these techniques of course do not “ensure” anything, additionally they are well established and it should be made clear that these techniques are standard.\n\n**Experimental**\n\nThe experiments focus on small benchmarks and none of the commonly studied benchmarks of [1] are considered. Especially in graph transformer literature these might be some of the most standard and common benchmarks that are analyzed, so it is somewhat disappointing to see them missing.\n\nOn ogb-molhiv, Table 3 does not represent the state of the art on the dataset, cf., https://ogb.stanford.edu/docs/leader_graphprop/#ogbg-molhiv.\n\n**Minor Notes:**\n\nPlease order the columns in Tables 2, 5 and 6 the same. \n\n[1] Dwivedi, Vijay Prakash, et al. \"Benchmarking graph neural networks.\" Journal of Machine Learning Research 24.43 (2023)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dkenRrztc5", "reviewer": "Reviewer_akWw", "summary": "This paper introduces TOPOFORMER, a novel framework that integrates topological data analysis (TDA) with transformer based graph learning. The central idea is to capture multi scale structural information from graphs using a topological sequence representation that can be directly processed by a transformer encoder.\n\nThe key component, TopoScan, converts a graph into an ordered sequence of topological tokens. Each token represents information extracted from overlapping slices of a graph filtration, such as degree or heat kernel signature. This design enables the model to preserve both local and global topological properties while avoiding the computational cost of traditional persistent homology.\n\nFormally, the authors prove an ℓ1 ,stability theorem showing that small perturbations in the filtration function lead to bounded changes in the generated sequence, ensuring robustness of the representation.\n\nThe resulting TopoFormer model combines TopoScan with a transformer backbone to perform graph classification and molecular property prediction. It achieves state of the art or near state of the art results on more than fifteen datasets, including IMDB, REDDIT, MUTAG, PROTEINS, and several MoleculeNet benchmarks.", "questions": "1. Theoretical clarification on representation power\nCould the authors explain in more detail how the TopoScan representation compares in expressiveness to persistent homology? Specifically, does the sequential encoding preserve the same critical topological information that persistence diagrams capture, or does it approximate it? A small empirical or theoretical comparison could help clarify what kind of information may be lost or transformed in the conversion to token sequences.\n\n2. Learnable or adaptive filtrations\nHave the authors considered learning the filtration function directly rather than fixing it a priori? For example, one could parameterise the filtration with trainable weights that adapt during training, similar to Graph Filtration Learning (Hofer et al., 2020). If so, what are the main challenges in integrating such a module with TopoScan, and could it potentially improve generalisation across datasets?\n\n3. Sensitivity to TopoScan parameters\nThe results depend on the number of slices N and the window width 𝑚 used in TopoScan. Could the authors provide a sensitivity analysis or heuristic guideline for selecting these parameters? Understanding whether performance is robust to parameter variation would increase confidence in the stability of the framework.\n\n4. Attention interpretability and visualisation\nCan the authors show examples of which graph substructures the transformer attends to when processing the topological token sequences? For instance, does the attention mechanism focus on regions corresponding to high curvature, cycles, or clusters? Such visual evidence would strengthen the claim that TopoFormer is both interpretable and topology aware.\n\n5. Broader evaluation and generalisability\nWould the authors consider evaluating TopoFormer on non molecular heterogeneous datasets such as citation or temporal graphs? Since TopoScan is proposed as a general representation, results on more diverse graph types could demonstrate broader applicability and robustness.\n\n6. Quantitative runtime and scalability analysis\nThe paper claims that TopoScan avoids the heavy computational cost of persistent homology. Could the authors provide explicit runtime benchmarks comparing TopoFormer with PH based baselines like PersLay or TopoGCL on large datasets? This would substantiate the claim of improved efficiency.\n\n7. Relation to existing topological and transformer models\nHow does TopoFormer conceptually differ from recent hybrid approaches such as TopoGCL (Zhao et al., 2023) or Graphormer (Ying et al., 2021)? A more explicit discussion of what new design principle TopoFormer introduces beyond these works would help clarify its unique contribution to the literature.\n\n8. Empirical validation of the stability theorem\nCan the authors provide an experiment demonstrating the stability property empirically, for example by perturbing the graph structure or filtration and measuring the variation in output embeddings or predictions? Such a demonstration would connect the theoretical result with observable robustness in practice.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Originality:\nThe paper introduces an inventive way to integrate topological reasoning with transformer architectures for graph learning. The proposed TopoScan mechanism is a fresh formulation that replaces persistence diagrams with sequential topological tokens derived from overlapping filtration slices. This approach is conceptually novel because it allows topological information to be represented in a transformer compatible format, overcoming long standing barriers between topological data analysis and deep attention models. The use of interlevel filtrations and the demonstration that they capture multi scale graph structure are creative contributions that extend beyond incremental improvement.\n\nQuality:\nThe paper demonstrates high technical and experimental quality. The theoretical section provides a sound stability guarantee (the ℓ1\\ell_1ℓ1​ stability theorem) that ensures robustness to small perturbations, while the empirical results convincingly show strong and consistent performance across both graph classification and molecular property prediction tasks. The authors also perform comprehensive ablations that isolate the effect of filtration type, window size, and token sequence length. Comparisons with a wide set of baselines, including topological and transformer based methods, confirm the reliability of the findings. The method is computationally efficient and well engineered.\nClarity:\nThe writing is clear, well organised, and pedagogical. The motivation is established early, mathematical definitions are presented cleanly, and diagrams effectively illustrate the construction of topological sequences and their flow into the transformer. The appendix provides sufficient detail for reproducibility. The balance between topological intuition and algorithmic implementation makes the paper accessible to both theoretical and applied audiences at ICLR.\nSignificance:\nThe work is significant in both theoretical and practical terms. It provides a general framework for incorporating topological information into neural architectures without requiring heavy persistent homology computation, making it scalable to large graphs. The approach has broad applicability beyond the tested datasets, offering a template for topology aware transformers in other relational domains such as biological networks, material science, and social graphs. By demonstrating that topological structure can be embedded as a sequence and effectively processed through self attention, the paper establishes a promising direction for future research in structure aware representation learning.\nOverall:\nA technically rigorous, clearly written, and conceptually innovative paper. It combines theoretical insight with practical relevance, resulting in a contribution that meaningfully advances the integration of topology and modern deep learning.", "weaknesses": "1. Limited theoretical depth beyond stability\nWhile the inclusion of an ℓ1 stability theorem demonstrates that TopoScan is robust to small perturbations, the theoretical analysis does not go far enough to explain why the proposed representation preserves meaningful topological invariants or how it compares in expressive power to standard persistent homology (PH) based methods. The paper would be stronger with a formal comparison of representational capacity, such as bounding the information loss between TopoScan token sequences and PH barcodes, or connecting the proposed construction to existing frameworks like Graph Filtration Learning (Hofer et al., NeurIPS 2020) or Stable Rank Vectors (Chazal et al., 2021). A clearer theoretical bridge between TopoScan and persistent homology would make the contribution more foundational rather than heuristic.\n\n2. Dependence on hand crafted filtration functions\nThe approach still relies on predefined filtrations (for example degree, curvature, or heat kernel). The choice of filtration has a notable influence on performance, as shown in the ablation studies, yet no adaptive mechanism is proposed. This limits generality and introduces dataset specific tuning. The authors could strengthen the work by introducing a learnable or task aware filtration module, or at least by exploring gradient based parameterisation of the filtration functions to allow end to end optimisation.\n\n3. Limited interpretability analysis\nAlthough the paper claims that TopoScan tokens are interpretable and capture multi scale structures, there is no concrete analysis showing what the model learns. For instance, visualising the transformer’s attention weights mapped back to graph substructures (for example motifs, cycles, or communities) would provide stronger evidence that the model captures meaningful topology. A few case studies or qualitative visualisations would greatly improve interpretability and help validate the topological claims.\n\n4. Incomplete generalisation evaluation\nThe experiments focus primarily on molecule and social graph benchmarks, which are standard but relatively homogeneous. Testing on non molecular heterogeneous graphs (for example citation networks or dynamic temporal graphs) would help confirm that the method generalises to other graph structures. Since TopoScan claims to be a general topological sequence generator, demonstrating this versatility would make the paper’s impact broader.\n\n5. Efficiency claims lack concrete quantitative support\nThe paper argues that TopoScan avoids the cubic computational complexity of persistent homology, yet runtime improvements are reported qualitatively rather than quantitatively. Providing explicit runtime tables or scaling plots, such as comparing training and inference times against PH based baselines like PersLay or TopoGCL, would substantiate the scalability advantage and enhance the practical credibility of the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rqJmSOoJ3G", "reviewer": "Reviewer_iGU1", "summary": "The paper proposes a sliding window compared to the standard sublevel set filtration for the computation of persistent homology. The resulting  sequence of simplicial complexes (graphs in this case) necessarily form an *ordered* sequence which can be processed with a transformer architecture for downstream tasks. \nThe sequence of graphs  is summarized in a set of topological features (Betti numbers and number of nodes / edges) and used as the input tokes for a transformer architecture, which is a natural choice.", "questions": "- Choice of vectorization scheme, as three hard coded numbers. Have you considered using either learned filter functions or more elaborate vectorizations such as persistent images? \n- What motivated you to use the Ollivier–Ricci curvature and Heat Kernel Signature? Have you considered other methods as well, such as learning the filtrations?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Overall, the idea, though maybe simple, well thought out, natural, elegantly executed and thoroughly evaluated with great care for details. The code is provided, well documented and easily accessible, which is great to see!  The reviewer believes that the use of a sliding window for the computation of persistent homology (though no longer persistent perse) is interesting and novel, which is encouraging given the results on the benchmark datasets. This perspective is also interesting beyond the scope of the specific choices made in the paper which is very interesting.", "weaknesses": "Overall im very positive about this work. The only two weaknesses of the method could be the choice of vectorization scheme (choose a more expressive statistic instead of $\\beta_{0}$ or $\\beta_1$). \nAdding a discussion there could be good. The second minor weak point is that a discussion on the current limitations is missing and some remarks on either future work and the current challenges could be beneficial to the readers. For instance, most methods in TDA have been specifically developed for filtrations and would no longer hold in this scenario.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)