TopER: Topological Embeddings in Graph Representation Learning
Authors: Astrit Tola
Funmilola Mary Taiwo
Cuneyt Gurcan Akcora
Baris Coskunuzer
Venue: NeurIPS 2025 poster ()
Keywords: TopER
Graph Embeddings
Topological Data Analysis
Interpretability
Graph Visualization
Graph Classification
Graph Clustering
Low-Dimensional Representations
Primary area: general_machine_learning
TL;DR: TopER is a low-dimensional, interpretable graph embedding method based on topological evolution rates, enabling intuitive visualization and strong performance on clustering and classification tasks.

Abstract: Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. 

In this work, we introduce Topological Evolution Rate (TopER), a novel, low-dimensional embedding approach grounded in topological data analysis. TopER simplifies a key topological approach, Persistent Homology, by calculating the evolution rate of graph substructures, resulting in intuitive and interpretable visualizations of graph data. This approach not only enhances the exploration of graph datasets but also delivers competitive performance in graph clustering and classification tasks. Our TopER-based models achieve or surpass state-of-the-art results across molecular, biological, and social network datasets in tasks such as classification, clustering, and visualization.

TopER: Topological Embeddings in Graph Representation Learning
Astrit Tola
Funmilola Mary Taiwo Department of Statistics University of Manitoba Winnipeg, Manitoba, Canada taiwom1@myuman @aanlammttmaanaaaaaaaa ca
Department of Mathematics Florida State University
Tallahassee, FL 32306 atola@fsu. edu
Cuneyt Gurcan Akcora
Baris Coskunuzer
Al Institute University of Central Florida Orlando, FL, 32816 cuneyt. akcora@ucf edu
Department of Mathematical Sciences University of Texas at Dallas Richardson, TX 75080 coskumuz@utdallas. edu
Abstract
Graph embeddings play a critical role in graph representation learning, allowing machine learning models explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization.
this work, we introduce Topological Evolution Rate (TopER), novel, low- dimensional embedding approach grounded in topological data analysis. TopER simplifies key topological approach, Persistent Homology, by calculating the evolution rate of graph substructures, resulting in intuitive and interpretable vi- sualizations of graph data. This approach not only enhances the exploration graph datasets but also delivers competitive performance in graph clustering and classification tasks. Our TopER- models achieve or surpass state-of-the-as results molecular, biological, and social network datasets in tasks such classification, clustering, and visualization.
pip install toper
Introduction
Graphs are fundamental data structure utilized extensively model complex interactions within various domains, such as social networks [LBKTO8], molecular structures [YLY+ 18). and trans- portation systems [DCS+22] Their inherent flexibility, however, introduces significant challenges when applied to machine learning (ML) tasks, primarily due to their irregular and high-diment sish--imensionall nature. Graph data lacks inherent ordering and consistent dimensionality, making it challenging for traditional ML methods designed for vector spaces.
Graph Neural Networks (GNNs) have emerged the state-of-the-art models for tackling graph machine learning tasks due to their ability to learn effectively from graph-structured data [KW17) (7]. In the predominant paradigm of sassage-passing GNNs, the process begins by generating node embed- dings [BHG+21]. These embeddings can then be used in tasks such as node classification or link prediction. However, for graph-related tasks, such molecular property prediction, the embeddings
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

14-
2.25
12-
gowe
0,75
growth
1.25
MUTAG PROTEINS
Class
Class A
COX2
Class
Class 0
50
40
50
20
10
10
15
100
60
60
40
30
DIVCE 15l
pivot (5)
pivot Ta)
(a) 3 DATASETS
(b) MUTAG
(c) IMDB-B
Figure I: TopER Visualizations. Each data point represents an individual graph. On the left. TopER is applied to three benchmark compound datasets using closeness sublevel filtration. The middle panel zooms in the red point cloud from the left, demonstrating TopER's 's effectiveness in distinguishing between classes within the MUTAG dataset. On the right, a TopER visualization for the IMDB-B dataset is displayed.
muss be aggregated through a pooling layer to form graph-level representations [WKK This method is computationally intensive, mainly because generating and managing node embeddings as intermediate steps substantially increases the overall computational burden. Ideally, an approach would allow for the direct creation of graph embeddings, circumventing the need to generate node- level representations first. Furthermore, these graph embeddings must be both low-dimensional and interpretable to maximize their practical utility and efficiency various applications.
Topological Data Analysis (TDA) is well-suited for directly constructing graph representations without costly node embeddings. Topology studies the shape of data, and TDA primarily focuses on the qualitative properties of space, such as continuity and connectivity [CA24]. A particularly effective technique in TDA is Persistent Homology (PH), which tracks topological features, like connected components and cycles, across various scales via a process known as filtration. Filtration is adept at revealing both local and glohal structures within graphs. It proves exceptionally useful for comparing graphs of different sizes that maintain the same inherent structure, which may suggest similar properties in graph datasets. For instance, similar substructures in protein interaction networks across different species may indicate comparable biological functions. By focusing on data shape, PH proves invaluable in graph tasks that benefit from a graph-centric approach, offering insights that might not be as apparent when focusing on individual node analysis. This shift from a node- centric graph-centrin perspective can dramatically improve the understanding and application of graph data in fields like bioinformatics and network analysis. However, the utility of Persistent Homology is limited by the high computational demands involved in extracting topological features during the filtration process, mainly due to its cubic time complexity [OPT+ 17]. This constraint reduces its practicality for large-scale graphs and has restricted the broader integration of PH in graph representation learning.
With this work, we take significant step forward in addressing the challenges of topological graph representation learning and introduce Topological Evolution Rate (TopER). This novel approach refines the Persistent Homology process to efficiently capture graph substructures, thereby mitigating the significant computational demands of calculating complex topological features. As graph represen- tation learning aligns naturally with Topological Data Analysis, TopER excels in graph clustering and classification tasks, where it achieves the best rank in experiments. Furthermore, simplifying graph data into a low-dimensional space, TopER creates intuitive visualizations that reveal clusters, outliers, and other essential topological features, as demonstrated in Figure 1. As result, TopER merges interpretability with efficiency in graph representation learning, providing an ideal balance that can scale to large graphs. To our knowledge, TopER is the first topology-bast based graph representation method that can create low-dimensional, efficient, and scalable graph representations.
Our contributions can be summarized as follows:
New Graph Representation: We introduce TopER, a principled and computationally feasible graph representation designed to capture the structural evolution of a graph.
Inspied Design: Rather than computing persistent diagrams or Betti numbers, TopER directly leverages the filtration process to provide efficient, low-dimensional summaries rooted in topological intuition.

---

Experiments on benchmark graph classification and clustering tasks demonstrate that TRpER achieves competitive results compared to more complex, state-of-the- models, while offering superior interpretability.
Competitive Performance:
Interpretable Visualizations: TopER produces low-dimensional outputs that support intuitive visualization of clusters and structural outliers within and across graph datasets, enabling form of visual model comparison often lacking in traditional embeddings.
Robustness Under Perturbations: We provide stability analysis showing that TopER's represen- tations are consistent under small changes to the filtration function, reinforcing suitability for practical and comparative applications.
2 Background
2.1 Related Work
Graph-level Embedding Methods. Graph representation learning, including GNNs and Graph Pooling techniques, is a dynamic subfield of machine learning, focusing on transforming graph data into efficient, low-dimensional vector representations that encapsulate essential features of the data [Ham20, GHT 19]. These representations facilitate a deeper analytical understanding of graphs, which is critical for various applications such molecular graph property prediction [DTRF19]. GNNs have revolutionized the analysis of graph data, drawing parallels with the success of Con- volutional Neural Networks in image processing [EPBM20]. GNNs utilize spectral and spatial approaches to graph convolutions based on the graph Laplacian and direct graph convolutions, respectively [BZSL14, DBVI6, KW17J. Despite their success, GNNs often suffer from issues like over-smoothing and a lack of transparency, making them less ideal for applications requiring interpretability [Günn2]].
Graph Pooling emerged as a key component in GNN architectures, drawing parallels to the role of pooling in Convolutional Neural Networks [EPBM20]. Pooling aims to deduce into meaningful graph embeddings through node aggregation (mean, max, and add pooling [XHLJ19]) or hierar- chical pooling (node selection: Top-k [GJ19] and SAGPool [LLK19]; node clustering: DiffPool [YYM+ 18] and MinCutPool [BGA19]). Both pooling groups have their challenges. Node aggrega- tion methods may lose the structural information by treating each node equally, while the hierarchical pooling methods can be computationally heavy, and loss of information can occur if important nodes discarded.
Eapology-Inspiree Representations. TDA provides a robust and computationally efficient frame- work to address the interpretability and over-smoothing issues present in GNNs [AAF19]. Persistent Homology, a key technique in TDA, has been applied successfully to graph data, demonstrating potential to match or even exceed the performance of traditional methods in classification and clus- tering tasks [HMR21, DCG 22, HILL+24 ISG23, CSA+ L3]. However, the computational intensity of PH limits its scalability [HKN19, ZYCW20, AKGC22).
Visualization Techniques. Graph embedding techniques, including spectral methods, random walk- based approaches, and deep learning-based models, transform graph data into vector representations to support tasks like visualization and machine learning (CZC18, GF18, Xu21J. Approaches such as Laplacian Eigenmaps and Walk have been particularly effective in revealing clusters within graphs [BN01, PAS14]. However, these methods are predominantly applied to visualize a single graph in node classification tasks, focusing on cluster identification [WCZ16, MK20, TPPM23]. Furthermore, they often overlook domain-specific information, which can limit their effectiveness in more specialized applications [JZ20].
TopER addresses these challenges by combining the interpretative benefits of TDA with the analytical strength of modern graph machine learning. Distinct from current approaches, TopER employs simplified filtration process to create embeddings that are both interpretable and computationally efficient. By extending the filtration to multiple functions, TopER stands out as one of the first methods to offer effective and interpretable visualizations of graph datasets, while also achieving superior performance in clustering and classification tasks.
3

---

2.2 Persistent Homology for Graphs
Topological Data Analysis (TDA) offers a powerful framework for graph representation Iearn- ing [AAF19], with persistent homology (PH) being especially effective at capturing multi-scale topological features [CA24]. While PH typically involves filtrations, persistence diagrams, and vectorization, our model focuses on the filtration step, reformulating the evolution of topological features in novel and efficient way.
In the crucial filtration step, PH decomposes graph into a nested sequence of subgraphs Gi S 92 ... S 9n = G. For each 9i an abstract simplicial complex G defined, forming filtration of simplicial complexes. Clique complexes are typical choices, where each (I: + co-mmlee subgraph in 9 corresponds to a k-simplex [AAF19].
Filtration. Utilizing relevant filtration functions is essential to obtain effective filtrations. For given graph G (V,E), a common approach to define a node filtration function f V - R, which establishes a hierarchy among the nodes. By selecting a monotone increasing set of thresholds I {er i= this method generates subgraphs G. = (V;, E,) where V; = {u E, is the set of edges in & with endpoints in V. This is called sublevel filtration induced by f (See Figure 2). Also, superlevel filtrations can be constructed by defining V: = {u E V f(v) 2 e for decreasing thresholds [AAF19].
f(v) S c and
edge filtration functions g : E R to define such filtration. We define & = E E g(exx) S h. and V; as all the endpoints of & to create nested sequence {G.) )==) Especially for weighted graphs, this method is highly preferable as weights naturally define an edge filtration function. The common node filtration functions are degree, betweenness, centrality, heat kernel signatures [BK10], and node functions coming from the domain of the datasets (e.g.. atomic number for molecular graphs) Common edge filtration functions are Ollivier and Forman Ricci curvatures [LLYI]] and edge weights (e.g,, transaction amounts for financial networks).
Similarly, one
In addition to existing approaches, we in- troduce
filtration function, Popu- larity, which extends the idea of degree- based ranking [New03]. While the de- gree function measures the number of di- rect neighbors node has, Popularity cap- tures the average degree of those neighbors. The underlying intuition is that, whereas degree reflects how many connections node has. Popularity emphasizes node influ- ence through association with high-degree nodes.
5
5
2
1.5
1.5
1.5
1.5
3.5
5
5
2
2
G
G2
Figure Filtration. For 9 = G3 both examples, the top figure illustrates superlevel filtration with node degree function for thresholds (1, 2, 3}. Similarly, the bottom figure illustrates sublevel filtration for edge weights with thresholds {1.5, 1.8, 2.1}.
Formally, for each node v, we define popu-
larity as: P(t) = deg(v) + ΣmENio deg(u) [W(w)
where deg(v) is the node's degree and ((v) its set of neighbors. This function incorporates 2-neighborhood information by weighting high-degree neighbors more heavily, making it a refined version of the degree function.
3 TopER: Topological Evolution Rate
TopER is inspired by the foundational idea that topology can capture the shape of graph data, and that this shape can be observed through the evolution of graph during the filtration process. In this sense, TopER is applloy--nnspird However, TopER diverges from traditional PH in how tracks the shape. We first achieve computationally efficient alternative to persistent homology by simplifying its filtration-base perspective, and second, we develop a low-dimensional, interpretable representation of graphs that enables both effective classification and intuitive visualization across graph datasets.
Our reformulation reduces the computational overhead typically required for topological feature ex- traction. Unlike Persistent Homology, which extracts costly topological features, TopER summarizes the filtration process through two key parameters derived via regression: filtration sequences and evolution.

---

Filtration Sequences. We first decompose a graph & into a nested sequence of subgraphs (filtration graphs) 9 C .2... S In g by using a filtration function, such as node degree or closeness. Let 9 C G. Vi represent nodes in 9i and & represent the edges. Next, we compute x; = =Vi as the count of nodes, and y; = |& as the count of edges. Then, for each filtration graph G. we obtain the pair (z4, V1) RR, which creates two monotone sequences, I1 S 22 S ... S 2n and 91 S 122 ... S yn. Hence, TopER yields two ordered sets X2 describing the evolution of the filtration graphs Gi S ... S Gn = G. X (21, .2,...,, ,x)) and ) (y), 1/2, .2...... ,....)....)..............)...... Here, n corresponds to number of thresholds { used in the filtration step. Consider the top row of Figure 2 where we have three filtration graphs (i.e., n=3); we have X' (2, 3, 7) for node counts and y = (0, 2, 6) for edge counts.
Fileation and line Sotin
Evolution. In the next step, PH would typically compute topological features on each filtration and create a persistence dia- gram to summarize the features. Not only is it costly, but the approach would re- quire efforts to vectorize the persistence diagrams. We circumvent this computa- tionally costly step and analyze how the number of edges {y: relates to the number of nodes {a,} throughout the filtration se- quence. We use line fitting to characterize this relationship as follows. Simple linear regression, often applied through the least squares method [JWH+23) is standard approach in regression analysis for fitting a linear equation to set of data points {(2;, y:}} IR22 This method calculates the line L(x) = a + be that best fits the data by minimizing the loss function E = [_(,)( - y.]². The regression coefficients (a, 6) capture the graph's 's structural evolution through filtration (see descriptor step in Figure 3). The full TopER method is outlined in Algorithm in the Appendix.
Deserpttess
14 22 bxx
as + bix
b
Figure 3: TopER steps. The filtration process on three different graphs using node or edge filtration. The graphs undergo filtration, and for each graph, a best-fit line deter- mined through the filtration data. The coefficients of these best-fit lines are then used as descriptors for the graphs
With evolution on filtration sequences, we define the topological evolution rate of graph as follows:
Definition 3.1 (Topological Evolution Rate (TopER)). Let f V + R be filtration function on graph 9 and I = ) be the threshold set. Let G; = (V,, E,) be the induced filtration. Let I, [V, and y [& Let L(x)=
bæ be the best fitting line to {(x,, y))];= Then, we define the TopER vector of g wrt. f as TE, = (a, b). We call a the pivor and the growth of G.
Remark 3.2 (Why a Linear Fit?). Although one could consider higher-order polynomials, our experiments show that coefficients beyond the first degree are negligible (Table 15), revealing an essentially linear relationship. Due to space constraints, we defer a more detailed comparison of linear, polynomial, and higher-order fits Appendix B, where we also provide visual examples of cvolution rates (Figure 6) and fundamental pattern types (Figure 7).
Remark 3.3 (On the Name TopER). Although TopER does not compute classical topological invariants such as persistence diagrams or Betti numbers, its name is grounded in two topology-inspired principles. First, it mirrors the TDA notion of a filtration, a nested sequence of subgraphs {G:) } that progressively unveils structural features across scales, Second, it is directly tied to the evolution of the Euler characteristic x(G.) |V.|- [& number. Equivalently, one could track { y/ =z)) ((VA), }. but we use {(x,, }:)} for notational simplicity. The resulting slope thus quantifies how topological complexity changes over the filtration, justifying the term Topological Evolution Rate.
(,.) 3 (G.), where 3, denotes the ith Betti
We emphasize that the term "topological" reflects the conceptual roots of TopER in topological data analysis: the method uses a filtration to reveal structural patterns in the graph, analogous to the filtration process in persistent homology. While we do not compute full persistence diagrams or Betti numbers, the slope of the Euler characteristic across the filtration captures essential topological information, justifying the terminology.
3.1 Computational Complexity
The primary computational steps in TopER include constructing filtration graphs and performing regression on node and edge counts, which incur the following costs.
5

---

Analyzing each node and edge across filtration thresholds typically requires O(n x ((V + (&)) operations, where V and & denote the numbers of vertices and edges, respectively. The regression step involves fitting a line to the pairs (ri; yi) using the least squares method. The complexity of calculating the necessary sums for this regression is (n), and solving for the regression coefficients (slope and intercept) from these sums involves constant amount of additional computation.
Thus, the overall complexity of TopER predominantly hinges on the graph filtration process, summing up to (n x (IV) + (E])) where [V] > n.. As we will show in the next section, the runtime costs of TopER are notably low, making it practical and efficient for large-scale applications.
3.2 Stability Results
This section states our theorems on the stability of TopER. the following, Wp(-.-) represents Wasserstein distance, and PD&(X, represents 1thh persistence diagram of X with sublevel filtration with respect to Similarly, II-IIp represents L"-norm and dp(..-) represents lp-distance in R We fix a threshold set = {4 [ for both functions to keep the exposition simple. Further, to keep the setting general, we the pairs {(Bo(e)) (e:))) in IR² fit the least squares line y + bu defining TE (a,b).
Theorem 3.4. Let X a compact metric space, and J,g: X - be two filtration functions.. Then, for some C > 0,
(X) - (Eg(1) S C. (I)
By combining the above result with the stability result for sublevel filtrations, we obtain the stability with respect to filtration functions as follows.
Corollary 3.5. Let X be compact metric space, and f,g: X - R be two filration functions. Then, for some C > 0,
(X) TE, C. ||f glli
The following lemmas are essential for proving the theorem and corollary above.
Lemma 3.6. [ST20] Let X be compact metric space, and f.g X + R be two filraaton functions. Then, for any p 2 1, we have (pDDP (A,, (D, g)) S glpp
The next lemma is on the stability of Betti curves by [DG23] [Proposition 1].
Lemma 3.7. [DG23] Let Bx(X) is the tt Betti function obtained from the persistence module (M).
[x) B&(D)ll S (D(()))
By adapting the above results to the graph setting, when two metric graphs G1, G2 are close in the Gromov-Hausdore sense, obtain similar stability result for the filtrations of Gi induced by the same filtration function. Due to space limitations, details and the proofs are given Appendix C.
Experiments
Table 1:
Characteristics of the benchmark graph classification datasets.
We evaluate the performance of TopER in
Datasets
#Graphs
[V]
[S]
Classes
classification, clustering and visualization. Our Python implementation is available
BZR
405
35.75
38.36
COX2
467
41.22
43.45
MUTAG
188
17.93
19.79
https:/ /hgt://b./bbb.....// com/Astrittola//TOO
PROTEINS
1113
39.06
72.82
IMDB-B
1000
19.77
96.53
IMDB-M
1500
13.00
65.94
4.1 Experimental Setup
REDDIT-B
2000
429.65
497.75
REDDIT-SK
4999
508.52
594.87
We conduct experiments nine bench- mark datasets for graph classification. These are the molecule graphs of BZR, and COX2 [MV09]; (ii) the biological graphs of MUTAG and PRO- TEINS [KM12]; and (iii) the social graphs of IMDB-Binary (IMDB-B), IMDB-Mult (IMDB-M), REDDIT-Binary (REDDIT-B), and REDDIT-Multi-SK (REDDIT-5K) (YV15). Finally, the OGBG- MOLHIV is large molecular property prediction dataset, part of the open graph benchmark (OGB) datasets [HFZ+200 Data statistics given in Table 1.
Datasets.

---

Table 2: Graph Classification, Accuracy results on eight benchmark datasets. Best results are in bold blue, second- best are underlined. The final column shows each model's average deviation from the best per dataset.
Model
BZR
COX2
MUTAG
PROTEINS
IMDB-B
IMDB-M
REDDIT-B
REDDIT-KK
Avg.l
DiffPool IYYM 18]
83,93+441
79.664264
79.22=162
79,
73.63=3m
68. 38.60
45.70=340
79.00s1100
8.06
P-WL-C (RBB19)
10.51+134
75.27=033
2.08
SACiPoo [LLK19]
82.95+405
79.45+298
76.78=212
71.86=00
74.87=409
49.33s450
84.70s440
6.61
Top-k [G]19]
19.40=120
80.30+421
67.61+336
00:300
73. 73.17:4.54
48.80=319
79.40=7.4
9.70
1-GIN (GHL) [HGR 201
74.10=340
74.50=4.00
49.70=250
90.20=22
55, 25.7099290
2,09
6 GNNs [EPBM20]
80.42+2:07
75.80=170
71 71.20=1.00
49. 10=3.0
89.90=11M 10
56. 10
4.13
MinCutPool [BGA 19]
82.64a505
80.07
79, 164
76.62:
70.77
49.00
87. 17:00::00
5.82
DMP [BC1.21]
84.00.K40
15.30
L3.80
2.190
86.20..40
51.901210
PC-V [ORB2]]
a5.61
87.31
74.54
73.84
46. 30
89.4
4,01
SubMix [YSK22]
84.6 16.34
1:20
80,991.000
270
70.30
46.47_220
G-Mix [HULH22
64.15=2330
83.83+2.e
81.96:060
66.28
69.
69.40=1.30
46.40=2-0
6.91
RGCL [LWZ
84.5411
79.31
17.661101
1..0
11.85
49.31 1o44
90.34
56.38 10-40
AutoGCL [YWH+ 22]
86.27
79. 1970
88. 18.64
72.32:
50.60
88.58
$6.75
3.08
GCN [PAMF23]
1
1800
1.000
11
6..00
11.53
WWLS [FHSK23]
88.02
81.58
88, 1.2
75.35
95.0
$1.61=022
2.26
EPIC [HLAK24]
85,53+1. 88.78+220
82.44=0.0
69.06=1.00
71.70=1.00
47.93=1130
EMP (CSA 23]
88.79:983
72.78
74.44
48.01 =044
91.03
$4.41
2.97
HSM [3)
17.10
14.60
84.80.1 1.90
5.390
TopoCCL [CFG24)
87.17
81.45
20.
77.30
74.67
52.81 00.1
90,40
1.76
PGOT [QTL1224]
87.32_
12.98
92.63
13.21
6..90
51.33
RePHINE [ISG23]
71.25:110
69.40
5.86
GPSE [CLL. -24]
80.49
78.37
k7.19
72.15
69.30_141
47.40
80.40
TopER
90.13+414
82,01+459
90.99e6.4
74.58=1.0
73.20a3.41
50.00s4:0
92.70=2.3
56.51
1.60
Hardware. We ran experiments on a single machine with 12th Generation Intel Core i7- vPro Processor (E-cores up 3.50 GHz, P-cores up to 4.80 GHz), and 32GB of RAM (LPDDR5- 6400MHZ).
Runtime. TopER is highly scalable and can be applied to a 100K node graph in less than 2 minutes (see Figure 5). Our small network experiments took about two days in a shared resource setting, whereas the OGBG-MOLHIV experiments took 7,85 hours. One of the most demanding datasets, REDDIT-SK, requires 2.91 hours to calculate all node and edge functions. The runtime of our methods is dominated by the computation of node functions such as closeness and Riccis blue(see Appendix A. Using approximate values for centrality metrics instead could greatly decrease computation time [BP07]. Since this is not our current focus, we leave it as future work.
Model Setup and Metries. We employ a rigorous experimental setup to ensure a fair comparison and the selection of the best graph classification model. We begin by applying BatchNormalization to the input features to maintain consistent scaling. We employ a 90/10 train-test split, adopt the StratifiedkFold strategy, and present the average accuracy from ten-fold cross-vas cross-validation across all our models. We employ accuracy as the evaluation metric, a widely utilized performance measure within graph classification tasks [EPBM20].
Filtration Functions. In TopER, we use both node and edge filtrations (Definition 3.1). Alongside popularity, we apply degree, closeness, and degree centrality [EC22] as node filtration functions and Forman- and Ollivier-Riae Ricci functions [LLY [] as edge filtration functions. We also use atomic weight node function for molecular and biological datasets (BZR, COX2, and MUTAG), and node attributes (PROTEINS). We utilized the t-test to assess the statistical significance of each function and applied the Lasso method for regularization. Functions were retained in the model only if they achieved v-luuess less than 0.05 in the t-test and had non-zero coefficients in the Lasso model [JWH+23) 233. This approach ensures that the selected filtration functions contribute statistically significant and regularized features to the model. Incorporating additional filtration functions can enhance TopER's ' ability to analyze graphs from diverse perspectives. However, we will next illustrate in Table 5, TopER demonstrates strong performance even in its most basic form using the simple and scalable node degree function. This balance of performance and simplicity suits scalability philosophy; we avoid complex and costly schemes for learning -ataset-specific activation functions and homogenize the filtration step in all datasets.
Classifier. We utilize a Multilayer Perceptron (MIP) in our graph classification task, The hyperpa- rameters are detailed in Appendix A.7.
7

---

4.2 Graph Classification Results
Baselines. We compare our method with 22 state-of-the-art and recent models graph classification, including variants of graph neural networks: six GNNs including GCN, DGCNN, Diffpool, ECC, GIN, GraphSAGE which are compared in [EPBM20] (best results of these six GNNs are given in the 6 GNNs row), FF-GCN [PAMF23]; topological methods: DMP [BCI.21], FC-V [ORB21], WWL.S [FHSK23), MP-HSM [ISCC23] and EMP (CSA+23]) GNNs enhanced with data aug- mentation methods: SubMix [YSK22], G-Mix [HJLH22], and EPIC [HLAK24]]; GNNs enhanced with contrastive learning methods: RGCL [LWZ 12], AutoGCL (YWH+22), TopoGCL [CFG24];
prototype-based methods: PGOT [QTLL24]; pooling methods: Top-k [GJ19], SAGPool [LLKI9), DifiPool [YYM+ 18), MinCutPool 19]A19] and structural encoder: GPSE [CL
Table 2 shows the accuracy results for the given models. We use the reported results in the cor- responding references for each model. "_" entries in the table mean the reference did not report any result for that dataset. In [EPBM20], the authors compare the six most common GNNs on the graph classification task (see the GNNs row). The last column summarizes each model's overall performance. We report the average of the differences between each model's performance and the best performance in the column across all datasets. If a model's performance is missing for a dataset, it is excluded from the average computation for the model.
Out of eight datasets, TopER achieves the best results in two and ranks second in two other datasets. For the remaining four datasets, TopER's 's performance is within 4% of the SOTA results. For overall performance, TopER outperforms all other models with an average deviation of 1.60% from the best performances. The closest competitor is TopoGCL, which has an average deviation of 1.76%.
OGBG-MOLHIV results. To evaluate our model's per- formance on large datasets, we compare it with recently published models on the OGBG-MOLHIV dataset, as
Table 3: AUC results for OGBG- MOLHIV dataset.
shown in Table els are listed in chronological order based on their publication dates, with baseline performances reported from ICPWC23, YCL +21] or the respective model's ref- erences. In Appendix A.I, we give further details for TopER performance and the contribution of each function on this dataset. TopER achieves the second-best result on the MOLHIV dataset, while the pop-erforming model requires learning a significantly larger model with 119.5 million parameters.
The performances of these mod-
Model
AUC
GIN-VN [XHLJ19]
77. 17.80=1882
HGK-WL [TGL+19]
19.05=1130
(BGI+
75.58=11440
PNA [CCB+22
79.05
DGN [BPL+21]
79.70
GruphSNN [WW22]
79.72+1.8
GCN-GNorm [CLX 21]
78. 18.80
Graphormer [YCL*21]
80.51.10.53
Cy2C-GCN [CPWC23]
78. 18.02.10.60
GAWL [NV23]
78.34
LLM-GIN [ZZM244]
79.221NA
TopER vs. PH. TopER consistently outperforms Persis- tent Homology methods in both accuracy and computa- tional efficiency. As shown in Table 4, we compare against the best PH results reported in [Cai21]. which evaluates 16 combinations of four filtration functions and four vectorization techniques per dataset. TopER achieves higher accuracy on all six benchmarks. In terms of runtime, TopER is over 10 times faster than PH on large graphs like Reddit-5K, while maintaining high performance (see Table 8 and Appendix A.2).
GMoE-GIN [WJY*23]
76,90
TopER
80.21
Table 4: Accuracy results for TopER vs. Persistent Homology in graph classification tasks.
BZR
COX2
PROTEINS
IMDB-B
IMDB-M
RED-SK
PH
88.41:06
82.0=aa
74.0204
69.5.as
46.5203
54.1
TopER
90.1
82.0=46
74.6439
73.2+34
50.0
56.5
Ablation Studies. We have conducted three ablation studies. In the first one, we evaluated the individual performance of each function as well their combined effect on classification. As shown in Table 5, the common filtration functions we employ from TDA exhibit strong individual performance. Moreover, when combined, they synergistically enhance overall performance. This not surprising, as different filtration functions, such as atomic weight or Ricci curvature, generate distinct hierarchies and node-edge distributions, resulting in diverse connectivity patterns throughout the filtration sequence. This diversity is analogous viewing an object from multiple angles. Hence,
8

---

integrating these complementary perspectives improves performance by offering richer and more varied representation of the graph structure, allowing the model to capture more intricate features. The other two ablation studies are provided in the Appendix A.4, which examines the effect of the number of thresholds on TopER's performance, while Appendix A.5 analyzes the impact of the number of filtration functions used.
Table 5: Ablation Study. Individual and altogether performances of filtration functions with TopER.
Datasets
Degrec-cent.
Popularity
Closeness
Degree
F. Ricci
O. Ricci
Atom weight
TopER
BZR
82.22.12.13
82.20
$1.48-199
82.73
80.75.1..73
80.99
82.231212
90.1314.14
COX2
75.38+199
-9.
67.90-796
73.88+5.0
70.4647.2
73.03=421
69.82=827
82.01
MUTAG
76.61
77.66=6.12
4..88_4.19
74.97
80.85
82.46-784
a3.45
90.99
PROTEINS
67.66+3.16
70.71=441
69.01=424
69.01
72.964347
71.25=206
73,59+335
74.58
IMDB-B
73.00.4.49
71.90
42.60_4.20
13.1014.18
69
80.2.4
66.40.3335
73.20
IMDB-M
48.47+3.0
47.87=30
48.33=3.9
47.93+288
18.1344411
43.60=3317
50.00:4402
REDDIT-B
76.70=3.00
79.35=346
18.10_
79.55+2.3
22.35
68,20=208
92.70=2.38
REDDIT-5K
42.85
50.87=268
50.03=1.49
47.01
50.27+1.92
42.8
56.51:222
4.3 Graph Clustering Results
We employ cluster quality metrics to assess the embeddings of graphs sourced from all datasets in Table 2. The embeddings are labeled with their respective dataset memberships, and we assume that good embeddings will have graphs of the same dataset clustered together. We evaluate embeddings based on three widely used clustering metrics: Silhouette (SILH), Calinski-Harabusz (CH), and Davies-Bouldi (DB) [GBC21]. Table 6 compares the clustering performance of TopER and Spectral Zoo [JZ20]], which is, to our knowledge, the only model that allows low-dimensional graph embed- dings. Detailed results provided in Appendix A.3. The findings demonstrate that the embeddings generated by TopER outperform those created by Spectral Zoo. This is evident from the superior cluster quality metries observed for five out of eight datasets in the case of Silhouette and CH, and for all eight datasets in the case of DB.
Table 6: Clustering Performances. Comparison of Spectral Zoo vs. TopER. The detailed results are given Appendix A.3.
Metric
Method
BZR
COX2
MUTAG
PROT.
IMDB-B
IMDB-M
REDD-B
REDD-5K
Spec. Zoo TopER
0.050
0.049
0.344
0.050
0.097
-0.024
0.108
-0.121
Silh f
0.249
0.414
0.258
0.086
0.064
-0.032
0.196
-0.067
CH
Spec, Zoo TopER
3.51
6.13
120.73
38.77
85.24
30.98
269.94
119.81
42.58
26.00
72.52
151.64
60.52
11.77
446.12
1209.95
Spec. Zoo
7.25
6.07
0.95
4.55
2.78
10.73
2.20
25.74
TopER
1.93
2.29
0.88
1.54
2.19
6.87
1.32
2.78
4.4 Graph Visualization
In the case of a single filtration function, TopER creates 2D graph embeddings (a, b) that can be visualized with ease (see Figure 1). Traditional dimensionality reduction techniques such as PCA can be used to visualize point cloud data, but accurately depict- ing graph data has historically been significant challenge [GBGA20). To our knowledge, until TopER, the only model that allowed graph visualiza- tion was the GraphZoo [JZ20].
200
12-
275-
2
122
14-
10-
Class
Class A
425
Class
Class
TopER creates highly interpretable graph visualizations. To recall, the pair (a, b) represents the coefficients of the best-fitting function L(x)= a bu, where a is the pivot (y-intercept) and b is the growth (the slope). Specif- ically, the pivot reflects graph con- nectivity, while b reflects the growth rate of edges/nodes for the filtration function. In particular,
-20
piver 00
pive
(a) PROTEINS
(b) BZR
Figure 4: TopER visualizations of the PROTEINS dataset with O.Ricci edge filtration, and BZR dataset with degree centrality node filtration. Each point corresponds to an individual graph.

---

higher value of corresponds to a more interconnected graph. As we demonstrate in Appendix Figure 7, graph connectivity and community structure can be analyzed using three types of pivot behavior. In the following, we illustrate how these quantities can be employed to interpret our two-dimensional representations of the graph datasets.
In Figure 1(b) of the MUTAG dataset, class B has a higher growth rate and smaller pivot than the red class. This shows that the class is growing faster than class A with respect to the closeness function in the MUTAG dataset, i.e., the graph has a low diameter. Similarly, in contrast, in the PROTEINS dataset (Fig. 4(a)), the growth rates are similar for both classes (~ 1.5 1.7), but the pivot (initial graph size) is smaller in class A. This implies that class A has fewer edges in relation to the number of nodes. Such patterns, as described in Sec. B.4, can reveal key insights into graph topology. In similar vein, TopER visualizations can be used for anomaly detection. For example, in Fig. (aa). an outlier PROTHINS graph alone has a positive pivot and appears as the rightmost data point.
More importantly, TopER homogenizes graph representations, allowing us to compare graphs across datasets, which may open new paths training graph foundation models. To our knowl- edge, TopER is unique in directly producing interpretable 2D embeddings for cross-dataset vi- sualization without relying on learned high-dimensional encodings or opaque projections, unlike GPSE [CLL+24] and GFSE [CZW 25), which rely on learned nigh-ddmmesiioal embeddings. For example, Figure ((a) visualizes graphs of three datasets on the same panel, where we see that Mutag and COX2 differ in their pivot only. The similarity is not surprising; MUTAG and COX2 are datasets of molecular graphs where nodes are atoms and edges are chemical bonds. As the molecules both datasets have similar types of atoms and bond configurations (e.g., ring structures), TopER captures these similarities, leading to similar embeddings.
These examples highlight that, in many practical settings, TopER's 's interpretability outweighs modest performance differences compared to more expressive or data-driven models. As shown in Figure the pivot-growth representation captures fine-grained structural variations, such as changes in edge density, community organization, and filtration behavior, through simple geometric patterns that can be directly visualized and interpreted. This enables users to pinpoint the topological mecha- nisms responsible for observed differences between classes or datasets. Beyond classification, such interpretability makes TopER particularly well-suited for applications where structural understand- ing critical. Its training-free and computationally efficient design further allows deployment in large-scale or sitacsccace environments, as well as integration into hybrid pipelines where TopER embeddings serve as interpretable anchors for downstream learning models. Thus, while deep or spectral approaches may achieve marginally higher benchmark performance in some cases, TopER offers complementary framework that prioritizes clarity, scalability, and theoretical grounding in topological structure.
Limitations. While our approach is designed to be efficient and broadly applicable, its performance can vary depending the choice of filtration function, which may require domain knowledge in certain applications. In practice, we found the method to be robust across a variety of datasets, and further refinements to filtration strategies could enhance adaptability to new domains.
5 Conclusion
We have introduced a novel graph embedding method, TopER, leveraging Persistent Homology from Topological Data Analysis. TopER demonstrates strong performance in graph classification tasks, rivaling SOTA models. Furthermore, it naturally generates effective 2D visualizations of graph datasets, facilitating the identification of clusters and outliers. For future research, one promising direction is to extend TopER to temporal graph learning tasks, enabling the capture of dynamic graph trajectories that reflect evolving user behaviors over time. Another avenue the integration of TopER embeddings into graph foundation models, where the homogenization of graph structures could enhance the learning of transferable representations across different domains.
10

---

Acknowledgments. This work was partially supported by Canadian NSERC Discovery Grant RGPIN-2020-05665. Data Science on Blockchains, National Science Foundation under grants 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. http:// /www://wwwww.// tacc. wtw.useexee.o.. .du..
References
[AAF19] Mehmet Emin Aktas, Esra Akbas, and Ahmed EI Fatmaoui. Persistence homology of networks: methods and applications. Appl. Netw. Sci., 4(1):61:1-28, 2019.
[AKGC22] Cuneyt Gurcan Akcora, Murat Kantarcioglu, Yulia R. Gel, and Baris Coskunuzer. Reduction algorithms for persistence diagrams of networks: Coraltda and prunit In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28 December 9, 2022, 2022.
[BCL21] Cristian Bodnar, Catalina Cangea, and Pictro Liò. Deep graph mapper: Sceing graphs through the neural lens. Frontiers Big Data, 4:680535, 2021.
[BGA19] Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Mincut pooling in graph neural networks. arXiv, https::/arxiv.org/abb///// .0481111111 2019.
[B] Karsten M. Borgwardt, M. Elisabetta Ghisu, Felipe Llinares-López, Leslie O' B'aay, and Bastian Rieck. Graph kernels: State-of-the-arr and future challenges. Found. Trends Mach. Learn., 13(5-6), 2020.
[BHG 21] Muhammet Balcilar, Pierre Héroux, Benoit Gaüzère, Pascal Vasseur. Sébastien Adam, and Paul Honeine. Breaking the limits of message passing graph neural networks. In Proceedings of the 38th International Conference on Machine Learning, ICMI. 2021, 18- 24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pages 599-608. PMLR, 2021.
[BK10] Michael M. Bronstein and Iasonas Kokkinos. Scale-invariant heat kernel signatures for non-rigid shape recognition. In The Twenty-Third IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2010, San Francisco, CA, USA, 13-18 June 2010, pages 1704-1711. IEEE Computer Society, 2010.
[BN01] Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14 [Neural Information Processing Systems: Natural and Synthetic, NIPS 2001, December 3-8, 2001, Vancouver, British Columbia, Canada], pages 585-591. MIT Press, 2001.
[BP07] Ulrik Brandes and Christian Pich. Centrality estimation in large networks. Int. J. Bifure. Chaos, 17(7):2303-2318, 2007.
BPPL2211 Dominique Beaini, Saro Passaro, Vincent Létourneau, William L. Hamilton, Gabriele Corso, and Pietro Li6. Directional graph networks. In Proceedings of the 38th Interna- tional Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 133 of Proceedings of Machine Learming Research, pages 748-758 PMLR, 2021.
[B7SL14] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. In 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14- 46, 2014, Conference Track Proceedings, 2014.
[CA24] Baris Coskunuzer and Cüneyt Gürean Akçora. Topological methods in machine learning: A tutorial for practitioners. arXiv, http://arriivorr/ab///40009 02901, 2024.
[Cai21] Chen Cai. Sanity check for persistence diagrams. In JCLR 2021 Workshop on Geometri- cal and Topological Representation Learning (GTRL). OpenReview.net, 2021.
11

---

[CCB 20] Gabriele Corso, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Velickovic. Principal neighbourhood aggregation for graph nets. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NewrlPS 2020, December 6-12, 2020, virtual, 2020.
[CFG24] Yuzhou Chen, José Frías, and Yulia R. Gel. Topogel: Topological graph contrastive learning. In TiihthEEighth AAAI Conference on Artificial Intelligence, AAAI 2024, Soirty-Sixth Conference on Innovative Applications of Artificial Intelligence, IAAI 2024, Fourteenth Symposium on Educational Advances in Artificial Intelligence, EAAI 2014,
February 20-27, 2024, Vancouver, Canada. pages 11453-11466 AAAI Press, 2024.
[CHM12] Nikolai Chernov, Qiang Huang, and Hongwei Ma. Does the best-fitting curve always exist? International Scholarly Research Notices, 2012:895178, 2012.
[C] Semih Cantürk, Renming Liu, Olivier Lapointe-Gagné, Vincent Létourncau, Guy Wolf, Dominique Beaini, and Ladislav Rampásek. Graph positional and structural encoder. In Forty-first International Conference on Machine Learning, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024.
[CLX+21] Tianle Cai, Shengjie Luo, Keyulu Xu, Di He, Tie-Yan Liu, and Liwei Wang. Graphnorm: A principled approach to accelerating graph neural network training. In Proceedings of the 38th International Conference on Machine Learning, ICMI. 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pages 1204-1215. PMLR, 2021.
[CPWC23] 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 on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023.
[CSA+2] (SSA+23) 23] Yuzhou Chen, Ignacio Sggovia-Domingueee Cuneyt Gurcan Akcora, Zhiwei Zhen, Mu- rat Kantarcioglu, Yulia R. Gel, and Baris Coskunuzer. EMP: effective multidimensional persistence for graph representation learning. In Learning on Graphs Conference. 27- 30 November 2023. Virtual Event, volume 231 1 of Proceedings of Machine Learning Research, page 24. PMLR, 2023.
[CZC18] Hongyun Cai, Vincent W. Zheng, and Kevin Chen-Chuan Chang. A comprehensive survey of graph embedding: Problems, techniques, and applications. IEEE Trans. Knowl Data Eng., 300)):6166111333 2018.
[C5W+25] Jialin Chen, Haolan Zuo, Haoyu Peter Wang, Siqi Miao, Pan Li, and Rex Ying. Towards A universal graph structural encoder. arXiv, http://axiv.orr/abss550 1000110044011911,..11 2025.
[DBV16] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Infor- mation Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 3837-3845, 2016.
[DCG+22] Andac Demir, Baris Coskunuzer, Yulia R. Gel, Ignacio DoooinueeD Yuzhou Chen, and Bulent Kiziltan. Todd: Topological compound fingerprinting in computer- aided drug discovery. In Advances in Neural information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022. NeurlPS 2022, New Orleans, LA, USA, November 28- December 9, 2022, 2022.
[DCS+22] Youxiang Duan, Ning Chen, Shigen Shen, Peiying Zhang, Youyang Qu, and Shui Yu. FDSA-STG: fully dynamic self-attention spatio-temporal graph networks for intelligent traffic flow prediction. IEEE Trans. Veh. Technol. (1)952250-,200, 2022.
[DG23] Pawel Dlotko and Davide Gurmar.. Euler characteristic curves and profiles: A stable shape invariant for big data problems. GigaScience, 12:giad094, 2023.
[DTRF19] Xiaowen Dong, Dorina Thanou, Michael G. Rabbat, and Pascal Frossard. Learning graphs from data: signal representation perspective. IEEE Signal Process. Mag., 36(3):44-63, 2019.
12

---

[EC22] Tim S. Evans and Bingsheng Chen. Linking the network centrality measures closeness and degree. Communications Physies, 5(1):172, July 2022.
[PPBM20] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification. In 8th International Conference on Learning Representations, JCLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020.
[FHSK23] Zhongxi Fang, Jianming Huang, Xun Su, and Hiroyuki Kasai. Wasserstein graph distance based on 11-approximated tree edit distance between weisfeiler-lehman subtrees. In "hirty-Seventi AAAI Conference on Artificial Intelligence, AAAI 2023, Thirty-Fifh Conference on Innovative Applications of Artificial Intelligence, IAAI 2023, Thirteenth Symposium on Educational Advances in Artificial Intelligence, EAAI 2023, Washington, DC, USA, February 7-14, 2023, pages 7539-7549. AAAI Press, 2023.
[GBC21] Marek Gagolewski. Maciej Bartoszuk, and Anna Cena. Are cluster validity measures (in) valid? Inf. Sci., 58 0881:620-6366 2021.
[GBGA20] Loann Giovannangeli, Romain Bourqui, Romain Giot, and David Auber. Toward automatic comparison of visualization techniques: Application to graph visualization. Vis. Informatics, 4(2):86-98, 2020.
[GF18] Palash Goyal and Emilio Ferrara. Graph embedding techniques, applications, and performance: survey. Knowl. Based Syst., 151:78 99, 2018.
[GHT+19] 19] Xiang Gao, Wei Hu, Jiaxiang Tang, Jiaying Liu, and Zongming Guo. Optimized akeleton-based action recognition via sparsified graph regression. In Proceedings of the 27th ACM International Conference on Multimedia, MM 2019, Nice, France, October 21-25, 2019, pages 601-610, ACM, 2019.
[GJ19] Hongyang Gao and Shuiwang Ji. Graph u-nets. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 2083-2092. PMLR, 2019.
[Gün22] Stephan Günnemann. Graph neural networks: Adversarial robustness. In Graph Neural Networks: Foundations, Frontiers, and Applications, pages 149-176. Springer Nature Singapore, Singapore, 2022.
[Hmm20] William L. Hamilton. Graph Representation Learning. Synthesis Lectures Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers, 2020.
[HI Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurlPS 2020, December 6-12, 2020, virtual, 2020.
[HGR+20] Christoph D. Hofer, Florian Graf, Bastian Rieck, Mare Niethammer, and Roland Kwitt. Graph filtration learning. In Proceedings of the 37th International Conference on Ma- chine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 4314 4323. PMLR, 2020.
[HIL+24] Yasuaki Hiraoka, Yusuke Imoto, Théo Lacombe, Killian Meehan, and Toshiaki Yachimura. Topological node2vec: Enhanced graph embedding via persistent homology. J. Mach. Learn. Res., 25:134:1 1-:1:11226, 2024.
(HJLH22) Xiaotian Han, Zhimeng Jiang, Ninghao Liu, and Xia Hu. G-mixup: Graph data augmentation for graph classification. In International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA, volume 162 of Proceedings of Machine Learning Research, pages 8230-8248. PMLR, 2022.
[HK02] Petter Holme and Beom Jun Kim. Growing scale-free networks with tunable clustering. Physical Review E, 65(2):026177, January 2002.
13

---

[HKN19] Christoph D. Hofer, Roland Kwitt, and Mare Niethammer. Learning representations of persistence barcodes. J. Mach. Learn. Res., 20:126:: 2019.
[HLAK24) Jaeseung Heo, Seungbeom Lee, Sungsoo Ahn, and Dongwoo Kim. EPIC: graph augmentation with edit path interpolation via learnable cost. In Proceedings of the Thirty-Thine International Joint Conference on Artificial Intelligence, IJCAI 2024, Jeju, South Korea, August 3-9, 2024, pages 41 4116-4126. ojcai.org, 2024.
[HMR21] Felix Hensel, Michael Moor, and Bastian Rieck. A survey of topological machine learning methods. Frontiers Artif. Intell., 4:681 108, 2021.
[ISG23] Johanna Immonen, Amauri H. Souza, and Vikas Garg. Going beyond persistent homol- ogy using persistent homology. In Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurlPS 2023, New Orleans, LA, USA, December 10 16, 2023, 2023.
[JWH 23] Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirami, and Jonathan Taylor. An Introduction to Statistical Learning: with Applications in Python. Springer Texts in Statistics. Springer International Publishing, Cham, 2023.
[JZ20] Shengmin Jin and Reza Zafarani. The spectral zoo of networks: Embedding and visualizing networks with spectral moments. In KDD '20: The 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, CA, USA, August 23-27, 2020, pages 1426-1434, ACM, 2020.
[KM12] Nils M. Kriege and Petra Mutzel. Subgraph matching kemels for attributed graphs. In Proceedings of the 29th International Conference on Machine Learning, ICML. 2012, Edinburgh, Scotland, UK, June -6lJuly July 1, 2012. ieml.cc / Omnipress, 2012.
[KW17] Thomas N. Kipf and Max Welling. Semi-superriscd classification with graph con- volutional networks. In Sth International Conference on Learming Representations, ICLR 2017, Toulon, France, April 26-26, 2017, Conference Track Proceedings. Open- Review.net, 2017.
[LBKT08] Jure Leskovec, Lars Backstrom, Ravi Kumar, and Andrew Tomkins. Microscopic evolution of social networks. In Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Las Vegas, Nevada, USA, August 24-27, 2008, pages 462 462-470. ACM, 2008
[LLK19] Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In Proceed- ings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 3734-3743. PMLR, 2019.
[LLY11] Yong Lin, Linyuan Lu, and Shing-Tung Mathematical Journal, (3(4))605-62) 2011.
Tung Yau. Ricci curvature of graphs. Tohoku
[3SC+23] David Loiseaux, Luis Scoccola, Mathieu Carrière, Magnus Bakke Botnan, and Steve Oudot. Stable vectorization of multiparameter persistent homology using signed bar- codes measures. In Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Or- leans, LA, USA, December 10 16, 2023, 2023.
[LWZ+22] 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. ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA. volume 162 of Proceedings of Machine Learning Research, pages 13052-13065. PMLR, 2022.
[MK20] Costas Mavromatis and George Karypis. Graph infoclust: Leveraging cluster- level node information for unsupervised graph representation learning. arXiv, http://arri..rg//////09....44 2020.
14

---

[MV09] Pierre Mahé and Jean- Phinlppi Vert. Graph kernels based on tree pattemss for molecules. Mach. Learn., 75(1):3-35, 2009.
[New03] Mark E. J. Newman. The structure and function of complex networks. SIAM Rev. 45(2):167-256, 2003.
[NV23] Giannis Nikolentzos and Michalis Vazirgiannis. Graph alignment kernels using we- isfeiler and leman hierarchies. In International Conference on Artificial Intelligence and Statistics, 25-27 April 2023, Palau de Congressos, Valencia, Spain, volume 206 of Proceedings of Machine Learning Research, pages 2034. PMLR, 2023.
[OPT+17] 17] Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod, and Heather A. Harring- ton. A roadmap for the computation of persistent homology. EPJ Data Sci., 6(1):17, 2017.
[ORB21] Leslie O' O'Bray, Bastian Rieck, and Karsten M. Borgwardt. Filtration curves for graph representation. KDD '21: The 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, Singapore, August 14-18, 2021, pages 1267-1275. ACM, 2021.
[PAMF23] Daniele Paliotta, Mathieu Alain, Bálint Máté, and François Fleuret. Graph neural networks go forward-forward In NeurlPS 2023 Workshop on Graph Learning Frontiers, 2023. Workshop Poster.
[PAS14] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: online learning of social representations. In The 20th ACM SIGKDD International Conference Knowledge Discovery and Data Mining, KDD) '14 New York, NY, USA August 24 27, 2014, pages 701-710. ACM, 2014.
[QTLL24) 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, KDD 2024, Barcelona, Spain, August 25-29, 2024, pages 2444 2454. ACM, 2024.
[RBB19] Bastian Rieck, Christian Bock, and Karsten M. Borgwardt. A persistent weisfeiler- lchman procedure for graph classification. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 5448 5448-5458. PMLR, 2019.
(SOG09) Jian Sun, Maks Ovsjanikov, and Leonidas J. Guibas. A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum, 28(5):1383 1392, 2009.
[ST20] Primož Skraba and Katharine Turner. Wasserstein stability for persistence diagrams. arXiv, http://arxiv.or//ab///000 1680bb//220000.6666.... 16824, 2020.
[TGL+19] Matteo Togninalli, M. Elisabetta Ghisu, Felipe Llinares-L-ópez, Bastian Rieck, and Karsten M. Borgwardt. Wasserstein weisfeiler-lehman graph kernels. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 6436-6446, 2019.
[TPPM23] Anton Tsitsulin, John Palowitch, Bryan Perozzi, and Emmanuel Müller. Graph clustering with graph neural networks. J. Mach. Learn. Res., 24:127:1 2023.
[WCZ16] Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. In Pro- ceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pages 1225-1234. ACM, 2016.
15

---

[3]Y+23] Haotao Wang, Ziyu Jiang, Yuning You, Yan Han, Gaowen Liu, Jayanth Srinivasa, Ramana Kompella, and Zhangyang Wang. Graph mixture of experts: Learning on large-scale graphs with explicit diversity modeling. In Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurlPS 2023, New Orleans, LA, USA, December 10 16, 2023, 2023.
[WKK*20] 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, 37:1-12 2020.
[WW22] Asiri Wijesinghe and Qing Wang. A new perspective on "how graph neural networks go beyond weisfeiler-lhhman" In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022.
[XHLJ19] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019.
[Xu21] Mengjia Xu. Understanding graph embedding methods and their applications. SIAM Rev., 63(4))825-853, 2021.
[YCL+21] 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? In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 28877-28888, 2021.
[YLY+18] 18] Jiaxuan You, Bowen Liu, Zhitao Ying, Vijay S. Pande, and Jure Leskovec. Graph convolutional policy network for goal-directed molecular graph generation. In Advances Neural Information Processing Systems 31: Annual Conference on Neural Informa- tion Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pages 6412-6422, 2018
[YSK22] Jaemin Yoo, Sooyeon Shim, and U Kang. Model-agnostic augmentation for accurate graph classification. In WWW '22 The ACM Web Conference 2022, Virtual Event, Lyon, France, April 25 29, 2022, pages 1281-1299 ACM, 2022.
[YV15] Pinar Yanardag and S. V. N. Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD International Conference Knowledge Discovery and Data Mining, Sydney, NSW, Australia, August 10-13, 2015, pages 1365-1374. ACM, 2015.
[YWH+22) 22] Yihang Yin, Qingzhong Wang. Siyu Huang, Haoyi Xiong, and Xiang Zhang. Autogel: Automated graph contrastive learning via learnable view generators. In Phirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 March 2022, pages 8892-8900. AAAI Press, 2022.
[YYM+18] 18] Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, William L. Hamilton, and Jure Leskovec. Hierarchical graph representation learning with differentiable pooling. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pages
2018.
[ZYCW20] Qi Zhao, Ze Ye, Chao Chen, and Yusu Wang. Persistence enhanced graph neural network. In The 23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy], volume 108 of Proceedings of Machine Learning Research, pages 1896-2906 PMLR, 2020.
[ZZM24] Zhiqiang Zhong, Kuangyu Zhou, and Davide Mottin. Benchmarking large language models for molecule prediction tasks. arXiv, http://aaxii.orr/hss/40...55.5.0 2024.
16

---

NeurIPS Paper Checklist
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The claims in the abstract and introduction accurately reflect the theoretical, methodological, and experimental contributions presented in the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: The limitations discussed in dedicated paragraph, noting areas such as dependence on the filtration function and computational scalability to very large graphs.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
Justification: Section 4 states the full set of assumptions for each theorem, and the complete proofs are provided in the Appendix.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: The main paper and appendix include detailed descriptions of experimental setups, and an anonymous code repository is provided to ensure full reproducibility.
5. Open access to data and code
Question: Does the paper provide open access the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: The code provided via an anonymous link in the supplementary materials, along with detailed instructions for reproducing all key experiments.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Answer: [Yes]
Justification: Training/test splits, model configurations, optimizer settings, and hyperparam- eter values are all specified either in the main text or in the appendix.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: All reported results include standard deviations across multiple runs, with details on how error bars were computed described in the experimental section.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
17

---

Answer: [Yes]
Justification: The appendix provides details about the computational environment used for all experiments, including GPU type and training time.
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: : /ttr//eurrppp.e. httpp///i///tiiiccciieeeooeellee
Answer: [Yes]
Justification: The research complies fully with the NeurIPS Code Ethics and does not involve sensitive data, human subjects, or high-risk applications.
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
The Broader Impact section discusses potential benefits of interpretability and efficiency scientific applications, while noting ethical considerations related to graph data usage.
Justification:
11. Safeguards
Question: Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Answer: [NA]
Justification: The models and datasets used in the paper do not present high risk for misuse and thus do not require additional safeguards.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: All datasets and external codebases are cited properly and used under their respective open-source licenses.
13. New assets
Question: Are assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
The paper introduces a new method and accompanying code, which are thoroughly documented and made available via anonymous link.
Justification:
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots. if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: The paper does not involve crowdsourcing or any research involving human subjects.
15. Institutional review board (IRB) approvals or equivalent for rescarch with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
18

---

Answer: [NA]
Justification: The research does not involve human participants and does not require IRB approval.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality the research, declaration not required.
Answer: [NA]
Justification: No large language models were used in the methodology experiments: any language assistance was limited to standard editing support.
19

---

Appendix
A Further Experimental Details
A.1 OGBG-MOLHIV Results
For the OGBG-MOLHIV dataset, we fur- ther evaluated the improvements of TopER with the addition of new filtration func- tions. Table 7 provides the performance of each TopER- where i represents num- ber of filtration functions used in the model, i.e. TopER-i uses {(a), bi..... 0g bx)} as graph embedding where (a,, bx) the pivot and growth for function fi. We used XG- Boost rank the importance of filtration functions first, and the functions are added iteratively with this ranking. We fixed max-
Table 7: TopER- -.
Results for OGBG-MOLHIV of each
Method
Added Function
Valid. AUC
Test AUC
TopER-1
degreo-centrality
72.76+033
74.44=00:0
TopER-2
atomic weight
71.8940.2
74.25
TopER-3
O. Ricci
70.11
76.79=0024
TopER-4
F. Ricci
71.76:10.18
18.15
TopER-5
degree
71.79+0035
79.26=0014
TopER-6
popularity
72.27+t0.29
79.88=0.24
TopER-7
closeness
71.3010.18
80.21 =.115
imum tree depth = 3, learning rates 0.035, subsample ratios = 0.95, the number of estimators 1000, and the regularization parameter lambda 45, where the objective function rank:pairwise, with log loss as the evaluation metric. The seed is set to be 16.
A.2 Time Experiments for TopER vs. PH
To compare the time efficiency and performance of TopER and persistent homology (PH),
We conducted experiments using the same filtration func- tion, the sublevel degree filtration. For PH, we applied Betti vectorization. Our results, summarized below, show that TopER is significantly faster than PH. Al- though both methods use the same filtration function, a key distinction lies in their embeddings: TopER gener- ates 2D embeddings, whereas PH produces a vector with dimensionality equal the number of thresholds in the filtration. Despite the considerable difference in dimen- sionality, TopER's performance with 2D embeddings remains comparable to that of PH.
Beat Thas tarondd
130
500
1300
5000
70000
50000
150200
Node Count
Figure 5: Scalability. TopER run time synthetic power law graphs [ПКO2]] with node degree filtration. The mean node degree is 30, and 100 filtration steps used.
Figure 5 shows that TopER scales efficiently with graph size, maintaining low runtime even with 100 filtration steps and high node degree. It processes graphs with up to 100,000 nodes in just over minute, demonstrating its suitability for large-scale applications.
A.3 Clustering Performances
In Table 9, showcase our clustering performance across eight benchmark graph classification datasets using three widely adopted clustering metrics: Silhouette, Kalinski-Harabass and Davies- Bouldin. These metrics serve as evaluative measures for assessing the efficacy of clustering algorithms
Table Comparison of TopER-1 (only one filtration function) and PH in terms of time and accuracy across different datasets.
TopER-1
PH
Dataset
Time
Accuracy
Time
Accuracy
# Thresholds
BZR
1.14
82.73 # 2.12
5.99
83.70 + 3.51
4
IMDB-B
3.27
73.10 4.18
319.95
71.00 4 4.07
65 501
REDDIT-B
107.65
79.55 + 2.20
9173.37 s
84.50 + 2.51
20

---

Table 9: The clustering performances of Spectral Embeddings and TopER with different metries. Best performances are given in blue.
Silhouette Scores (1)
Method
BZR
COX2
MUTAG
PROT. IMDB-B
IMDB-M
REDD-B
REDD-5K
Spec Zoo
0.050
0.049
0.344
0,050
0.097
-0.024
0,108
-0.121
degree
-0.108
0,414
0.258
0.048
0.030
-0.032
0.049
-0,169
popularity
0.249
-0.015
0.134
-0.000
0.008
-0.159
0.196
-0.173
closeness
0.019
0.036
0.036
0.086
nan
nan
0.087
-0.185
degree
0.084
0.030
0.017
0.065
0.056
-0.075
0,034
-0.067
scores (*)
Method
BZR
COX2
MUTAG
PROT.
IMDB-B IMDB-M REDD-B REDD-5K
Spec 7oo
3.51
6.13
120.73
38.77
85.24
30.98
269.94
119.81
degree
0.42
1.06
11.29
130.07
60.52
3.92
97.85
1209.95
popularity
13.85
26.00
36.13
77.22
12.89
11.77
446.12
619.37
closeness
42.58
40.04
73.51
10.17
0.30
188.10
689.27
F.iicii
4.92
0.48
11.82
151.64
11.68
1.03
92.14
454.34
Davies-Bouldin scores (L)
Method
BZR
COX2
MUTAG PROT. IMDB-B IMDB-M
REDD-B
REDD-5K
Spee Zoo
7.25
6.07
0.95
4.55
2.78
10.73
2.20
25.74
degree
9.84
2.29
1.95
4.92
46.46
2.32
3.27
popularity
4.16
37.87
2.11
25.25
6.87
3.46
closeness
1.93
26.44
1.41
2.25
4.99
37.51
1.95
3.09
F.Ricci
4.19
7.20
1.27
1.54
2.19
10.35
1.83
5.41
in partitioning datasets into meaningful clusters. They gauge the degree of similarity or dissimilarity within and between clusters, offering insights into the quality of clustering outcomes. For precise definitions of Silhouette, Selinski-Harabasee and Davies-Bouldin metrics, as well as additional details on clustering measures, refer to [GBC21].
A.4 Number of Thresholds
In our experiments, we utilized large number of thresholds capture finer-grained information, as the model is computationally efficient and the additional cost of increasing the number of thresholds minimal. Furthermore, in Table 10, we evaluated the model's performance with fewer thresholds and observed that it remains robust and highly effective even in such scenarios.
Table 10: The accuracy results of TopER with different numbers of thresholds.
# Thresholds
PROTEINS
REDDIT-5K REDDIT-B
10
72.78
90.55
55.99=1.97
20
74.31
91.20
55.91
50
74.76
92.05 1.96
55.39-2.10
100
73.85
92.85 1.18
55.51=261
200
75.47
93. 03.15
56.51=204
500
74.58
92.70
56.51
A.5 Combining Filtration Functions
To assess the impact of embedding dimensions, we conducted new experiments evaluating the performance of the TopER model by progressively adding each filtration function step by step. This analysis provides insights into how the inclusion of additional filtration functions influences the model's performance. Table 11, the TopER-n model represents the TopER utilizing n-filtration functions (2n features).
21

---

Table 11: Performance improvements achieved by integrating filtration functions into the TopER model. Here, TopER-n denotes the TopER model with n filtration functions.
Dataset
TopER-1
TopER-2
TopER-3
TopER-4
BZR
82,48:: 1.98
84,7042.86
85.66
35.66155.00
86.68-13.81
COX2
1
49.26:14.6
79.04
80.30::3.91
MUTAG
86. 6.33
88.3343.88
86.75:+4.78
88,30::4.63
PROTEINS
74.03:12.71
74.6742.73
75.21
75.65
IMDB-B
73.00++4.40
74.20++4.26
74.50=33.50
74.70+33.95
IMDB-M
48.73:44.8
49.8042294
49.73
49.87
REDDIT-B
81.95
90.45
91.054:2.2
91.50+22.1
REDDIT-5K
50,21+141
54.1142243
56,1942240
56,3342274
A.6 TopER filtrations runtimes and substitute
Filtration Timing Results. In Table 12, we report the computation times (in seconds) for the TOPER filtration functions across various datasets. We also include the timings for the Heat Kernel Signature (HKS) [SOG09], which can as efficient substitute for the Ollivier-Rii Ricci curvature in certain scenarios due to its faster computation while preserving relevant structural information about the graphs. The table below summarizes the observed computation times for each filtration type across multiple benchmark datasets.
Table 12: Computation times (in seconds) of different filtration functions across datasets.
Filiration
BZR
COX2
MUTAG
PROTEINS
IMDE-B
IMDE-M
REDDIT-B
REDDIT:KK
Degrec Centrality
0.86
1.67
0.40
13.82
13.32
13,03
115.09
447.72
Popularity
0.73
1.36
0.58
5.81
13.21
15.60
111.11
414.04
Closeness
2.29
4.40
0.77
15.97
5.85
6.63
399.2
1274.00
Forman Ricci
0.80
1.56
0.58
4.63
8,29
7.38
258.32
693.18
Ollivier Ricci
130.04
121.42
48.98
313.21
277.66
428.32
1291.93
6640.95
Degree
1.14
1.16
0.61
2.62
3.27
107.65
363.32
Weight
2.25
2.80
0.90
12.89
IIKS
2.07
1.94
0.56
16.90
15.54
15.72
470.01
1007.50
We evaluate the performance of our TopER model against state-of-the-aar (SOTA) methods on benchmark graph classification datasets, including BZR, COX2, MUTAG, PROTEINS, IMDB-B, IMDB-M, REDDIT-B, and REDDIT-5K. Table 13 reports classification accuracy (mean : standard deviation) across multiple Ablation studies show the impact of Ricci curvature and Heat Kernel Signatures (HKS) on model performance. Notably, TopER without Ricci but with HKS recovers most of the performance lost when Ricci is removed, suggesting that HKS can serve as a viable replacement for Ollivier-Rieci curvature capturing structural information.
TopER generally achieves competitive results compared to SOTA.
Table 13: Graph classification accuracy (mean + std) for different models across benchmark datasets.
Highest scores per dataset are bold blue, second second-highest are underlined blue.
Model
BZR
COX2
PROTTINS
REDDITBB
REDDIT.SK
SOTA
85.532160
92.332.58
77.3020.39
75.0820.31
528126331
91.032022
36.7520 18
TopER
91.13+4.14
82.014455 59
90.99-6.64
74.58=3922
23.20+33.3
50.00s4.02
92.70+2.35 38
56.51+2.22
TopER R.
87.0044300 30
77.9648.
87.78:7.84
74.04=3.96 86
23.50:3.53
50.00:5.44
91.90+2.63
56.37+1
TopER wio O. R.& w/ HKS
89.634..55
81.58+3.54
92.0864.23
75.20+3.59
25.00+3.49
50,6745.58
92.75+2.47
$7.33+2.02
A.7 Hyperparameters
Our proposed MLP algorithm is constructed with single hidden layer. The output layer's activation function is set log softmax, and the loss function we used is Negative Log Likelihood Loss. The learning rate is chosen between 0.01 and 0.001. Subsequently, we investigate the impact of the number neurons the hidden layer, considering values from the set [16, 64, 128). The optimizer is set to be Adam, and the number of epochs is 500. To prevent large weights and overfitting, apply 1.2 regularization coefficients of le-3, le-4. The activation function for the hidden layer varies between ReLU, GeLU, and ELU. Lastly, consider the cases of adding not batch normalization
22

---

Algorithm TopER: Topological Evolution Rate
Input: Graph G. Filtration function f R, Threshold set I {e: |
Output: TopER vector (G,
Initialize lists X' = 0. =
for i = to rt do
G: f Induced subgraph of G where V, S ( ei))
24 [41
Vi < [&
Append 2; to X'
Append y/ y
end for
bo pairs (ri,3:) vt) from X' and using least squares Extract coefficients a and b
Fit line L(x)
Return (a, b) the TopER vector II,
layer to the output of the hidden layer and setting dropout values be 0.0 or 0.5. In Table 14, provide the details for each dataset. The last column shows the number of TopER features used for each dataset after the feature selection step.
Table 14: Employed hyperparameters for each dataset.
Dataset
Neurons
Dropout
Batch Norm. Decay
Learning rate
Activation
TopER Dim.
BZR
64
0.5
True
le-4
0.001
gelu
26
COX2
128
0
True
le-4
0.01
relu
26
MUTAG
16
0.5
False
le-3
0.01
gelu
20
PROTEINS
64
0.5
True
1e-3
IMDB-B
128
0
False
le-3
0.001
relu
20
IMDB-M
16
0
False
le-3
0.01
elu
20
REDDIT-B
64
0.5
False
le-3
0.01
relu
24
REDDIT-5K
128
False
le-3
0.01
B More on TopER
B.1 Refining the point set
While we have described the main steps of TopER in Section 3, due to the repetitions the points in A = ((r,, J)) C I22, there are some choices be made before defining the set A (i.e., X and Y) to get the best fitting function : X' + y. The main reason is that the set { y can contain repetitions of z-values (z; 2++1), repetitions of y-values (y/ 31+1) repetitions of both ((x,, 91) = (I++1, B1+1)) depending on the filtration function, the threshold set I, and the graph G.
For the filtrations induced by node filraatio functions, the number of edges can not change unless the number of nodes changes, i.e., x; F;+1 = y: V-+1- Hence, with this elimination, we still allow keeping y-values the same while rrvalues are increasing. This means there can be horizontal jumps A., In this paper, to eliminate all horizontal jumps for filtrations with node functions, we eliminate all repetitions of y-vuuues from A., In particular, we remove all the points with the same g-value and add a point with a mean of vavlues. other words, y, = 7/-1 =.... Wi-k = 7. we define f mean ( 24, F.+1...... F. Then, replace (k+1) points { 7), (),41,9),... ..... (Z++*. 7) with one point (3. 9) in A,, This process eliminates all repetitions and horizontal jumps in A, and we define our best- best-fitting line this refined set.
B.2 TopER with Alternative Quantities
While we use the most general quantities for graph-the count of vertices and edges our algorithm, depending on the problem, there might be other induced quantities (æ,, vi) for given subgraph G. which can give better vectors. keep the line-fitting approach meaningful in our model, long as the sequences {x,} and } are monotone like node-edge counts above, for given
23

---

dataset in a domain (e.g., biochemistry, finance), one can use other domain related quantities induced by substructure G as a (I,,, y4) pair obtain TopER vector.
BZR graph 34
REDDIT-BINARY graph 419
PROTEINS graph 245
32-
nember aogos.
1000
400-
600
230
280
433
500
number of hodes
numberoffodes
(a)
(b)
(c)
Figure 6: Linear Fit. TopER summarizes the growth behavior in the graph induced by filtration with linear fit.
B.3 Linear or Higher Order Fitting
In experiments, observe that linear fitting captures the growth information for node-edge pair ((c;, y well (See Figure 6), and quadratic fit and linear fit stay very close to each other. However, if one decides sequences } and {u.}. trying higher order fits (c.g., ar² + br + c) can be more meaningful. In Table 15, present the average of the coefficients of quadratic terms when we use quadratic fit for the datasets, i.e., if we fit = a + br + cx polynomial, we observe that the quadratic term cr² mostly negligible, and tends be linear fit.
other quantities described above and loses the monotonicity of the
Table 15: Average of x coefficient across datasets for quadratic fitting.
Dataset
BZR
COX2
MUTAG
REDDIT-5k
Average of x? Coefficient
4.71 x 10 6.61 10
1.16 x 10-2 1.78 10
B.4 Interpreting TopER
Our approach involves accurately modeling the evolution of a graph throughout the filtration process. One can easily identify clusters for each class and outliers in the other datasets given in Figure ((a) and make inferences about the different clusters and outliers. Furthermore, when the pivot a is positive or negative, it can be interpreted as graph density behavior in the filtration sequence (See Figure 7).
C Proofs of Stability Theorems
In this part, we prove the stability results for TopER.
Lemma 3.6. [ST20] Let be a compact metrie space, and f.g: X - be filtration functions. Then, for any 2 1, we have W,
g)) < |[ g!!n
The next lemma is the stability of Betti curves by [DG23] (Proposition 1].
Lemma 3.7. [DG23] Let 3& (X) the 4th Betti function obtained from the persistence module (X).
(x*) 8a()i)l < 2WI PDz())
Now, we are ready prove our stability result.
Theorem 3,4 Let be a compact metric space, and f.g X' - IR be two filtration functions. Then, for some C > 0,
(X)) TE, (2')l1 CC (X.,

---

Positive pivot
Zero pivot
Negative pivot
edges
edges
edges
nodes
nodes
nodes
Figure 7: Pivot Behavior. A graph can exhibit three distinct pivot behaviors. Positive pivot graphs display a cluster of vertices that are closely interconnected and appear early in the filtration process. On the other hand, negative pivot graphs feature loosely connected nodes where the edges enter the filtration at later stage. Graphs with zero pivot usually quasi-complete graphs.
Proof. We will utilize the stability theorems from topological data analysis given above. First, we employ the stability of Betti curves by Lemma 3.7.
(2) (x()i)1 < (PD
(1)
Hence to obtain (x) (ar, bε), we fit least squares line aa + bya to the set of N points in R i.e. 2 (8) of (4) Similarly, we obtain TEg (X) (ag, bg) by fitting least squares line to Zy (e), 89 (c.)) By Equation (I), we have
DH (2 2g) < (PP)
where Dz(Z,, ( 2g) represent Hausdorff distance between the point clouds Z, and Zg in R.2 Now, by the stability of least squares fit with respect Hausdorff distance ([CHM12) [Theorem 3.1J). we have
(T) TEg C DH, 2g)
Hence, when we combine Equations (2) and (3), we have
[XT (T < C- W;(PD((4),
The proof follows.
By combining the above result with Lemma 3.6, obtain the following corollary.
Corollary 3.5 Let X' be a compact metric space, and f.g R two filtration functions. Then, for some C > 0,
(X) TEg (X) < C gllı
Proof. By Lemma 3.6, have
(D&,,, S - g|1
(4)
By Theorem 3.4, we have
< - (D
(5)
By combining Equations (4) and (5), conclude
(E) (X) Č |f-gll: gll:
The proof follows.
25

---

D Synthetic Experiments
This section describes experiments performed on the Erdos-Renyi synthetic graph. We conducted two experiments by applying TopER on this graph and Principal Component Analysis (PCA). The goal is to compare the performance, runtime, and interpretability of the two models. From the plots shown in Figures 8, 9, and 10, we see that TopER is interpretable compared to when PCA is used.
Tracit your salk
TepFi Jows, A407
YedR jayer
TeySR (ap0
Groe
Phono
(b)
(c)
(d)
Teetf shgna JUX
TepdR jep.telyy.eee aged
TeER pepuarty
(c)
(f)
(g)
(h)
Figure 8: TopER plots showing pivot vs growth for each function function-degree centrality, closeness, degree, and popularity when threshold=50.
PTA
PM Peeeebodee PTAA
Congonest
Car Curarett
Carpotent
(a)
(b)
(c)
(d)
Witenbeddigg PCA, Jeppe
soding depes
IPAA, deppa
andeodings TCA Cc.ppeeae
(f)
(h)
Figure Persistent Homology PCA plots emboddings (Betti 0 and Betti showing component component 2 for each function closeness and degree when threshold=50 and the respective filtrations sublevel and superlevel.
E Broader Impact
This work advances the field of graph representation learning by introducing a topological approach that is both interpretable and scalable. By leveraging the structural insights of persistent homology without incurring prohibitive computational costs, TopER enables more efficient and insightful analysis of complex graph-structured data. This has the potential to benefit range of scientific and
26

---

(a)
(b)
(c)
(d)
bectiings (PAc pre wild
PMonthokingg 2PO4 5O
conteeting PPS B1 preietty vewo
(f)
g
(h)
Figure 10:
Persistent Homology plots embeddings (Betti and Betti 1) showing component vs
component 2 for each function degree centrality and popularity when threshold=50 and the respective filtrations -sublevel and superlevel.
industrial domains where graph data is prevalent, including bioinformatics, social network analysis, and infrastructure monitoring, In particular, the ability to generate low-dimensional and interpretable embeddings could assist researchers in visual analytics, pattern discovery, and model debugging. At the same time, acknowledge that the use of graph representations- or biological datasets -myy carry ethical concerns around data privacy, representational bias, or unintended consequences of automated decision-makee making. While TopER itself is an unsupervised and domain- domain-agnostic method, its application must be governed by domain-speeiffe ethical considerations. emphasize interpretability and transparency in design, and
in social networks
To support responsible release our code and visualizations to promote reproducibility and community oversight.
27


Reviews:
[{"review_id": "mUSuUCfYU3", "reviewer": "Reviewer_jFiC", "summary": "This paper proposed TopER, a topology-aware graph representation learning framework, that overcomes the efficiency problem of persistence homology by modeling the evolution of topological filtration, rather than actual persistence features. TopER generates low-dimensional, interpretable embeddings for whole graph by fitting a linear model on evolution of the graph. The method is evaluated on graph classification, clustering, and visualization tasks across several datasets, and is shown to be competitive, or close to state-of-the-art baselines.", "questions": "- How does the popularity filtration differ from PageRank? Could you include a discussion?\n- Are the report runtimes in lines 244-253 end-to-end, including precomputing filtration functions, or just the embedding and classification? How do these runtime compares to GPU-accelerated GNNs and pooling methods?\n- Does TopER scale better than Spectral Zoo?\n- In practice, does TopER's interpretability justify the performance loss in some datasets? What are the idea use-cases for TopER?", "limitations": "The authors have included limitations of their work.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 2, "originality": 4, "strengths_and_weaknesses": "### Strengths \nThe paper is well written and easy to follow. The paper identifies and addresses the lack of interpretability in high-dimensional data and graph embeddings, as well as the shortcomings of existing topological methods. Experiments are thorough and the paper provides ablation studies to showcase its performance. The 2D embeddings computed by TopER offer interpretability, allowing for exploratory data analysis, which is rarely a focus in other graph representation learning papers.\n\n### Weaknesses\n- From lines 27-31, the paper criticizes computational cost of pooling node embeddings in graphs and motivated direct graph embeddings, but does not mention DiffPool [1], SAGPool [2], and related recent pooling-based methods that address this problem. These works should at least be acknowledged as prior work in producing graph-level representations. \n- The phrase \"reducing computational complexity\" is over-used throughout the paper even though the reported runtimes seem too high for a relatively small graph (e.g. almost 1.5 days for Reddit-5k). I doubt on the claimed efficiency of TopER, especially when compared to graph neural network and related methods that can be GPU accelerated. \n- While DiffPool is included in the baselines, the results are not reported in Table 3. Furthermore, the authors need to compare more recent pooling methods like EdgePool [2], SAGPool.\n- On several benchmarks, TopER underperforms, but the paper does not adequately discuss why or what settings it excels in. The average deviation metric (“Avg$\\downarrow$” in Table 3) may obscure poor performance on some datasets and is somewhat misleading.\n- On lines 368–369, it is claimed that TopER homogenizes graph representations in a way that existing models cannot. However, models like GPSE [3] and GFSE [4] are designed to do exactly this by being pre-trained on a large dataset from different sources, this claim might be an overstatement.\n\n[1] Ying, Zhitao, et al. \"Hierarchical graph representation learning with differentiable pooling.\" Advances in neural information processing systems 31 (2018).\n\n[2] Lee, Junhyun, Inyeop Lee, and Jaewoo Kang. \"Self-attention graph pooling.\" International conference on machine learning. pmlr, 2019.\n\n[3] Cantürk, Semih, et al. \"Graph positional and structural encoder.\" arXiv preprint arXiv:2307.07107 (2023).\n\n[4] Chen, Jialin, et al. \"Towards A Universal Graph Structural Encoder.\" arXiv preprint arXiv:2504.10917 (2025).", "was_revised": true, "final_justification": "I found the core algorithm in this paper to be original and well-motivated from the beginning. However, I had concerns about its performance on some benchmarks and the high runtime, particularly due to the use of Ollivier-Ricci curvature. After an extended back-and-forth during the discussion phase, the authors introduced a variant of their method using the heat kernel signature (HKS) as a lighter filtration function. This update led to meaningful runtime reductions while preserving or even improving performance on several datasets. I’m now satisfied with the method's practicality and robustness, and have updated my score accordingly.", "rebuttal": "*We thank the reviewer for their insightful feedback. Below, we address each concern with new experiments and detailed clarifications that underscore our approach’s robustness and originality. We hope these revisions resolve any remaining questions and respectfully request your reevaluation in light of the additional evidence.*\n  \n---\n&nbsp;\n  \n## W1&W3. Comparison with Pooling Methods\n>...but does not mention DiffPool [1], SAGPool [2], and related recent pooling-based methods that address this problem\n\n**Response:** Thank you for this helpful suggestion. We have compared TopER with the mentioned methods. As shown in the table below, our model consistently outperforms or matches the performance of recent pooling-based methods such as DiffPool and SAGPool across a range of benchmarks. These results further support the effectiveness of our direct-embedding approach for graph-level representation.\n\nWe agree that DiffPool [1], SAGPool [2], and related methods should be acknowledged. In the revision, we will add a discussion of these approaches in the Related Work section and include their results in the main comparison table.\n\n| Method     | BZR   | COX2  | MUTAG | PROTEINS | IMDB-B | IMDB-M | REDDIT-B |\n|------------|-------|-------|--------|----------|--------|--------|-----------|\n| Top-K      | 79.40 | 80.30 | 67.61  | 69.60    | 73.17  | 48.80  | 79.40     |\n| MinCutPool | 82.64 | 80.07 | 79.17  | 76.52    | 70.77  | 49.00  | 87.20     |\n| DiffPool   | 83.93 | 79.66 | 79.22  | 73.63    | 68.60  | 45.70  | 79.00     |\n| SAGPool    | 82.95 | 79.45 | 76.78  | 71.86    | **74.87**  | 49.33  | 84.70     |\n| **TopER**   | **90.13** | **82.01** | **90.99**  | **74.58**    | 73.20  | **50.00**  | **92.70**     |\n\n\n---\n\n&nbsp;\n\n## W2. Computational Time\n>The phrase \"reducing computational complexity\" is over-used throughout the paper even though the reported runtimes seem too high..\n\n**Response:** We thank the reviewer for highlighting this important concern regarding computational time. We agree that the phrase *reducing computational complexity* should be used more precisely. While TopER itself is computationally lightweight, with linear complexity in the number of filtration steps and graph size ($O(n(|V| + |E|))$), the primary runtime bottleneck arises from the computation of certain filtration functions, most notably the Ollivier-Ricci curvature, which is known to be computationally demanding and is external to our core method.\n\nTo clarify this point, we reran our pipeline on the Reddit-5K and MolHIV datasets, excluding Ollivier-Ricci curvature. This reduced the total runtime **from 1.5 days to 6 hours and 20 minutes** on Reddit-5K, and **from 16.5 hours to 3 hours and 1 minute** on MolHIV (40K graphs), using the same hardware specified in the paper.\n\nMoreover, as shown in Figure 5 of the Appendix, TopER remains highly scalable: when using simple degree-based filtrations, it computes embeddings for a *100K-node graph in under 50 seconds*. In our revision, we will clarify this distinction throughout the paper, emphasizing that TopER’s embedding pipeline is efficient, while the choice of filtration function can significantly impact preprocessing time.\n\n\n---\n\n&nbsp;\n\n## W4. Classification Performance\n\n> On several benchmarks, TopER underperforms, but the paper does not adequately discuss why or what settings it excels in...\n\n**Response:** Thank you for the insightful comment. While TopER may not lead on every benchmark, it achieves the best performance across diverse graph domains despite using very few parameters. Our main goal is not maximizing classification accuracy, **but producing low-dimensional, interpretable embeddings** that are useful for **visualization** and can serve as **effective features for downstream GNNs**.\n\nIn general, TopER shows strong performance on datasets where structural graph features (e.g., degree, closeness, Ricci curvature) are informative and discriminative across classes. This includes molecular and social datasets where meaningful topological signals can be captured through filtration. On datasets with very subtle class differences or noisy graph structures (e.g., IMDB-M), where node attributes or learned representations may be more advantageous, TopER’s performance is understandably more limited. This is the main analysis on TopER performance, and we will include it in our discussion. Thank you for your suggestion.\n\nRegarding the average deviation metric (\"Avg\" in Table 3), we agree that it may mask underperformance on individual datasets. However, we include it to provide a compact, high-level comparison across many baselines and tasks. To avoid confusion, we will revise the manuscript to clarify this intent. Also, we will highlight cases where TopER does not perform best, and discuss dataset-specific characteristics (e.g., size, attribute availability) that may impact performance.\n\n\n---\n\n&nbsp;\n\n## W5. Missing Comparisons\n\n> ... However, models like GPSE [3] and GFSE [4] are designed to do exactly this...\n\n**Response:** We appreciate the reviewer’s attention to recent developments and thank them for pointing out GPSE [3] and GFSE [4]. We agree that both works contribute toward universal graph representations by leveraging large-scale pretraining. However, TopER pursues a distinct design objective: our goal is not universal pretraining but the construction of low-dimensional, interpretable, and visualizable embeddings, directly grounded in topological intuition.\n\nWhile models like GPSE and GFSE focus on transferability and scalability via large encoders and massive training corpora, TopER emphasizes simplicity and transparency. For example, TopER produces two-dimensional embeddings for each graph, which makes it uniquely suited for visual comparison and anomaly detection without relying on dimensionality reduction.\n\nOn graph visualization specifically, we would like to clarify that [3] (GPSE) visualizes graphs only after applying PCA to high-dimensional representations. This differs fundamentally from TopER’s direct 2D outputs, where the axes themselves are meaningful and interpretable (pivot and growth). As for [4] (GFSE), it was released on arXiv on April 15, during the NeurIPS submission period, and does not present any graph visualizations. Furthermore, we note that [4] is a very interesting work, but it has not yet been finalized yet, having different versions for NeurIPS and ICLR. We will be citing this work in our related articles.\n\nTo address the reviewer’s concern, we will revise the manuscript to soften the claim on lines 368–369. Rather than stating that existing models cannot homogenize representations, we will clarify that existing models do not provide low-dimensional, interpretable embeddings suitable for direct graph-level visualization across datasets, which remains a unique strength of TopER.\n\nFinally, while the GFSE code is not publicly available, we were able to run comparisons with GPSE. Across standard benchmarks, TopER consistently outperforms GPSE. We will include these results in the main comparison table in our revision.\n\n\n|           | BZR              | COX2             | MUTAG            | PROTEINS         | IMDB‑B           | IMDB‑M           |\n| --------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- | ---------------- |\n| **GPSE**  | 80.49 ± 4.18     | 78.37 ± 2.62     | 87.19 ± 8.66     | 72.15 ± 3.66     | 69.30 ± 3.61     | 47.40 ± 5.40     |\n| **TopER** | **90.13 ± 4.14** | **82.01 ± 4.59** | **90.99 ± 6.64** | **74.58 ± 3.92** | **73.20 ± 3.43** | **50.00 ± 4.02** |"}, {"review_id": "TtH4q62tim", "reviewer": "Reviewer_xFgk", "summary": "The paper introduces TopER, an efficient graph embedding method based on topological data analysis, which simplifies the computationally expensive Persistent Homology approach by tracking the evolution rate of graph substructures during filtration. The method produces low-dimensional, interpretable 2D embeddings (pivot and growth) and is validated on a diverse of graph classification benchmarks.", "questions": "- How can users systematically choose the most effective filtration function for a new dataset? Are there heuristics or automated strategies?\n\n- Could TopER be adapted for temporal graphs, where filtration sequences evolve over time?\n\n- How does TopER perform on noisy graphs? For example, graphs with noisy features or partly lacked edges.", "limitations": "See Weakness and Questions", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength**\n\n- The proposed TopER addresses the scalability limitations of PH, offering a practical solution for large-scale graph analysis.\n\n- The 2D embeddings provide clear visualizations, aiding in cluster analysis and outlier detection.\n\n**Weakness**\n\n- The choice of filtration function can visibly impact performance, and optimal selection may require domain expertise.\n\n- The linear regression model may oversimplify complex topological relationships, though the authors argue higher-order terms are negligible.\n\n- The 2D embeddings, while interpretable, may lose nuanced structural details compared to higher-dimensional methods.", "was_revised": true, "final_justification": "This paper presents an efficient 2D graph visualization tool that could be valuable for interpreting complex real-world graphs. During the rebuttal phase, the authors addressed most of my concerns by highlighting the competitive performance of this 2D embedding method. The experiments on temporal graphs demonstrate its potential for extended applications, though additional tests on noisy graphs would further strengthen the work in the robustness of the method. Therefore, I recommend acceptance, while I have to say my confidence is limited due to my lack of deep expertise in this specific subfield. \n\nI have reviewed Reviewer jFiC's comments and the authors' response. Some critiques are constructive, particularly: (1) A more thorough comparison with related works (_e.g._, DiffPool, SAGPool), and (2) A detailed breakdown of training time, possibly through an ablation study on each component's computational cost. On the other hand, I think the performance critiques may be overly stringent—given the method's lightweight nature, it is reasonable not to surpass more complex approaches on all benchmarks.", "rebuttal": "*We thank the reviewer for their insightful feedback. Below, we respond to each point with new experiments and detailed clarifications that reinforce the robustness and novelty of our approach. We hope these updates address all remaining concerns, and we would be glad to clarify any further questions.*\n  \n---\n&nbsp;\n\n## W1 & Q1. Choice of Filtration\n>The choice of filtration function can visibly impact performance, and optimal selection may require domain expertise.\n\n**Response:** Thank you for this valuable observation. We fully agree that the choice of filtration function can significantly impact performance and that domain knowledge can guide this selection. However, our goal is to design a method that performs well across diverse graph domains without relying on dataset-specific tuning.\n\nTo this end, we demonstrate that general-purpose functions, such as degree, centrality, and closeness, already achieve strong individual performance (Table 5). We further provide a broadly applicable set of functions commonly used in persistent homology, and **introduce an automated selection procedure**. Specifically, we apply t-tests to identify statistically significant functions ($p < 0.05$), followed by Lasso regularization to eliminate redundant features and retain only the most informative ones (Section 4.1).\n\nAs shown in Table 5, while the effectiveness of individual filtrations varies by dataset, TopER’s aggregated representation, guided by this selection process, consistently delivers strong results without requiring domain-specific choices. We believe this design strikes a balance between flexibility and practical usability, and we will clarify this point more explicitly in the revision.\n\n---\n&nbsp;\n\n## W2. Linear Fit\n> The linear regression model may oversimplify complex topological relationships, though the authors argue higher-order terms are negligible.\n\n**Response:** Thank you for this comment. We agree that higher-order relationships may exist in some graphs, but our experiments show that for the quantities TopER models, specifically, the growth of edges with respect to nodes during filtrations, linear fitting provides a strong and stable approximation. As shown in Appendix B.3 and Figure 6, the linear and quadratic fits are nearly indistinguishable, and **the average quadratic coefficients (Table 13) are negligible across datasets (Between 0.00002 and 0.01)**. This indicates that the underlying growth is inherently linear in the types of filtrations we use.\n\nMoreover, preserving this linearity helps ensure interpretability, robustness, and computational simplicity. That said, TopER is flexible: if future applications require fitting more complex, non-monotonic quantities, higher-order terms can be introduced without altering the framework. We will clarify this flexibility and empirical rationale in the revision.\n\n---\n&nbsp;\n\n## W3. 2D Embeddings losing info\n>The 2D embeddings, while interpretable, may lose nuanced structural details compared to higher-dimensional methods.\n\n**Response:** Thank you for raising this point. While we agree that a 2D embedding may not capture all the nuanced structural details that high-dimensional methods can, TopER is designed to address a different and equally important need: producing **interpretable, low-dimensional representations** that enable **graph-level visualization and insight.**\n\nThere is a significant gap in graph machine learning when it comes to effective graph-level visualization tools. TopER directly addresses this by providing compact, meaningful 2D embeddings that reveal structural patterns, class clusters, and outliers across datasets. Despite its simplicity, TopER also achieves **competitive classification performance**, demonstrating that interpretability does not have to come at the cost of utility.\n\nIn the revision, we will more explicitly highlight TopER’s value as both a visualization tool and a lightweight embedding method that complements more expressive models.\n\n---\n&nbsp;\n\n## Q2. Temporal Graphs?\n>Could TopER be adapted for temporal graphs, where filtration sequences evolve over time?\n\nThank you for this insightful suggestion. TopER naturally extends to temporal graphs by treating each filtration step as a 2D embedding: as the graph evolves, **these embeddings trace a trajectory in the plane that captures temporal changes**. Building on this idea, we have developed a temporal TopER variant (*TopERTemporal*) for discrete‐time graph property prediction.\n\nOur preliminary experiments on eight benchmark datasets (including blockchain and social networks) show that TopERTemporal significantly outperforms both static and temporal baselines:\n \n| Method      | Average ROC-AUC |\n|-------------|------------------|\n| GIN         | 0.4754           |\n| TDA-GIN     | 0.5314           |\n| EvolveGCN   | 0.7772           |\n| GRUGCN      | 0.7826           |\n| HTGN        | 0.7658           |\n| GraphPulse  | 0.8575           |\n| **TopERTemporal**     | **0.9416**           |\n\nWe are preparing a separate submission detailing this extension. In the revised conclusion of the current manuscript, we will include a brief discussion of these results and their implications. This work underlines TopER’s versatility and its potential for a wide range of temporal‐graph learning tasks.\n\n---\n\n&nbsp;\n\n### Q3. Noisy Graphs?\n> How does TopER perform on noisy graphs? For example, graphs with noisy features or partly lacked edges.\n\nWe thank the reviewer for raising this important question. TopER is fundamentally grounded in topological descriptors, and topology is well-known to be robust against small perturbations, such as the addition or deletion of a few edges or nodes. Intuitively, this is because the shape of the filtration sequence (i.e., how subgraphs evolve across thresholds) tends to remain stable under minor changes.\n\nUnder an edge insertion as perturbation noise scenario, the change in regression parameters is bounded (change in slope for an added edge is $$|b' - b| \\le \\frac{|x_j - \\bar{x}|(1 - \\frac{1}{n})}{\\sum_{i=1}^n (x_i - \\bar{x})^2}$$ and inversely proportional to the variance of node counts across filtration steps, which is typically nontrivial in practice. Therefore, TopER is provably stable under small structural perturbations, and we already support this claim in the paper via Theorem 3.4 and Corollary 3.5, which bound changes in TopER vectors in terms of the Wasserstein distance between filtrations or the $L^1$ norm of filtration functions."}, {"review_id": "2MkKDEC890", "reviewer": "Reviewer_2YLF", "summary": "This work focuses on the problem of graph-level representation learning. The paper proposes TopER which leverages topological data analysis—specifically a simplified version of persistent homology—by quantifying the evolution rate of graph substructures. This results in low-dimensional embeddings that are not only more interpretable but also visually intuitive. Despite its simplicity, TopER achieves competitive performance compared to state-of-the-art methods in tasks such as graph classification and clustering.", "questions": "1.\tHow many hyper-parameters do this method have? Are there too many?\n\n2.\tSee the “Weaknesses” above.", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1.\tThis paper proposes a novel graph-level representation learning method. \n\n3.\tIt tests on several widely-used datasets, and the experimental study is some kind of interesting.\n\n\nWeaknesses:\n\n1.\tThe related work section would benefit from a more focused discussion—ideally narrowing down to two or three key research directions.\n2.\tThe experimental performance is weak. For example, in the main experiment (Table 3), the baselines outperform the proposed method on five out of the seven datasets.\n3.\tThe theoretical analysis in Appendix C needs to be improved. First, the theorem names are ambiguous. It seems like Theorem 3.4 and Corollary 3.5 (in Section 3) are renamed as “Theorem C.3”and “Corollary C.4” in Appendix C. This may confuse the readers. Second, in Appendix C, as Lemma C.1 and C.2 are cited from other works ([ST20] and [DG23]), it is suggested to mention their origins before introducing them in the main text. This will help readers better follow your proofs.", "was_revised": false, "final_justification": "", "rebuttal": "*We thank the reviewer for their insightful feedback. Below, we address each concern with new experiments and detailed clarifications that underscore our approach’s robustness and originality. We hope these revisions resolve any remaining questions and respectfully request your reevaluation in light of the additional evidence.*\n\n---\n&nbsp;\n\n## W1. Related Work\n>The related work section would benefit from a more focused discussion—ideally narrowing down to two or three key research directions.\n\n**Response:** Thank you for the suggestion. In the revision, we will condense Related Work into three focused areas, topology-inspired representations, graph-level embedding methods, and visualization techniques, to better highlight TopER’s unique contributions.\n\n---\n&nbsp;\n\n## W2. Weak Classification Performance\nThe experimental performance is weak. For example, in the main experiment (Table 3), the baselines outperform the proposed method on five out of the seven datasets.\n\n\n**Response:** Thank you for the comment. While TopER does not outperform all baselines on every individual dataset, its average deviation from the best performances in Table 3 is **the lowest among all methods**, demonstrating its robust and consistent performance across diverse domains, from molecular graphs to social networks. Unlike task-specific, high-capacity models, TopER provides a **unified, lightweight, and interpretable 2D representation** that remains competitive while requiring minimal parameters. Our goal is not to replace  high-capacity (and high computational cost) models, but to offer an **efficient and scalable alternative** that supports **graph-level visualization, anomaly detection**, and broader analysis. We will clarify this intent and TopER’s practical value more explicitly in the revision.\n\n\n---\n&nbsp;\n\n\n## W3. Theoretical Analysis\n> The theoretical analysis in Appendix C needs to be improved...\n\n**Response:** Thank you for this helpful feedback. We agree that the labeling and citation structure in Appendix C can be clearer. In our revision, we will synchronize theorem and corollary numbering between the main text and the appendix, relocate Lemmas C.1 and C.2 into both the main manuscript and appendix for easier reference, and add explicit citations ([ST20] and [DG23]) immediately before each lemma to indicate that these results are adopted verbatim from their original sources. Since Lemmas C.1 and C.2 are well‑known community results included without modification to support our proofs, clarifying their provenance will help readers trace each statement back to its origin and follow the logical flow more easily.\n\n\n---\n&nbsp;\n\n## Q1. Hyperparameters\n> How many hyperparameters do this method have? Are there too many?\n\n\nResponse: We appreciate the reviewer’s question about hyperparameter complexity. In fact, TopER itself is entirely non‑parametric: it constructs each graph embedding via a deterministic filtration process, yielding exactly two scalar descriptors per function (pivot and growth rate). Thus, if you apply k filtration functions, you obtain a 2k‑dimensional embedding, there are no trainable weights or deep‑learning parameters in this core procedure.\n\nThere are only three sets of genuine “hyperparameters” in TopER, namely choosing the number of **filtration thresholds**, selecting which **filtration functions** to include, and configuring the **lightweight MLP** that sits on top of the TopER vectors for classification. These choices are transparent and interpretable: **Table 10** explores threshold counts, **Table 5 and 11** compares different filtration functions, and **Table 12** reports our grid‑search over MLP architecture and regularization settings (detailed further in Appendix A.4 and A.6). Because the MLP is small and standard, just a few weight matrices and biases, its hyperparameter space is modest and well documented. We will clarify this separation between the fixed, non‑learned TopER descriptors and the downstream classifier’s parameters in the revised manuscript.\n\n\n---"}, {"review_id": "2atDShMiN0", "reviewer": "Reviewer_tsuE", "summary": "In their paper \"TopER: topological embeddings in graph representation learning\", the authors suggest a filtration-based approach to summarize a graph with two features. The algorithm constructs a node filtration and fits a linear relationship between the number of nodes and the number of edges in each graph in the filtration. The intercept and the slope are the resulting summary statistics. The authors show that these features, pooled across several filtrations, outperform many existing graph representation methods (including GNN-based) for graph classification in benchmark datasets.\n\nNote: I happened to review this paper already twice before. I am not sure why OpenReview keeps assigning this paper to me, this is not even exactly my field. The paper has been improving every time, and I am recommending acceptance this time.", "questions": "MINOR COMMENTS\n\n* Section 4.1, Paragraph \"Filtration functions\". I did not understand the procedure to select the filtrations. You did a t-test and aplied the Lasso. What does it mean? T-test between what and what? Lasso was applied to predict what from what? Please provide a clearer rescription.\n\n* Do the results actually improve when using t-testing and lasso regression for feature selection? How would the method perform when simply taking all filtrations without any selection? Can this be added to Table 5 as an additional column?\n\n* Table 5 suggests that 6 or 7 filtrations were used, resulting in max 12 or 14 features. But Table 12 gives TopER dimensionality as going up to 26. Why? Where can I find the list of all filtrations used for each graph?\n\n* Section 4.1, Paragraph \"Filtration functions\". \"TopER demonstrates strong performance\" using only node degree function. Actually looking at Table 5, I see ~10 percentage point difference between TopER and the degree filtration alone. This is a huge difference! Please do not write \"strong performance\" without explicitly adding \"(albeit ~10 percentage points worse than the full TopER)\".", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: the suggested method is conceptually simple, the evaluation experiments seem comprehensive, and the suggested algorithm outperforms many competitors.\n\nWeaknesses: I think the \"topological embedding\" terminology is mostly marketing. I would rather say the method constructs filtration-based graph features. But it's a purely terminological point.", "was_revised": true, "final_justification": "I gave high score originally (5) and am maintaining this score. Most of my questions were clarification and suggestions for simple ablations, and authors' responses make sense. I also looked at the negative reviews and think that the authors' responses there are convincing.", "rebuttal": "*We thank the reviewer for their insightful feedback. Below, we respond to each point with new experiments and detailed clarifications that reinforce the robustness and novelty of our approach. We hope these updates address all remaining concerns, and we would be glad to clarify any further questions.* \n\n---\n&nbsp;\n\n## W1. Terminology\n> I think the \"topological embedding\" terminology is mostly marketing. I would rather say the method constructs filtration-based graph features. But it's a purely terminological point.\n\n**Response:** Thank you for the comment. Our use of the term *topological embedding* is not intended as marketing, but to properly acknowledge the conceptual roots of our method in **persistent homology**, where **filtration argument** serves as the central mechanism for revealing topological structure. While “filtration-based graph features” is indeed a more precise term, we use “topological” to give proper credit to the TDA framework that motivates our design. That said, as noted in Remark 3.3, the slope computed by TopER reflects the evolution of the **Euler characteristic** across the filtration, which relates to Betti numbers via $\\chi(G_i) = |V_i| - |E_i| = \\beta_0(G_i) - \\beta_1(G_i).$ This connection underpins our use of the term Topological Evolution Rate, even though we do not compute full persistence diagrams. We will clarify this motivation more explicitly in the revision to avoid potential confusion.\n\n\n---\n&nbsp;\n\n## Q1. How to use Lasso\n\n>  the procedure to select the filtrations...\n\n**Response:** We start with a **28-dimensional feature encoding** generated from our filtration functions. To identify the most relevant dimensions, we apply two selection techniques: a **univariate t-test** and **Lasso regression** to our training data.\n\n* We perform a **t-test for each feature individually**, comparing the distributions of that feature across the two target classes. This helps us assess whether the means differ significantly, indicating class-discriminative power.\n\n* Separately, we apply **Lasso regression** (LassoCV from scikit-learn), which selects features by learning a sparse linear model that predicts the target variable from the full feature set. The L1 penalty in Lasso naturally eliminates irrelevant features by assigning them zero coefficients.\n\nWe then **retain only the features selected by both methods**,  those with a statistically significant t-test (p < 0.05) and non-zero Lasso coefficients. This dual-filtering strategy helps ensure the selected filtrations are both **statistically relevant** and **predictively useful**.\n\n\n---\n&nbsp;\n\n\n## Q2. Improvements from feature selection?\n> Do the results actually improve when using t-testing and lasso regression for feature selection? \n\n**Response:** Thank you for this suggestion. To evaluate the impact of our dual‐filtering strategy, we ran TopER using all 28 filtration dimensions (“TopER‑all”) and compared it to the filtered version (“TopER”). Across all eight datasets, feature selection consistently improves performance:\n\nDataset \t| TopER-all     \t| TopER\n------------|-------------------|-----------------------\nBZR     \t| 89.15 ± 4.65  \t| **90.13 ± 4.14**\nCOX2    \t| 80.07 ± 5.18  \t| **82.01 ± 4.59**\nMUTAG   \t| 85.61 ± 7.88  \t| **90.99 ± 6.64**\nPROTEINS\t| 73.95 ± 4.09  \t| **74.58 ± 3.92**\nIMDB-B  \t| 71.70 ± 3.69  \t| **73.20 ± 3.43**\nIMDB-M  \t| 48.93 ± 4.71  \t| **50.00 ± 4.02**\nREDDIT-B\t| 92.30 ± 2.10  \t| **92.70 ± 2.38**\nREDDIT-5K   | 56.39 ± 1.80  \t| **56.51 ± 2.22**\n\nAs the table shows, applying t‑testing and Lasso yields small but consistent gains on every benchmark. We will include a “TopER‑all” column in Table 5 of the revised manuscript to make these comparisons explicit. Thanks again for this suggestion.\n\n\n---\n&nbsp;\n\n## Q3. Number of Features\n> Table 5 suggests that 6 or 7 filtrations were used, resulting in max 12 or 14 features. But Table 12 gives TopER dimensionality as going up to 26. Why? Where can I find the list of all filtrations used for each graph?\n\n**Response:** Thank you for the question. In graph representation learning, sublevel and superlevel filtrations often capture complementary structural information and can lead to notably different performance. While one could use extended persistence to unify them for persistent homology [1], we chose a simpler and more suitable approach for our method: for each filtration function, we compute both sublevel and superlevel filtrations, each producing a 2D feature (slope and intercept), totaling 4 features per function.\n\nPrior to feature selection, we begin with 6 commonly used filtration functions, each applied in both sub/superlevel directions, giving 6 × 2 × 2 = 24 features per graph. We further include the number of nodes |V| and number of edges |E| as global features, bringing the total to 26. For molecular datasets, we add an additional filtration based on atomic number, increasing the total to 28 features. As such, the final feature count may vary depending on the dataset. Table 5 in the paper provides a summary of the filtration functions used per dataset. We will explicitly highlight this dataset-dependent variation in feature counts in the revised manuscript.\n\n[1] Edelsbrunner, H. and Morozov, D., 2017. Persistent homology. In Handbook of Discrete and Computational Geometry (pp. 637-661). Chapman and Hall/CRC.\n\n\n---\n&nbsp;\n\n## Q4. Overclaim of strong performance\n> Section 4.1, Paragraph \"Filtration functions\". \"TopER demonstrates strong performance\" using only node degree function...\n\n**Response:** Thank you for the helpful suggestion. You're right that the current phrasing overstates the performance of individual filtrations. Our enthusiasm is due to the fact that a simple few-feature model based on degree could achieve robust performance. For example, TopER with degree alone beats influential and recent models such as RepHINE 2024 in the Proteins and IMDB-B datasets and MP-HSM 2024 and EPIC 2024 in the IMDB-M dataset. However, we agree with your review, and in the revision, we will clarify this by writing:\n\n > While individual filtration functions, such as node degree, already yield solid performance, combining multiple functions leads to significantly improved accuracy, resulting in highly competitive performance with state-of-the-art models.\n\nThis more accurately reflects the results in Table 5 and underscores the value of TopER’s multi-filtration design."}]

Author rebuttal:

Meta-review: The paper proposes a low-dimensional, interpretable topological embedding for graphs that estimates the topological evolution rate of graph substructures as a simpler, scalable alternative to persistent homology. It supports visualization and yields competitive performance on graph clustering and classification across a variety of datasets.

The authors engaged constructively and provided meaningful rebuttal updates: they replaced Ollivier-Ricci curvature with the Heat Kernel Signature, yielding up to ~100x speedup; expanded comparisons to strong baselines to clarify accuracy vs. efficiency trade-offs; added deeper analyses linking performance to graph structure; and clarified distinctions from related approaches (such as, universal structural encoders). Including these updates into the camera-ready and polishing the theoretical section (e.g., clearly stating the assumptions) will further strengthen the paper. Overall, the combination of practical interpretability, strong aggregate accuracy (despite not always surpassing every baseline on individual datasets) and improved scalability makes this work of interest to the NeurIPS community.
Decision: Accept (poster)