Topology-aware Graph Diffusion Model with Persistent Homology
Authors: Joonhyuk Park
Donghyun Lee
Yujee Song
Guorong Wu
Won Hwa Kim
Venue: NeurIPS 2025 poster ()
Keywords: Graph Generation
Diffusion
Topology
Brain Network
Primary area: deep_learning
TL;DR: We propose a diffusion-based topology-aware graph generation method that aims to closely resemble the structural characteristics of the original graph by leveraging persistent homology from topological data analysis (TDA).

Abstract: Generating realistic graphs faces challenges in estimating accurate distribution of graphs in an embedding space while preserving structural characteristics. However, existing graph generation methods primarily focus on approximating the joint distribution of nodes and edges, often overlooking topological properties such as connected components and loops, hindering accurate representation of global structures. To address this issue, we propose a Topology-Aware diffusion-based Graph Generation (TAGG), which aims to sample synthetic graphs that closely resemble the structural characteristics of the original graph based on persistent homology. Specifically, we suggest two core components: 1) Persistence Diagram Matching (PDM) loss which ensures high topological fidelity of generated graphs, and 2) topology-aware attention module (TAM) which induces the denoising network to capture the homological characteristics of the original graphs. Extensive experiments on conventional graph benchmarks demonstrate the effectiveness of our approach demonstrating high generation performance across various metrics, while achieving closer alignment with the distribution of topological features observed in the original graphs. Furthermore, application to real brain network data showcases its potential for complex and real graph applications.

Topology-aware Graph Diffusion Model with Persistent Homology
Joonhyuk Park Donghyun Lee"r, Yujee Song", Guorong Wu Won Hwa Kim :POSTECH 2Samsung Eletronics *UNC-Chapel Hill
{pjh1023, xonhwa)@postech. ac.kr
Abstract
Generating realistic graphs faces challenges in estimating accurate distribution graphs in an embedding space while preserving structural characteristies. However, existing graph generation methods primarily focus on approximating the joint distribution of nodes and edges, often overlooking topological properties such as connected components and loops, hindering accurate representation of global structures. To address this issue. we propose a Topology-Aware diffusion- Graph Generation (TAGG), which aims to sample synthetic graphs that closely resemble the structural characteristics of the original graph based on persistent homology. Specifically, we suggest two core components: 1) Persistence Diagram Matching (PDM) loss which ensures high topological fidelity of generated graphs, and 2) Topology-awe aware Attention Module (TAM) which induces the denoising network to capture the homological characteristics of the original graphs. Extensive experiments conventional graph benchmarks demonstrate the effectiveness of our approach indicating high generation performance across various metrics, while achieving closer alignment with the distribution of topological features observed in the original graphs. Furthermore, application to real brain network data showcases its potential for complex and real graph applications.
Introduction
The major goal of graph generation is to achieve high resemblance between generated graphs and their reference counterparts. To achieve this goal, various graph generation approaches have been studied based on conventional generative models, e.g., recurrent neural networks [47], variational autoencoders (38) and diffusion models 23 43 and each exhibited promising results. Despite the achievement in context of the quantitative measures, e.g., similarity distributions of graph characteristics such as degree and clustering coefficients, there remains limitation on generating graphs coherent to the graph structure via the lens of graph topology.
Brain network is perhaps suitable example to demonstrate the challenges above. A brain network characterizes intricate wiring system the brain, which represented graph with anatomical regions of interest (ROIs) defining its nodes and the connectivity between different ROIs serving as edges 7 16] It is often large and dense, and its topological properties are well-knows eritical biomarkers [35 39 Moreover, brain networks are expensive; acquiring diffusion magnetic resonance images (dMRI) and processing them via tractography 42 to obtain structural brain connectome costly in both cost and labor. In this regime, generating realistic graphs (e.g., brain networks) that preserve their inherent connectivity as well as global structures highly demanding.
In recent years, various graph generative models have been heavily studied, but they often fall short in capturing essential topological features crucial for modeling interconnected brain regions with high
"J. Park and D. Lee contributed equally. Work done while D. Lee" and Y. Song" were at POSTECH.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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fidelity. The methods in [23 and domain, originally defined for images high computational cost, as the forward and reverse diffusion process is performed on infinitesimal continuous time point. Moreover, the uniformly added Gaussian noise results in noisy and complete graph, which causes the loss of structural information, e.g., sparsity of graph. Later, [43] proposed a discrete diffusion method, applying additive noise to each node and edge independently for graphs, nevertheless, existing methods overlook the topologically invariant characteristics, e.g., geometric shape and connectivity, limiting the generation.
32 proposed score-based diffusion methods in a continuous time However. the continuous diffusion methods suffer from
To overcome such issues, we propose novel Topology-Awar Graph Generation (TAGG), in which the sampled graphs resemble not only in the distributions of the original graphs in the embedding space but also in the homological features of the original graphs. Conventionally, topological data analysis (TDA) from algebraic topology has been studied in various graph analyses 6 21
to investigate topological features, and we bridge the gap between TDA and graph diffusion model to generate topologically realistic graphs. We define Persistence Diagram Matching (PDM) loss with persistence homology, which regularizes homological features of the reference graphs to be incorporated in graph generation process via I-Wasserstein distance. Furthermore, we introduce a Topology-awan Attention Module (TAM) which utilizes persistence landscape 5 of given graph to foster the denoising network with global structural information.
Contributions. To this end, our main contributions are summarized as follows: 1) We propose a novel topelogy-aware graph generation method that yields homologically similar graphs with high structural fidelity. 2) We propose PDM loss, utilizing persistent homology to encode the graph topology. 3) We propose Topology-aware Attention Module (TAM) that leverages persistence landscape to enhance the denoising network in capturing graph topology.
Our model demonstrates superior performance on real and synthetic graph generation, with intuitive visualizations for topological comparisons. Especially with the application on brain network gen- eration from Alzheimer's ' Disease Neuroimaging Initiative (ADNI), our method demonstrates its adaptability to diverse real-world graph generation tasks.
2 Related work
Graph Generation. Graph generation has been developed in two major branches; autoregressive and one-shot. Auto-regressive methods 22 25
recursively capture the intricate graph dependencies, and sequentially generate the graph structure conditioned on the current incomplete graph. In spite of their impressive performance, rugo-segrese approaches exhibit considerable computational demands due to the increasing number of generation steps along with the graph size. Also, they face a challenge stemming from the absence of an inherent node generation order. Conversely, one-shot methods 12 [7 28 48) generate the whole graph, i.e., every node and edge, at once. By doing so, they reduce computational requirements while facing performance degradation as the dataset scale grows. Recently, diffusion- methods [4] 23 29 30 43) showed promising capability in graph generation, by defining the forward and reverse diffusion processes and training a neural network that mimics the reverse process to reconstruct the graphs.
Persistent Homology. Persistent homology from computational topology studies the topological features of given objects, such as the number of holes [15]. It provides way to capture and quantify the shapes and global structures by computing homological features of objects. By treating graph as a topological object, the concept of persistent homology can be utilized to analyze the global structure of graphs, which leads improvements for graph classification [8 20 49) and link prediction (45.. Furthermore, persistent homology has been successfully applied in biology 6 10 33), signal processing [44), and point cloud (31), demonstrating its versatility and robustness.
3 Preliminaries
We briefly review persistent homology, which extracts homological properties from objects. We refer readers to g and [15]. should further questions arise regarding topological data analysis.
Simplicial Complex. Let V be a non-empty set. A simplicial complex K a collection of non-empty finite subsets of V which satisfies the following two properties; (2) if E K and r S a, then 7 E K. An element of K is called a simplex and the dimension of a simplex is determined by the length of its elements. For example, an element 7 E K with |r| k +
(1) for any E V. (v) E K, and

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is a k-simplex whose dimension is k. The dimension of a simplicial complex K is defined by the highest-dimension of its simplices.
Graph as a Simplicial Complex. Consider an undirected graph G = (V, F.), where V is a set of N nodes and F. C V V is set of edges. Then, graph G can be interpreted I-dimensional simplicial complex Kg whose O-simplices are the nodes and I-simplices are the edges, i.c.
(I) The homological properties of graph G can be characterized by its connected components and loops, each corresponding to the 0- and 1-dimensional features, respectively. These properties are fundamental in capturing the underlying topological strueture of the graph. The quantity of such properties in a given dimension k is referred to as the Betri number Bk. which serves as a key descriptor of the topological characteristics of the graph.
G = KG = {{0} : V U. E.
Filtration. Filtration of graph G is sequence of nested subgraphs of G, ie., 0 = G S G C
((2) -... S G(N-1) S G(N) G. Specifically, the filtration of G can also be defined using O-simplex (i.e., vertex) filter function f V + [0, 00) on node degree deg(-), as in [19]
Vob E G, f((u3) deg( (v)) max (deg((v')) (er)EG
(2)
Suppose that the computed filter values a, are given in ascending order dg dj 02 < aN, where a; E {f([x) : {u} E G}. To define the filtration of a graph G, we adopt non-negative scale parameter e, which is incrementally increased from O. Upon reaching we construct G from GA. = 0 by adding the node d1. When e subsequently reaches at d2. we extend G G by adding the node 1/ and the edge connecting 02 and the nodes in previous subgraph, i.e., G if the edge exists. By repeating this process, we systemically define the sublevel set filtration induced by f for 0 S S N as:
ai
G = for E G: max f(u) S a=)) = f (() DEo
(3)
0= G Gl 1 G c... c. CI..N-
Persistent Homology. The homological features can be extended via tracking the filtration of G. Filtration leads the notion persistent homology by monitoring the formmttin homological features in each G along the filtration, thereby allowing us to obtain their homological relevance in a given dimension, i.e., how long each homological feature persists. Specifically, if homological feature, e.g., a connected component, first appears at G and disappears at Gha in O-dimension, its birth and death are defined as i and j. respectively.
Representation of Persistent Homology. Suppose that a homological feature is born at G and dies at Gh., i.e., that persists from to j. can be denoted as a tuple of birth and death pair, i.e., (i,j), known as persistence barcode. By considering each and as coordinates and plotting the barcodes (i,j) on the 2 plane, we can obtain persistence diagram De of the graph G:
DG = {(b, d) (b, d) is a barcode of G} C T.2.
Note that persistence diagram De can be obtained separately with respect to the dimension of the barcodes, i.c., the dimension of the homological feature barcode encodes. For every points (b, d) E Do, i.c., |Dal n, we define a piece-wise linear function f(b.d) - o0), as:
if
$ (b. d) x bb
fo.a)
x
b if
(5)
-
x E 2
Persistence landscape [5] of persistence diagram De can be established as a sequence of functions A IR + [0, oo) for I E N, where A1(zz) denotes the 1th largest value of the functions d.),.;)(2) for i < n. We obtain points by dividing the domain of the function (z where >ı(z) 0 into (s + 1) equal sub-intervals. With chosen L and N, we can obtain a homological feature vector
MG [g I-SSS for given graph G' as follows:
Pac (M((1)) ..... (Z.)) M(xs)] E S
Pc = jua H ,o....,, . ER-S
where 1 < L, and 1 < S. The persistence diagram DG and the homological feature vector HG encode the entire information about persistent homology of a given graph [5] [5]
3

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Go (Vo, Fo)
Go (Vo En)
Persistence Diagram Do
Backward Pavan
Bo
PH
Persistence Landscupe
Cco Po.)
Lpon Go. Po,)
Denolsing Network 6,
Pro
Mon
Pro
PEo
Dopology-awant Attention Module
PH
V Eg
(TAM)
Pco PK.)
Persistence Diagram D
Piio
Figure 1: Training of the denoising network du (Gt, This network takes noisy graph G. and the embedding PG, obtained from persistence landscape of the original graph Go input, and outputs the probability vector nodes and edges, Pvv and PE. to predict Go. During training, we utilize these predictions in two ways: 1) cross-entropy loss LcE over all nodes and edges, and 2) Persistence Diagram Matching loss LPDM which computes the discrepancy between persistence diagrams of Go and pCo-
4 TAGG: Topology-aware graph generation
Unlike conventional diffusion-based graph generation methods which utilize Gaussian noise in continuous space [3 32), perform the diffusion process in discrete space preserve the sparsity of the graph for every diffusion time steps. We follow the settings in [43. which successfully expanded the method
3 for generating graphs with categorical node and edge attributes by treating each node and edge as categorical random variable.
Let Gu (Vo: En) be the original graph with N nodes, where Vo RNNxFv and Eg E NXNXXF are the node and edge matrices with Fy and FE attributes, respectively. At time step t, we denote the attribute of the i-th node v² as one-hot vector 2/ E R Fv and that of the edge e between of and Dd as one-hot vector edd Fis Thus, each element of V and Eg is represented one-hot vector.
4.1 Forward process
Considering node and edge attributes one-hot vectors, we follow the settings of 3 and 45 to define the forward and reverse process of diffusion acting on the node and edge attributes. We denote the forward diffusion process of each time step to impose noise as transition matrices Q,, where t = 1,2,... . T, and each element of the matrices, [Qth represents the probability that state
changes to state as the time step changes from (- to i.e. [QY], v EV q(v" = p--1=11"
and [Q]]] == = q(e' _ cl-l
11)
From time step f 1, a noised graph G, can be obtained by sampling the type of nodes and edges from the categorical distribution after transition, which is derived as:
q (Ge G.-1) = .V--Q(".EE-11F))
Specifically, the transition matrix Q is determined by the dimension of node categories, i.e., Q (1 1B)B ( my. where Fv is an identity matrix and J Fy E Fv is matrix of ones, and Be a real value in range [0, 1]. The transition matrix for edge Qf is determined in the same manner.
Assuming Markovian property of the process, we can derive the transition matrix from time to time by simply multiplying each transition matrices: Q = QY QY ...QQ." and Of O Q Of Then, similar to Eq. (7). the noised graph G, can also be obtained from time 0 by sampling from q (Ge Go) (Vo E.Q.)) Details on the forward process provided in Appendix A.1
4.2 Topology-aware denoising network
In this section, we introduce a topology-aware graph denoising network de parametrized by 0, which estimates the probability vector of the nodes and edges of the original graph Go, i.e., PG. = (fv. ). In addition to the noisy graph Gr, we leverage the homological feature lGG, of Go obtained via Eq. (6) as an input of d9 to retain the topological structure of the original graph during estimation.

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For this, we introduce novel Topology-awarr Attention Module (TAM), which serves as a fun- damental component within the denoising net- work Φ0. Given noisy graph G, and JAGa- TAM is iteratively applied to refine the node and edge attributes within dg. TAM operates as illustrated in Fig. 2 at the m-th iteration, node and edge at- tributes tith and ETTh are inputted to TAM, which produces refined output V.=+1 and E as (V/=+1, E = TA", E,", (4G.). Ulti- mately, the denoising network dg estimates the categorical probability vector of the original graph, i.e., Pc. (ro), by iteratively applying TAM M times as:
V/"
Lin,
12-
Welphee Sum
Ni.
Ling
Litty
X7
Z,,""
No
Ling
Lity
No.
times
Figure 2: Topologyyawar Attention Module (TAM). 6 and @ denote the outer and element product. The
blue rectangle is the topology-awee aware attention score A,""
(pv, PF,) @o(G,, (G,)) = TAM M (V. F.e MGo);
(8)
where 0, M- 1, and the initial conditions are given by V2 = V, and E. E,.
In TAM, the node attribute Vimh is passed through three distinet linear transformations to produce the query @ key KF"", and value 1/2h matrices. Similarly, the edge embedding Z and the homological embedding HG. are obtained by applying a linear transformation to ET" and JAG, respectively. After obtaining these representations, the topology-ave aware attention score Al is computed as:
Qtt -
ATh softmax
ZgT
HG.
(9)
dx
where de is the dimension of the query and key matrices.
Using the topology-aware attention score AT" we obtain the updated node and edge attributes, i.e. v.m++1 and Em++1 with additional linear transformations Ny and NE as:
m NVv V.)
ET E = NN (AAT))
(10)
Notice that TAM further adds the homological embedding Ha. global attention bias term to induce the attention to capture the graph's topological features. Conventional graph transformers multiply the edge embeddings to the node-wise attention computation, i.c., multiplication of the query and key matrices, thereby leveraging edge information [14]. However, as edge information represent local structures, utilizing the edge embedding alone may be insufficient for incorporating global structural information into the attention score. To address this limitation, TAM integrates the homological embedding Ho. bias term to inject global structural information into the attention.
Since jcG, encodes the homological information by tracking every subgraph in the filtration of Go (from Eq. 3 it contains rich structural information that is challenging for the network to capture solely from the final subgraph of the filtration, i.e., the original graph. Therefore, incorporating HG, the homological embedding obtained from J4G, as a global attention bias term enables TAM to estimate topology-aware node and edge embeddings, facilitating the final estimation of the topology- aware probability vector PG.. Consequently, the integration of (G., bolsters the denoising network de in generating more realistic graphs.
Moreover, it is noteworthy that the homological feature vector μGG, can be pre-computed during the data preprocessing step, thereby minimizing the need for additional computation for training. We empirically demonstrate in Sec. thtt the homological feature vector MGG, helps the model to better learn the original distribution of nodes and edges.
4.3 Training objective of TAGG
To produce accurate estimation of the probability vector PG,, we optimize the denoising network de with two loss terms: 1) Persistence Diagram Matching loss, which aligns the homological features of the generated graphs, and 2) Cross Entropy (CE), which ensures the node and edge attributes of the generated graphs resemble those of original graphs.
Let us first introduce Persistence Diagram Matching loss LPDM. As discussed in Sec. 3 persistence diagrams hold comprehensive homological information of graphs obtained via persistent homology.
5

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In order to ensure the generated graphs to resemble the homological features of the original graphs, which is our main contribution, we aim minimize the discrepancy between the persistence diagrams of the original and the generated graph.
Given the original graph Go and the estimate PG.. the persistence diagrams Da and D Dic, can be computed, as defined in Eq. 4 Considering the diagrams as distributions (26), we compute the discrepancy between the two distributions via I-Wasserstein distance Wi Wi(-) (-) as:
LPDM (G.) Wi (D(Go Drcn) = inf Σ DDD
|| x(x))
(11)
for any bijection DG - Dpa.'
Note that the bijection π between two persistence diagrams holds the following two challenges: 1) the bijection does not exist if the persistence diagrams have different number of points, which is true in most cases, and 2) the matching between points from two different persistence diagrams may be misleading, i.e., the two matched points may hold homological features of different dimension, such as matching a connected component to a loop. Hence, following the common approach in TDA (24), we pad the diagrams with points on the diagonal to ensure a proper matching between points of the two persistence diagrams. Also, as described in Sec. 3 persistence diagrams for each dimension can be acquired separately by plotting the barcodes of each dimension. Therefore, the PDM loss is computed over "D., and Dic, of the same dimension, with the bijection guaranteed to match points with homological features the same dimension. The final LPDM is determined by summing the distances across all dimensions.
In addition, guide the probability vectors from 60 to approximate the ground truth attributes of Go- This is conventionally done by minimizing the Cross Entropy CE(-) loss over all nodes and edges:
LCE (Go, PG.) = Cee (Go.PG.) + CEL (Go.PGo)
= >CE CE (v'.Pe;) + +
E CE ISASNN
( (ch..Bat))
(12)
ISIEN
where a1 E (0, 1] is real value. The final training objective linearly combines the two losses: Caual LCE(Go, PG.) + a2CpDas(Ga (G.. for a real value as (0. 1]. During the training process, LrDM helps the network to learn the topological structure of the original graphs. The overall training framework is shown in Fig. and the effect of LYDAS is validated in Sec. 3 For further details of TAGG, refer to Appendix A.2
4.4 Reverse process
After optimizing the denoising network do utilize the reverse process in 43 to generate new graphs. By iteratively estimating the denoised graph Pc. from a noisy graph G, and imposing noise to the estimated graphs PG, by q (G.-1 Go) from T' we can synthesize new graph. Note that unlike in the training step, where each homological feature vector μcG, is derived from its corresponding original graph Go. the matching of μc. its original graph Go cannot be defined in the reverse process, thus requiring homological feature vectors based on graphs from the training dataset. Hence, we utilize the averaged homological feature vector μc E LLS. where L and are the number of piece-wiss linear functions and the number of sub-intervals as introduced in Eq. (6). The average operation is performed over all training graphs.
Experiments
5.1 Dataset and experimental settings
Graph Benchmark. To obtain a coherent analysis of graph generation performance, we adopt three conventional benchmarks of real and synthetic graphs: (1) Community small: 200 synthetic graphs with 12 < [V| < 20 generated from a stochastic block model with two communities, (2) Ego-small: 200 small sub-graphs of the Citeseer network dataset (37 with 4 < [V| < 18, and (3) ENZYMES: 600 protein tertiary structures of the enzymes in graphs from the BRENDA database [36]. Additional experiments other datasets, i..,, SBM and Planar graphs, are included in the Appendix A.5

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Table 1: Quantitative comparison with baseline models on synthetic and real graph datasets. The best and second best results are highlighted in bold and underline, with lower values indicating better performance.
ADNI
ENZYMES
Community-smin
Ego-small
Method
Real, [V]
-
100
Real, 10 s [V] S 125
Synthetic, 12
[M] 20
Real, 4 4 [V] 18
Deg.i
Clus. Orbit,
Avg.
Deg.l Clus. Orbil, Avg.l
Deg. Clus.. Orbit), Avg.L
Deg.1
Clus.) Orbitl
Avg-1
GmphRNN
1.392
0.916
0.153
0.820
0.183
0.182
0.159
0.069
0.090
0.052
0.071
EDP-GNN
32
1,063
1.430
0,626
1.09
0.052
0.895
0.474
0.474
0.056
0.038
0.069
0.054
0,029
0,046
0.008
0.028
GDSS 23
0.949
1.104
0.165
0.739
0.314
0.506
0.084
0.301
0.033
0.112
0.009
0.051
0.045
0.076
0.008
0.043
DiGress 43
0.504
1.168
0.379
0.683
0.023
0.051
0.205
0.093
0.089
0.091
0.049
0.076
0.026
0.090
0.023
0.046
LocalPPGN
0.777
0.149
0.954
0.627
0.037
0.068
0.048
0.051
0.034
0.218
0.018
0.090
0.014
0.091
0.006
0.037
TAGG
0.160
0.886
0.237
0.427
0.003
0.045
0.015
0.021
0.048
0.068
0.004
0.040
0.012 0.024 0.010
0.015
Brain Network. To validate practicability of TAGG, we use brain connectivity from Alzheimer's Disease Neuroimaging Initiative (ADNI). In house tractography pipeline was applied to Diffusion Weighted Imaging (DWI) of healthy subjects from ADNI adhering to the Destrieux atlas [ with 160 regions of interest (ROIs) comprising 148 cortical and 12 sub-cortical regions. The dataset is composed of 844 undirected weighted graphs, i.e., structural brain connectivity, where the edge weights represent the number of fiber tracts connecting different ROIs The edges were thresholded by removing those below 5%p of the maximum edge weights to obtain sparsity.
Baselines. We used the following one-shot deep generative methods as baselines: EDP-GNN 32
GDSS 23 DiGress 43), and the one-shot version of LocalPPGN 4 In addition, a conventional auto-regressive generation method, i.e., GraphRNN [47. is used for comparison.
Evaluation Setting. The (dis similarryy between distributions of graph statistics on the same number of generated and test graphs were computed using the maximum mean discrepancy (MMD) [17. Specifically, we compared the distributions of degree, clustering coefficient, and the number of occurrences of orbits with nodes, as in [23] and [32]. For consistency, we adhered to the train/test split reference from (3 and performed three replicate experiments to report averaged performance.
5.2
Quantitative analysis
The comparison between the baselines and the proposed method is shown in Table I Note that although the models for the best and the second best performances vary in each metric, TAGG steadily shows highly promising performance, especially in the averaged value of the three MMD metrics. Specifically, we observed that DiGress, the referenced discrete diffusion based graph generation method, did not perform well on small-scale graphs. showing 0.076 (4th) and 0.046 (5th) averaged MMD on Community-smal and Ego-small dataset, whereas TAGG showed superior performance across all metrics, including the averaged MMD score of 0.040 (Ist) and 0.015 (Ist).
Results.
Ablation study on M.G. We conducted ablation study to evaluate the impact of the homological feature MG.- In Table 2 we show that the feature J4G, introduced in Sec. 4.2 enhances the denoising network generate quantitatively improved realistic graphs by providing the underlying global structural information. Note that utilizing PGo provides performance gain (i.e., from 0.061 to 0.096 decrease in averaged MMD gains) in most categories in Table 2
Alation study on LPDM- To evaluate the effectiveness of our PDM loss LPDM. we also provide an ablation study in Tab. 3 Similar to the method explained in Sec. 4.3 applied the LPDM on the persistence diagrams of the original graph Go and the estimate pc. on the one-shot graph generation baselines to observe the general performance gap when utilizing LPDM. Shown in Table 3 we empirically demonstrate that CPDM successfully guides baseline networks to produce topologically reliable graphs. Notably, the average metric values improved by 1.09 ~ 3.29 times across all methods, highlighting the impact of LPDM-
Table 2: Ablation study on PG.- Gain refers to the performance gain obtained by adding (G to the denoising network with lower MMD discrepancy. [V] denotes the averaged number of nodes.
ADNI
ENZYMES
Community small
Ego small
Metric
0 -
160. ([V]
100)
[P]
32.63, (10 S [V| S 125)
P 15.28 (12 S [V] S 20)
6.41, S [V] 18)
Avg. Gain
wio PCs
with PG:
Gain
w/o Pcs
with PG:
Gain
w/o PG:
with RCs
Gain
w/o Rd.
with PCs
Gain
Dog.
0.399
0.160
-0.239
0.017
0.003
-0.04
0.106
0.048
<0.058
(1007
(1012
0.005
.0,0%
Clus.]
1.006
0.986
-0.120
0.049
0.045
-0.04
0.123
0.068
-0.055
0.039
0.024
-0.015
.0.044
Orhit)
0.267
0.237
0.146
-0.131
0.020
-0.016
0.010
-0.011
.0.061
Avg.l
0.557
0.127
-0.130
0
0.021
-0.550
0.083
0.040
-0.043
0.022
0.015
0.007
-0,080
7

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Table Ablation study on LPDM. The superior values are highlighted bold.
ADNI
ENZYMES
Community-smin
Ego-small
Deg.
Clus.]
Orbit]
Avg.
Deg.
Clus.]
Orbit.
Avg.
Deg.
Clus.),
Orbici
Avg.
Deg.
Clus.p
Orbit]
Avg.
EDP-GNN
1,063
1,40
0.626
1,089
0.052
0,895
0.474
0.474
0.056
0,038
0.069
0.054
0.029
0.046
0.008
0.028
EDP-GNN+CPDX
0.041
GDSS
0.949
1.104
0.165
0.739
0.314
0.506
0.084
0.301
0.053
0.112
0.009
0.051
0.045
0.076
0,008
0.043
GDSS+CrDM
0.403
0.675
0.539
0.539
0.127
0.529
0.058
0.238
0.026
0.096
0.045
0.043
0.040
0,059
0.010
0.036
DiGress
0.504
1,168
0.683
0.023
0.051
0.003
0.091
0.049
0.076
0.026
0.090
M.Grresst&Cooas
0.399
1.006
0.267
0.557
0.017
0.049
0.146
01071
0.059
0.089
0.045
0.064
0.007
0,039
0,021
0.022
LocalPPGN
LocaPPGNN+CPDM
0,374
0.120
0.695
0.396
0.025
0.061
0.031
0.039
0.030
0.154
0.019
0.068
0.012
0.081
0.009
0.034
TAGG (w/o Cppas)
0.379
1.263
0.247
0.630
0.010
0.049
0.148
0.069
0.047
0.079
0.028
0.051
0.005
0.090
0.017
0.027
TAGG
0.160
0.886
0..37
0.427
0.003
0.045
0,015
0.021
0.048
0.068
0,004
0,040
0.012
0.024
0,010
0.015
Regardless of the size of the graph, the ablation studies presented in Tab. and Tab. 3 demonstrate that our proposed method, which utilize the topology-awan self-attention module and PDM loss, consistently improves performance across the overall dataset. However, in the case small graphs, its limited number of simplices results in restricted set of topological features, which may diminish the effectiveness our method when compared to larger graph datasets. As a result, the MMD metric results (degree, clustering, orbit) in Tab. [ may not show significant differences. Nevertheless, the averaged MMD scores outperform all baselines on both the community -mall and ego-small datasets, highlighting the meaningful impact of our method generation performance of small graphs.
Moreover, the improvements clustering and orbit metries emphasize the effectiveness of μcG, and LPDM in preserving essential topological structures. The improved clustering metrics demonstrate TAGG's ability capture local connectivity patterns, while the improved orbit metrics reflect its capacity to preserve structural patterns in each node's local subgraphs. These findings validate the significant contribution of the proposed topology-aware framework in generating high-fidelity graphs.
5.3 Qualitative analysis
Visualization of generated graphs. We qualitatively compare TAGG with other baselines via visualization of the generated samples. As seen in Fig. [3] TAGG produces more realistic graphs that closely resemble test graphs across various datasets. More visualization of the generated graphs of
TAGG on benchmark datasets can be found in Appendix [ Baseline models [4] 23 32 43 often failed to capture the nuanced patterns present benchmark datasets. Particularly, TAGG better
(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
(c) TAGG (Ours)
(f) Test Dataset
Figure Visualization of the averaged brain network 50 samples from (a) EDP-GNN, (b) GDSS. (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. The close-up box (red) highlights the inter-hemisphere connections. a key structural property of brain network. The global structure, e.g. the sparsity and the inter-hemisphere connectivity, are well preserved using TAGG.

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Tesk dakect
detsse
demeet
(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
(c) TAGG
Figure 4: ATOL visualization homological features plane, where cach point represent samples from
test (Blue) and generated (Pink) ADNI datasct. Graphs generated with TAGG shows the highest resemblance.
represents critical characteristics compared to baselines, i.e., sparse inter- sheer-heispphere connections and symmetry of hemispheres. Although the brain networks generated from the other discrete diffusion method, i... DiGress 43 also exhibit high quality, they failed to capture the inter- anter-hemisphere connection, subtle but critical structural information of brain network. Also in a topological perspective, the existence of inter-hemisphere connection determines the number of connected components, and TAGG is capable of capturing such topological structures. This demonstrates the effectiveness of topology-away learning in capturing both global and detailed structural properties.
5.4 Homological assessment
To further validate the homological (dis)similarity between the test and the generated graphs, we utilize three distinct measures: 1) Automatic Topologically-Orin -riilnte Learning (ATOL) [34, 2) Persistence Image (PI) (2). and 3) Curvature Filtrations (CF) 41 ATOL and are established vectorization methods in TDA that encode the homological features from the persistence diagrams into vectors of desired dimensions, and CF quantifies the (dis)similarity between graph distributions based on their topological structures. Below are the ADNI results. See Appendix F for more results.
Quantification. In the evaluation setting, we compute multivariate kernel density estimation with Kullback- Leibler (KL) divergence for ATOL, and Mean Squared Error (MSE) for PI, to quantify the distance between ref- erence and generated graph distributions. Forman-Rice Curvature was utilized in CF as the basis for curvature filtration to measure the topological differences.
Table 4: Quantified homological assessment.
ATOL J
PI
CF 4
EDP-GNN
16.25
15.39
1115.00
GDSS
1.43
3.18
154.62
DiGress
0.33
0.67
31.68
LocalPPGN
0.60
0.91
45.75
As shown in Tab. TAGG consistently exhibited the lowest dissimilarity values across all three metrics, indi- cating superior topological alignment with the reference graphs compared to all bascline methods. Specifically, compared to the second best method, i.e., DiGress, TAGG showed 30.3%, 11.9% and 11.7% reduction on ATOL KL-divergence, PI MSE, and CF, respectively.
TAGG
0.23
0.59
27.96
Additionally, discussed in Sec. Betti numbers descriptor of the topological characteristics a graph. To quan- titatively assess this aspect, report the averaged Betti numbers (mean of 50 generated graphs) for each model, as shown in Table
the key
Table 5: Averaged Betti numbers of 0-, 1-dimension, i.c., Po. 31.
3
31
Given that the real brain network exhibits 3o = 2.98 and B1 225.64, TAGG closely approximates these real topological features, yielding 3 2.92 and 31 229.96, thereby demonstrating strong homological similarity. In contrast, most other models generally yield higher 3g values (around 5-6) and exhibit wide variation in 31 values, deviating more significantly from the real data.
Test
2.98
225.64
EDPGNN
1.00
1120.64
GDSS
6.54
90.98
DiGress
5.76
238
LocalPPGN
6.00
148.50
TAGG
2.92
229.96
Visualization.
We visualize the homological feature vectors of each sample obtained by ATOL on a 2-dimensional plane to evaluate the impact of our topology-aware learning. As shown in Fig. 4 the proximity between the blue (test) and pink (generated) points in (e) far closer than (a)-(d), indicating higher topological alignment and showcasing the impact of our topology-awe aware generation.

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Conclusion
In this study, we proposed novel graph generation framework preserving the intricate topology of the network. Through the proposed topology-awae attention module and the Persistence Diagram Matching Loss, we achieve high generation performance while maintaining the essential topological features of the original graphs. This approach improves the fidelity of generated graphs and provides valuable insights into their structure for both synthetic and real graph datasets. Our research addresses critical challenge in complex real-world graph generation, particularly in the context of brain networks, and pave the way for practical graph generation with topological consistency.
Limitation. Despite the various methods on visually and empirically assessing the effect of topology- aware learning, c.g., ATOL, and ablation studies, it is hard track the influence of individual topological features on the training of the denoising network. In addition, we utilized some of the most representative vectorization methods on persistence homology, i.e., persistence landscapes, which can be changed into other methods. Therefore, more thorough investigation regarding other various vectorization methods may be done for future studies. (Partially discussed in Appendix C
Acknowledgments and Disclosure of Funding
This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea govemmeen(MMSIT) (No. 55-2002--0221133 50%; RS- 25-2222--R0004866 10%). Institute of Information & communications Technology Planning & Evaluation RAS-2022-I1220290 10%; IITP-2025-PP 011-22--RS222--0447555 10%), AI Graduate Program at POSTECH 11S-1999- 10%), and Samsung Electronies Co., Ltd (Project Code:
10%).
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References
[1] Abu-Aisheh, Z., Raveaux, R., Ramel, J.Y., Martineau, P.: An exact graph edit distance algo- rithm for solving pattern recognition problems. In: 4th International Conference on Pattern Recognition Applications and Methods 2015 (2015)
[2] Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research (2017)
[3] Austin, J., Johnson, D.D., Ho, J., Tarlow, D., Van Den Berg. R.: Structured denoising diffusion models in discrete state-spaces. Advances in Neural Information Processing Systems (2021)
[4] Bergmeister, A., Martinkus, K., Perraudin, N., Wattenhofer, R.: Efficient and scalable graph generation through iterative local expansion. In: International Conference on Learning Repre- sentations (2024)
[5] Bubenik, P:: Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research (2015)
[6] Bukkuri, A., Andor, N., Darcy, I.K.: Applications topological data analysis in oncology. Frontiers in artificial intelligence (2021)
[7] Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nature reviews neuroscience (2009)
[8] Carlsson, G.: Topology and data. Bulletin the American Mathematical Society (2009)
[9] Carlsson, G.: Topological methods for data modelling. Nature Reviews Physics (2020)
[10] Chen, Y., Volié, I:: Topological data analysis model for the spread of the coronavirus. Plos one (2021)
[11] Cho, H., Jeong, M., Jeon, S., Ahn, S., Kim, W.H.: Multi-resolution spectral coherence for graph generation with score-based diffusion. Advances in Neural Information Processing Systems (2023)
[12] De Cao, N., Kipf, T.: MolGAN: An implicit generative model for small molecular graphs. ICML 2018 workshop on Theoretical Foundations and Applications of Deep Generative Models (2018)
[13] Destrieux, C., Fischl, B., Dale, A., Halgren, R.: Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature. Neuroimage (2010)
[14] Dwivedi, V.P., Bresson, X.: A generalization of transformer networks to graphs. AAAI Work- shop on Deep Learning on Graphs: Methods and Applications (2021)
[15) Edelsbrunner, H., Harer, J.L.: Computational topology: an introduction. American Mathematical Society (2022)
[16] Farahani, F.V., Karwowski, W., Lighthall, N.R.: Application of graph theory for identifying connectivity patterns in human brain networks: systematic review. frontiers in Neuroscience (2019)
[17] Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B., Smola, A.: A kernel two-sample test. Journal of Machine Learning Research (2012)
[18] Hofer, C., Graf, F. Rieck, B., Niethammer, M.. Kwitt, R.: Graph filtration learning. In: International Conference on Machine Learning (2020)
[19] Hofer, C., Kwitt, R., Niethammer, M., Uhl, A.: Deep learning with topological signatures.
Advances in Neural Information Processing Systems (2017)
[20] Horn, M., De Brouwer, E., Moor, M., Moreau, Y., Rieck, B., Borgwardt, K.: Topological graph neural networks. In: International Conference on Learning Representations (2022)
[21] Hwang, Y. Hwang, S., Wu, G., Kim, W.H.: Multi-order Simplex- -ased Graph Neural Network for Brain Network Analysis In: proceedings of Medical Image Computing and Computer Assisted Intervention MICCAI 2024 (2024)
[22] Jin, W., Barzilay, R., Jaakkola, T.: Hierarchical generation of molecular graphs using structural motifs. In: International Conference Machine Learning (2020)

---

[23] Jo, J., Lee, S., Hwang, S.J.: Score-based generative modeling of graphs via the system of stochastic differential equations. In: International Conference on Machine Learning (2022)
[24] Kerber, M., Morozov, D., Nigmetov, A.: Geometry helps to compare persistence diagrams. ACM Exp. Algorithmies (2017)
[25] Kong, L., Cui, J., Sun, H., Zhuang, Y., Prakash, B.A., Zhang, C.: Autoregressive diffusion model for graph generation. In: International Conference on Machine Learning (2023)
[26] Lacombe, T., Cuturi, M., Oudot, S.: Large scale computation of means and clusters for persistence diagrams using optimal transport. Advances in Neural Information Processing Systems (2018)
[27] Ma, T., Chen, J., Xiao, C.: Constrained generation of semantically valid graphs via regularizing variational autoencoders. Advances in Neural Information Processing Systems (2018)
[28] Madhawa, K., Ishiguro, K., Nakago, K., Abe, M.: Graphnyp: An invertible flow model for generating molecular graphs. arXiv preprint arXiv: 1905. 1600 (2019)
[29] Martinkus, K., Loukas, A., Perraudin, N., Wattenhofer, R.: Spectre: Spectral conditioning helps to overcome the expressivity limits of one-shot graph generators. In: International Conference on Machine Leanning pp. 15159-15179, PMLR (2022)
[30] Minello, G., Bicciato, A., Rossi, L., Torsello, A., Cosmo, L.: Generating graphs via spectral diffusion. In: The Thirteenth International Conference on Learning Representations
[31] Nishikawa, N., Ike, Y., Yamanishi, K.: Adaptive topological feature via persistent homology: Filtration learning for point clouds. Advances in Neural Information Processing Systems (2024)
[32] Niu, C., Song, Y., Song, J., Zhao, S., Grover, A., Ermon, S.: Permutation invariant graph generation via score-based generative modeling. In: International Conference on Artificial Intelligence and Statistics (2020)
[33] Qiu, Y., Wei, G. W.: Artificial intelligence-aided protein engineering: from topological data analysis to deep protein language models. Briefings in Bioinformatics (2023)
[34] Royer, M., Chazal, F., Levrard, C., Umeda, Y., Ike, Y.: Atol: measure vectorization for automatic nopologically-orinnee orrenneed learning. In: International Conference on Artificial Intelligence and Statistics (2021)
[35] Saggar, M., Sporns, O., CoonzllezCasstille J., Bandettini, P.A., Carlsson, G., Glover, G., Reiss, A.I.: Towards new approach to reveal dynamical organization of the brain using topological data analysis. Nature communications (2018)
[36] Schomburg, I., Chang, A., Ebeling, C., Gremse, M., Heldt, C., Huhn, G., Schomburg, D.: Brenda, the enzyme database: updates and major new developments. Nucleic acids research (2004)
[37] Sen, P., Namata, G., Bilgic. M.. Getoor, L., Galligher, B., Eliassi-Rad, T.: Collective classifica- tion in network data. Al magazine (2008)
[38] Simonovsky, M., Komodakis, N.: Graphvae: Towards generation of small graphs using varia-
tional autoencoders. In: International Conference on Artificial Neural Networks (2018)
[39] Sizemore, A.F., Phillips-Cremins J.F., Ghrist, R., Bassett, D.S.: The importance of the whole: Topological data analysis for the network neuroscientist. Network Neuroscience (2019)
[40] Song, Y., Sohl-Dickstein, J., Kingma, D.P., Kumar, A., Ermon, S., Poole, B.: Score-based generative modeling through stochastic differential equations. In: International Conference on Learning Representations (2020)
[41] Southern, J., Wayland, J., Bronstein, M., Rieck, B.: Curvature filtrations for graph generative model evaluation. Advances in Neural Information Processing Systems 36, 55036-63061 (2023)
[42] Sporns, O., Tononi, G., Kötter, R.: The human connectome: a structural description of the human brain. PLoS computational biology (2005)
[43] Vignac, C., Krawczuk, I., Siraudin, A., Wang, B., Cevher, V., Frossard, P.: Digress: Discrete
denoising diffusion for graph generation. In: International Conference on Learning Representa- tions (2023)
[44] Xu, X., Drougard, N., Roy, R.N.: Topological data analysis as new tool for eeg processing. Frontiers in Neuroscience (2021)
12

---

[45] Yan, Z., Ma, T., Gao, L., Tang, Z., Chen, C.: Link prediction with persistent homology: An interactive view. In: International Conference on Machine Learning (2021)
[46] You, J., Liu, B., Ying, Z., Pande, V., Leskovec, J.: Graph convolutional policy network for
goal-directed molecular graph generation. Advances in Neural Information Processing Systems (2018)
[47] You, J., Ying, R., Ren, X., Hamilton, W., Leskovec. J.: Graphrnn: Generating realistic graphs with deep auto-regressive models. In: International Conference on Machine Learning (2018)
[48] Zang, C., Wang, Moflow: an invertible flow model for generating molecular graphs. In: 26th ACM SIGKDD international conference on knowledge discovery & data mining (2020)
[49] Zhang, S., Mukherjee, S., Dey, T.K.: GEFL: extended filtration learning for graph classification. In: Learning on Graphs Conference (2022)
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The abstract and the introduction include the problem statements, where conventional graph generation methods often overlook topological features of graph, which can be critical factor in graph generation. We claim that applying topological data analysis can help the neural network learn topological features and produce appoloyy-aike graphs.
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20

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A Additional details
A.1 On forward process
The additional details of the forward process described in Sec. 4.1
i.c., obtaining the noised graph G, from the original graph Go. are provided. The notations below are reused from those introduced in Sec. 4 Note that we denoted the forward diffusion process at each time step, which imposes noise, as transition matrices Qt. where
1.2....,7? T. Each element of the matrices, [Qely.5 represents the probability that state 7 changes to state & as the time step changes from t to i.e. [Q)] gy q(0² =5 y² = ) and [Q[]] J = q(e' = e' =
==""==n") n²).
A noised graph Gt (V,, E1) from the previous graph Gt-1 can be obtained by sampling the type of nodes and edges from the categorical distribution after transition, which is derived as:
q (G, G.-1) = (V--1Q% Ez-1
(13)
More precisely, each row vector of Vi-1 Nx contains the node attributes of the graph Gt-1. as a form one-hot row vector. By performing matrix multiplication Vi-1Q we impose the noise V and obtaín new matrix, where each row vector represents the categorical distribution after the transition. Subsequently, we finally obtain the noised node attributes V/2 by sampling the type of nodes from the calculated categorical distribution V. The transition of the edge attributes is processed similarly to that of the node attributes.
The forward process from time t 1 to time t is mathematically formulated follows:
q (G+ Gt-1) (V- (f)
1,1 Cat ( 1
Cat (et.M, ; Q1))
Cat (ut: (01:
Cat (vz iv
Cat (e) N.1 N.1 (et ie
Cat
QF E ;e--1
N.N. N.N
(14)
where w)) denotes the categorical distribution over one-hot row vector with probability vector 4:, and the dimensions of w are Fy and Fg for node and edge, respectively.
Moreover, assuming the Markovian property of the process, we can derive the transition matrix from time 0 to time by simply multiplying the individual transition matrices: Q = QY Q3 ... Q
and Of of Q ......QQ Of Then, similar to Eq. (14), the noised graph G. can also be obtained from time 0 by sampling from the distribution q (G. Go), which mathematically formulated as follows:
q (G. Go) = (VOQ),EEQFF)
Cat eett0"
I.N Cat (et :e0 Q..
Cat (u):))
Cat ( Q()
Cat (cM N. N.1 1.1.)
N,N N.N (.FQ Cat (et
N.NN
;cn
(15)
21

---

A.2 On training algorithm of TAGG
The overall scheme of TAGG is demonstrated in Algorithm I The homological feature vector pG. is obtained via persistence landscape of the original graph Go. which is derived from the filtration of Go. Using the resultant μC, as an additional input, the denoising network of TAGG iteratively utilize the topology-awan attention module (TAM) update the node and edge embeddings. The homological feature P.a enhance the attention module to estimate topology-aware node and edge embeddings, i.e., V; and Eg, which leads to high fidelity of generated graphs. Consequently, the denoising network outputs probability vector PGo (Dvo-PE, of the original graph, which is then optimized using the Cross-Entropy and Persistence Diagram Matching loss.
Algorithm Overall scheme of TAGG
1: Input: Original graph Go = (Vo, Eb), number of diffusion step T', number of TAM M, hyperpa- rameter 01 and ag.
2: 1. Obtain homological feature P.G.
3: phc, + Filtration (Go)
4: Obtain persistence barcodes and persistence diagram Da, from phc.
5: PG. PorsistonceLLandsappd (Da.)
b Homological feature of the given graph Go-
6: 2. Training TAGG
7: Sample t ~ (41,,,....
8: Sample noisy graph G. = (V, Ex) ~ E
9: Model input: G; = (V,, Et), PGo
2-1. Estimate PG, via TAGG
11: Given V/70 = 1/2 and E.-- Eg, for m
M do 2 , Kr" V. <- Lin,(( (Vi'),, Lima (ViT),, Lin,
13:
da is the dimension of Q, and K.,
14:
Zv. (2 @ KTT)) /V
Self-attention of node features.
15:
Z E.. Hc. Ling (ET") (ing)
16:
AT + Zym @ Zg + HG.
b Utilize μAG, as a global attention bias term.
Ath < Softmax(AA) (At))
D Compute topology-aware attention score.
18:
W.n+1 Em - Nv @ VITH,, (E(A?))
19:
end for
PG. (pv, PE.) (LayerNormi V.M), LayerNorm(EM))
Model output: probability of denoised graph PG.
22: 2-2. Training Objective
23: Linal LVB PGo,PG.) + ai CILEE (Go. + agL
(Go,
22

---

A.3 On implementation
We provide additional details of experiment settings used in TAGG, As explained in Sec. the training objective of TAGG has two real valued hyperparameters used control the cross-entropy loss of edges L and the persistence diagram matching (PDM) loss LPDM, respectively. The hyperparameters (1 and 02 were chosen through grid search of values in (1, 0.1, 0.01, 0.001, 0.0001) ) on each dataset, and the settings are shown in Tab. 6 We followed the hyperparameters provided in the original papers for the baseline methods. For the datasets that were included the original papers, we conducted the same hyperparameter search as with TAGG to ensure fair comparison. Additionally, after the reverse diffusion process to sample the generated graphs, we quantize the entries the adjacency matrices using the operator 1y50.5. resulting in binary adjacency matrix.
E (0, 1] and 0g E (0, 1 each
Table 6: Hyperparameters of TAGG on different datasets
Hyperparameter ADNI ENZYMES
Community-smal
Ego-small
On
1 0.0001
0.001
0.01
022
0.001
0.001
0.0001
A.4 On additional ablation studies
We conducted an additional ablation study on PDM to provide homological assessment to disentangle the effects of our core components as below. As mentioned in Sec. 5.4 each metric measures the topological difference, meaning that a lower score better.
Table 7: Quantitative comparison of ATOL, PI, and CF metrics across models. Lower values indicate better performance.
Method
ATOL
PI
CF
EDP-GNN
16.25
15.39
1115.00
EDP-GNN+CPDM
16.22
14.91
1038.69
GDSS
1.43
3.18
154.62
GDSS+CPDM
1.28
2.90
124.30
DiGress
0.33
0.67
31.68
DiGress+LPDM
0.27
0.63
28.80
LocalPPGN
0.60
0.91
45.75
LocalPPGN+&rDM
0.52
0.89
42.20
The result clearly shows that PDM loss ensures topological alignment, leading to improvement in all metrics, which supports the results of the manuscript.
23

---

A.5 On additional experiments
To further demonstrate the robustness and superiority, we have conducted additional experiments on the standard Planar and SBM datasets.
Table 8: Performance comparison on Planar and SBM datasets. Lower is better.
Planar
SBM
Method
Deg.
Clus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
GraphRNN
0.234
1.429
1.239
0.967
1.459
1.797
0.988
1.414
GDSS
0.232
1.032
1.137
0,800
0.991
1.705
0.484
1.060
DiGress
0.106
0.314
1.430
0.617
0.242
0.744
1.344
0.777
TAGG
0.058
0.311
1.056
0.475
0.112
0.476
0.798
0.462
As evident from the table, TAGG consistently achieves superior or highly comparable performance across all evaluated metrics on both the Planar and SBM datasets. This further supports and strength- ens the experimental results presented in the main manuscript.
In addition, validity, novelty, and uniqueness are common metrics used in molecular graph generation, mainly to validate their molecular properties. The table below presents these additional metrics (results of SPECTRE are from its original paper (2
Table 9: Comparison of additional metries on Planar and SBM datasets. Higher better.
Planar
SBM
Method
Val.
Uniq.
Nov.
Val.
Uniq.
Nov.
GDSS
33.3
100
100
33.7
100
99.0
DiGress
38.8
100
93.6
23.1
100
100
SPECTRE
47.5*
100*
100"
60.0"
100*
100"
TAGG
62.5
100
100
63.6
100
100
Additionally, further demonstrate our model's capabilities, we evaluated the diversity of TAGG- generated ADNI graphs using averaged Graph Edit Distance (GED) and Average Pairwise Distance (APD). GED quantifies structural diversity by measuring the number of operations (node/edge additions or deletions) required to transform one graph into another [ APD measures pairwise distances based on graph properties, such degree distributions, where higher values indicate greater diversity. The results are as follows: TAGG-generaad
graphs show (GED: 1.55 APD: 0.23), while test graphs show (GED: 1.12 / APD: 0.15). The increased GED and APD of TAGG- graphs (~ 40%) compared to test graphs suggests that the generated graphs exhibit sufficient diversity while remaining within a valid distribution.
24

---

B Analysis of computational and practical feasibility
We provide detailed feasibility analysis including the graph generation process and persistent homology component. We report the measured time (sec/epoch) and memory usage (in MB) for TAGG and baseline diffusion models, i.e., ConGress (continuous) and DiGress (discrete) 4 on four datasets of varying scale. All experiments were conducted on a single GeForce RTX 3090 with 24GB of GPU memory, with batch size
Table 10: Averaged sec/epoch of 10 epochs.
sec/epoch
ADNI
ENZYMES
Community-sme
Ego-small
ConGress [43
11.2
9.4
1.4
2.7
DiGress
[3
7.7
7.6
1.1
1.9
TAGG 77.1
65.0
2.4
3.0
The primary sources of computational burden in TAGG can be decomposed into three aspects: 1) the computation of the homological feature μr, 2) persistent homology, and 3) the Wasserstein distance in the PDM loss. While y can be pre-computed, requiring minimum additional computation during training. the computational cost of persistent homology and the Wasserstein distance is comparatively high, accounting for most of the time differences shown in the table above. This limitation is primarily due to the CPU-based implementation of both methods, significant challenge faced by researchers in the TDA community, as none the existing Python libraries currently support GPU computation. Despite the well-known high computational complexity of persistent homology, we believe this is still one of the highly attractive research topics to investigate in the graph machine learning community.
In the case for large graph datasets (e.g., ENZYMES with up to 125 nodes, and ADNI with 160 nodes), the number of simplices increases leading the additional costs for computing the persistent homology, e.g., the graph filtration. Despite this, the training times on ENZYMES and ADNI dataset 64.8 and 78.1 seconds per epoch, respectively, and fully training the TAGG (1000 epochs) be done within a day (< 22h) on both datasets.
Table 11: Averaged inference time (in scc) for generating 4 samples 5 individual runs
sec
ADNI
ENZYMES Community-small
Ego-small
DiGress 43
63.18 64.19
14.48
14.27
13.92
TAGG
15.04
14.44
14.99
To verify the practicality of our model, we also report the average inference time (sec) for generating 4 graph samples over independent runs across all datasets, using a single GPU. As shown in Tab. II the inference time of TAGG is comparable to that of the baseline model, i.e., Digress, across all datasets. This demonstrates that TAGG achieves superior performance and fidelity in graph generation, as reported in Sec without incurring any significant additional computational cost.
In addition, as shown in Tab. 12 the memory requirements for TAGG are not substantial. Considering that these results were achieved using single GPU, we think that the computational demand is practically feasible for real-world applications especially when multiple GPUs are employed.
Table 12: Required memory for training (MB), hatch size 4
MB
ADNI
ENZYMES
Community
Ego-small
DiGress 43
5680
5988
1570
1532
TAGG
6186
6352
1580
1542
25

---

C Topological design choices
In TAGG, utilized node degree the O-simplex filter function as in Eq. and persistent landscape as the vectorization method of persistence diagrams. However, we do not claim in any way that such design choice the optimal or state-of-the-ar framework for topology-aware graph generation. While the choices can vary, the change in generation performance is minor, which we will cover in following subsections.
C.1 Choice of filter functions
Below, we report the generation performance using different filtration functions: degree, clustering coefficient, and betweenness centrality.
Table 13: Comparison of different filter functions.
ADNI
Community-smi small
Ego-small
Function
Deg-
Clus.
Orh.
Avg.
Deg.
Clus.
Orb.
Avg.
Deg.
Clus.
Orb.
Avg.
degree
0.160
0.886
0.237
0.427
0.048
0.068
0.004
0.040
0.012
0.024
0.010
0.015
clust coef
0.242
0.802
0.198
0.414
0.061
0.057
0.008
0.042
0.016
0.030
0.015
0.020
btw_cent
0.252
0.847
0.236
0.445
0.051
0.059
0.017
0.042
0.015
0.036
0.010
0.020
As be seen, the choice of filtration function does not have crucial impact the overall performance.
C.2 Choice of vectorization methods
We also evaluated both persistence landscapes (PL) and persistence images (PI) on Community-smal and Ego-small datasets.
Table 14: Comparison of different vectorization methods.
Community
Ego-small
Deg.
Clus.
Orb.
Avg.
Deg.
Clus.
Orb.
Avg.
PL
0.048
0.068
0.004
0.040
0.012
0.024
0.010
0.015
PI
0.050
0.079
0.004
0.044
0.011
0.025
0.016
0.017
Results indicate minimal performance differences, suggesting method is robust to different vectorization choices.
26

---

D Additional visualization
In this section, provide additional visualizations that have been included in the main paper due to space limit.
D.1 Generated graph samples via GraphRNN
For example, Fig. | includes the (a) brain network and (b) ENZYMES graphs generated via GraphRNN [47]. a baseline method of our paper. Align with the qualitative comparison results shown in Sec. of the main paper, TAGG better represents critical characteristics compared to GraphRNN.
(a) ADNI
(b) ENZYMES
Figure 5: Visualization of the generated graphs using GraphRNN; (a) Brain network from ADNI dataset and (b) ENZYMES graphs.
D.2
Generated graph samples via TAGG
Also, Fig. 6 8 shows additional graphs generated using TAGG to validate the consistency of the generation performance.
Figure 6: Visualization of the brain network generated using TAGG. TAGG successfully generates homologically reliable brain network, preserving the symmetry of brain network and the edges interconnecting left and right hemisphere.
27

---

(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
(c) TAGG (Ours)
(f) Test Datasct
Figure 7: Visualization of the generated Ego-small graphs from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset.
28

---

*
(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
(c) TAGG (Ours)
(f) Test Dataset
Figure 8: Visualization of the generated Community-sman graphs from (a) EDP-GNN, (b) GDSS, (c) DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. Compared to the baselines, TAGG generates the most topologically equivalent graphs. TAGG successfully generates edge that connects two communities.
29

---

(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
(c) TAGG (Ours)
(f) Test Dataset
Figure 9: Visualization of the generated ENZYMES graphs from (a) EDP-GNN, (b) GDSS, (c)
DiGress, (d) LocalPPGN, (e) TAGG, and (f) test dataset. TAGG generate topologically equivalent graphs.
30

---

D.3 ATOL result of all baseline methods and TAGG
Text datsett
Text detinet
Text dataer Diore
Text deinet TAGG
Tos detavet ED0-ONN
detaer pitress
Text dastoo TAGG
opss
LocalPPNN
(a) EDP-GNN
(b) GDSS
(c) DiGress
(d) LocalPPGN
TAGG (Ours)
Figure 10: ATOL visualization of homological features derived from the test (Pink) and generated graphs (Blue). Top: ADNI dataset, Bottom: ENZYMES dataset. The distribution of features from sample from TAGG exhibit the best similarity with ground truth.
31

---

E Visualization of hidden representations
600
base w/ Mao
base
300
Meo
400
w/ Lynn Proposed
200
Proposed
LynM
100
200
-100
-20
-300
-100
-400
-200
(a) Ego small
(b) ADNI
Figure 11: t-SNE visualization of the trained features from the topology-aware attention module (a) Ego-small and (b) ADNI dataset. Colors denote the t-SNE results from trained features under different model settings; Blue: baseline, Orange: TAGG without PA. Green: TAGG without LPDM. and Red: TAGG.
In order to investigate the effect of the topology-awan learning framework, Fig. demonstrates the t-SNE visualization of the trained features from the topology-awan attention module on different datasets. Specifically, the hidden features from the final layer the attention module were extracted and projected onto 2-dimensional plane using t-SNE, providing visual representation of the trained hidden features the latent space. evaluate the individual contributions of the topology-aware attention module and the PDM loss of TAGG, t-SNE results were obtained using the same models from the ablation study, along with a baseline model that excludes both μlG, and LPDM. Notably, incorporating either the homological feature (G., the PDM loss, both consistently improved performance across all datasets. In line with the enhanced quantitative results, the visualization reveals differences in the latent space between the cases where neither method was applied and where both were utilized, indicating that the graph features were optimized into a more desirable latent space.
F Additional quantified homological assessment
Table 15: Quantified homological assessment.
ADNI
ENZYMES
Methods
ATOL (KL-div)
PI (MSEN
CF 1
ATOL (KL-div)
PI (MSE)
CF
EDP-GNN
16.25
15.39
1115,00
2.98
21.35
87.66
GDSS
1.43
3.18
154.62
1.12
0.13
72.76
DiGress
0.33
0.67
31.68
0.55
0.06
32.76
LocalPPGN
0.60
0.91
45.75
0.61
0.04
55.74
TAGG
0.23
0.59
27.96
0.29
0.01
21.29
To further validate the homological assessment of TAGG, we provide additional quantified result on the ADNI and ENZYMES datasets. As shown in Tab. [5] TAGG consistently exhibited the lowest dissimilarity values across all three metrics both datasets, showcasing superior topological alignment with the reference graphs compared to all baseline methods. Specifically, on the ADNI dataset, TAGG showed about 30.3%, 11.9% and 11.7% reduction ATOL KL-divergence, PI MSE, and CF, respectively, compared to the second best scores. Similarly, on the ENZYMES dataset, TAGG yielded improvements of around 52.7%, 25.0%, and 64.9% the same metrics, demonstrating consistent topological performance.
32


Reviews:
[{"review_id": "rYwsAy2nks", "reviewer": "Reviewer_Z9nJ", "summary": "The authors introduce a graph generation method based on discrete diffusion, defined as topology-aware because it also preserves the structural features of the original graphs. The approach uses a loss function based on Persistence Diagram Matching (PDM) and integrates a Topology-aware Attention Module within the denoising network.\nSeveral experiments are presented, with a focus on the application to complex brain network data, showing the method’s effectiveness in real-world scenarios.", "questions": "Please see above", "limitations": "Missing runtime analysis", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "*Strengths*\n- The issue of reconstructing the homological features of the original graphs is very important, and the proposed method is therefore interesting because it explicitly addresses this aspect. The experiments on homological assessment are also very interesting.\n\n*Weaknesses*\n\n- Persistent homology works better and makes more sense on graphs that are not small. The experiments on Ego-Small are therefore particularly challenging and appreciated. However, I would have included, at least in the appendix, experiments on standard datasets such as SBM and Planar. I expect the model to achieve performance at least on par with other methods on these datasets.\n- The computational cost analysis is done in comparison with DiGress, which is known to be a discrete diffusion method and relatively slow. To have a more informative comparison, I suggest including at least one continuous diffusion method. This point should also be stressed in the limitations.\n- Persistent homology is always a difficult topic, and since it is allowed, I would add in the appendix a richer explanation in terms of graphical support for Section 3 (Preliminaries), to help readers who are less familiar with topology.\n- The experiments are missing standard metrics such as Validity, Novelty, and Uniqueness, which are common in graph generation works. Even if they are not directly related to topology, they are useful to provide a more complete picture of the generative model’s capabilities.\n- In the sentence \n>“Recently, diffusion-based methods [3, 20, 38] showed promising capability in graph generation, by defining the forward and reverse diffusion processes and training a neural network that mimics the reverse process to reconstruct the graphs.”\n\nI suggest adding two recent references that propose methods based on diffusion in the graph spectrum, which also capture both local and global structural information:\n\n- Martinkus, K., Loukas, A., Perraudin, N., & Wattenhofer, R. (2022, June). Spectre: Spectral conditioning helps to overcome the expressivity limits of one-shot graph generators. In ICML.\n- Minello, G., Bicciato, A., Rossi, L., Torsello, A., & Cosmo, L. (2025). Generating graphs via spectral diffusion. In ICLR.", "was_revised": true, "final_justification": "After considering the submission and my initial review, I confirm my accept recommendation. \nThe approach is well-motivated and the experiments, particularly on complex brain networks, demonstrate its effectiveness in real-world scenarios.", "rebuttal": "We sincerely thank the reviewer Z9nJ for the constructive comments. In the following, we will address all the reviewer's remaining doubts.\n\n[1] SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators. ICML, 2022.  \n[2] An exact graph edit distance algorithm for solving pattern recognition problems. ICPRAM, 2015.  \n[3] Generating graphs via spectral diffusion. ICLR, 2025.\n\n---\n**W1) Additional experiments on standard datasets, as PH works better on large graphs.**\n\n**A)** Great insight on the effect of PH, as mentioned in Line 285-287. Our initial choice of datasets was driven by the goal of demonstrating TAGG’s scalability across a diverse range of graph sizes (average nodes ranging from 6.41 to 160). To further demonstrate its robustness and superiority as requested, we have conducted additional experiments on the standard SBM and Planar datasets.\n\n|    |   Planar   ||||   SBM   ||||\n|----|------------|---|---|----|------------|---|---|----|\n|    |Deg.|Clus.|Orbit|Avg.|Deg.|Clus.|Orbit|Avg.|\n|GraphRNN|0.234|1.429|1.239|0.967|1.459|1.797|0.988|1.414|\n|GDSS|0.232|1.032|1.137|0.800|0.991|1.705|**0.484**|1.060|\n|DiGress|0.106|0.314|1.430|0.617|0.242|0.744|1.344|0.777|\n|TAGG|**0.058**|**0.311**|1.056|**0.475**|**0.112**|**0.476**|0.798|**0.462**|\n\nAs evident from the table, TAGG consistently achieves superior or highly comparable performance across all evaluated metrics on both the Planar and SBM datasets. This further supports and strengthens the experimental results presented in the main manuscript. We will include these results in the Appendix of the revised version.\n\n---\n**W2) To have a more informative comparison, include at least one continuous diffusion method.**\n\n**A)** We thank the reviewer for this valuable suggestion regarding our computational cost analysis. We clarify that the purpose of the cost analysis was not to emphasize TAGG's superior efficiency, but rather to demonstrate its practical feasibility and applicability even with the computation of topological features. As requested, we have now included ConGress, a continuous diffusion method, to the comparison table below (results are averaged sec/epoch of 10 epochs):\n\n| |comm-small|ego-small|ENZYMES|ADNI|\n|---|---|---|---|---|\n|ConGress|1.4|2.7|9.4|11.2|\n|DiGress|1.1|1.9|7.6|7.7|\n|TAGG|2.4|3|65|77.1|\n\nIt is important to note that discrete diffusion models, like DiGress, generally require fewer computation steps compared to continuous diffusion models, which typically involve discretizing numerous small steps to simulate the continuous process. This fundamental distinction often results in discrete methods having faster training and inference speeds, as our expanded comparison illustrates. We will elaborate on these points in the limitations section and Appendix B of the revised version.\n\n---\n**W3) Adding a richer explanation, i.e., graphical support, of PH in the appendix will help readers’ understandings.**\n\n**A)** We fully agree that adding more explanations will improve the reader’s understanding of the contribution of our work and its effect on graph generation. Frankly, the explanation of PH is never enough, which is why we referred to the solid paper on TDA in Line 84. However, the reviewer’s  suggestion to add graphical support is perhaps a very good idea, and we will add figures demonstrating the filtration process and the computation of persistent diagrams in the Appendix.\n\n---\n**W4) Missing standard metrics such as Validity, Novelty, and Uniqueness.**\n\n**A)** We thank the reviewer for pointing out these additional standard metrics. We are aware that Validity, Novelty, and Uniqueness are common in molecular graph generation, especially for validating their molecular properties. Following the reviewer’s suggestion and acknowledging that some works (e.g., [1]) also report these metrics on synthetic datasets, we have now included results for Validity, Novelty, and Uniqueness on the SBM and Planar datasets. The table below presents these additional metrics (results of SPECTRE are from its original paper):\n\n|   |Planar|||SBM|||\n|---|---|---|---|---|---|---|\n|   |Val|Uniq|Nov|Val|Uniq|Nov|\n|GDSS|33.3|100|100|33.7|100|99.0|\n|DiGress|38.8|100|93.55|23.1|100|100|\n|SPECTRE|47.5*|100*|100*|60*|100*|100*|\n|TAGG|62.5|100|100|63.6|100|100|\n\nAdditionally, to further demonstrate our model's capabilities, we evaluated the diversity of TAGG-generated ADNI graphs using averaged Graph Edit Distance (GED) and Average Pairwise Distance (APD). GED quantifies structural diversity by measuring the number of operations (node/edge additions or deletions) required to transform one graph into another [2]. APD measures pairwise distances based on graph properties, such as degree distributions, where higher values indicate greater diversity. The results are as follows: TAGG-generated graphs show (GED: 1.55 / APD: 0.23), while test graphs show (GED: 1.12 / APD: 0.15). The increased GED and APD of TAGG-generated graphs (~40%) compared to test graphs suggests that the generated graphs exhibit sufficient diversity while remaining within a valid distribution.\n\n---\n**W5) Suggest adding two recent references that propose methods based on diffusion in the graph spectrum.**\n\n**A)** We thank the reviewer for this valuable suggestion. We agree that incorporating these two recent references regarding diffusion methods in the graph spectrum [1,3] will enrich the discussion in our related work section (Line 103), providing readers with a more comprehensive overview of graph diffusion models. We will include these references in the revised manuscript.\n\n---\n**L1) Missing runtime analysis**\n\n**A)** Please refer to the W2) and Appendix B."}, {"review_id": "TPTuLZMk9S", "reviewer": "Reviewer_7g2a", "summary": "Classical methods for operations on graphs, such as random graph generation or graph denoising, sometimes scramble important graph features that we need. One of these features is the topology of the graph. The paper does a good job addressing this problem.", "questions": "Can these methods be generalized to general simplicial complexes rather than just graphs?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is well written and seems to make an important contribution.", "was_revised": false, "final_justification": "", "rebuttal": "**Q1) Can these methods be generalized to general simplicial complexes?**\n\n**A)** We thank the reviewer for this insightful question. While our current framework is designed for graphs, viewed as 1-dimensional simplicial complexes, the underlying methodology can, in principle, be extended to more general simplicial complexes. For example, in settings such as 3D point cloud data, one could construct higher-dimensional simplicial complexes and compute higher-order persistent homology accordingly. Although a direct application is currently not feasible due to the input constraints of the network architecture, the core idea of discrete diffusion-based generation guided by topological alignment remains applicable in broader settings with suitable architectural adaptations."}, {"review_id": "vLmJKkTJ9q", "reviewer": "Reviewer_hBMW", "summary": "The paper presents TAGG, a topology-aware graph generation framework that integrates persistent homology into diffusion models to better account for topological/structural information of undirected graphs. In particular, the authors introduce a novel loss function ( persistence diagram matching PDM), which enforces topological similarity between generated and original graphs, and a topology-aware attention mechanism that augments the self-attention module with homological features. The proposed approach is evaluated across four datasets (two synthetic ones), including brain networks.", "questions": "1. Regarding homological similarity, could you report the distributions of betti_0 and betti_1 for the generated graphs from different models and test data? \n2. How does the choice of vectorization and filtration function impact the overall performance? The method should work for any choice of vectorization, no?! What is the motivation for persistence landscapes?\n3. Does the incorporation of $\\mu_{G_0}$ in the attention layer lead to a constant additive term in Eq. (9) across all diffusion steps? \n4. Could the authors report results with random $\\mu_{G_0}$? I would like to know the sensitivity to that choice.\n5. How does the proposed model compare with Digress on their evaluation setup?\n6. In line 112, what if the homological feature does not disappear?", "limitations": "The paper briefly discusses limitations in Section 6.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths \n\n- The paper is well-written and clear;\n- The application on brain networks is very interesting with many nice visualizations;\n- The authors provide ablation studies to validate the effectiveness of their loss function and topology-aware attention mechanism.\n\nWeaknesses\n\n**Novelty**. I consider the novelty limited as:\n\n- The core generation process closely follows the method in [1];\n- Topology-aware loss functions based on PH have been abundant in graph representation learning [e.g., 2,3];\n- TAM consists of a minor modification of a standard self-attention module, incorporating topological features.\n\n[1] Digress: Discrete denoising diffusion for graph generation. ICLR, 2023. \n\n[2] Improving Self-supervised Molecular Representation Learning using Persistent Homology. NeurIPS, 2023.\n\n[3] Boosting Graph Pooling with Persistent Homology. NeurIPS, 2024.\n\n**Experiments**. I have concerns regarding the completeness and significance of the experiments. In particular:\n- The paper says it \"performed three replicate experiments to report averaged performance\". Could you report the variances?\n- Important evaluation metrics are missing, such as novelty & uniqueness or spectrum (see [4] for others).\n- Also, the paper lacks baselines for one-shot graph generation, such as SPECTRE [4].\n- Overall, this paper would benefit from considering more datasets. I would like to see more results on molecular graphs, and assess how the proposal compares with SOTA for molecule generation (e.g., see [1]).\n\n[4] SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators. ICML, 2022.\n\n**Motivation**. Overall, modeling graph distributions requires capturing a diverse range of structural patterns. I am not entirely convinced that better accounting for homological features (btw, I believe the paper refers to persistent (detailed) homological features) would lead to better models in general, considering the diversity of structural patterns. This is more pronounced because the topological information in the proposal is derived from a fixed filtration and a specific choice of vectorization. These design choices are not adequately motivated. In addition, only the average topological embedding (from training) is used --- it is unclear to me why such a simple modeling choice would work across different graph distributions.", "was_revised": true, "final_justification": "The proposed method demonstrates empirical improvements over DiGress and exhibits potential for advancing the current art in graph generation. I would like to see more experiments on molecular data, including additional evaluation metrics. Also, reporting results using different filtration functions and vectorization schemes might be useful to understand the sensitivity of the model to those choices.", "rebuttal": "We sincerely appreciate your careful review. In the following, we will address all the reviewer's remaining doubts.\n\n[1] Digress: Discrete denoising diffusion for graph generation. ICLR, 2023.\n\n[2] SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators. ICML, 2022.\n\n[3] Graph filtration learning. ICML, 2020.\n\n---\n**W-N1) The core generation process follows [1].**\n\n**A)** Our novelty does not lie in introducing discrete diffusion but in proposing a topological perspective for graph generation. While we adopt a discrete diffusion backbone as [1], TAGG explicitly incorporates TDA tools via PDM loss and TAM to ensure topological alignment between generated and real graphs. This crucial capability, which preserves properties like connected components and loops, fundamentally distinguishes TAGG from prior works.\n\n---\n**W-N2) Topology-aware loss functions have been abundant in graph representation learning.**\n\n**A)** We acknowledge that PH-based loss functions have been previously utilized. However, we claim that our PDM loss contributes a novel approach to graph generation by focusing on preserving global homological information during the generative process. Unlike previous methods primarily optimizing node-edge distributions, PDM ensures topological resemblance, capturing intrinsic shape and connectivity patterns crucial for high fidelity graph generation. Experiments validate that generated graphs retain key topological properties grounded in TDA.\n\n---\n**W-N3) TAM consists of a minor modification of a standard self-attention module.**\n\n**A)** We humbly disagree that a modification of a standard self-attention module necessarily diminishes the novelty of our work. What information is integrated into attention, how it is computed, and its downstream impact are critical to model design. We emphasize that graph topology, specifically in terms of persistent homological features of various dimensions, has been largely unaddressed in the context of graph generation models. By carefully integrating these topological features into the attention mechanism, our TAM enables the module to explicitly account for global graph structure. This allows for the computation of attention scores that directly contribute to superior generation performance, as rigorously demonstrated in our experiments and ablation study.\n\n---\n**W-E1) Variance of performance.**\n\n**A)** As requested, we provide the standard deviation for the performance metrics of our proposed TAGG model in the table below.\n\n||Deg.|Clus.|Orbit||Deg.|Clus.|Orbit|\n|---|---|---|---|---|---|---|---|\n|ADNI|0.160±0.021|0.886±0.012|0.237±0.039|ENZYMES|0.003±0.001|0.045±0.002|0.015±0.004|\n|Com-s|0.048±0.021|0.068±0.006|0.040±0.009|Ego-s|0.012±0.006|0.024±0.011|0.010±0.005|\n\nAs these results indicate, TAGG demonstrates consistently stable performance across different datasets and metrics. Due to character limits, we'll include the standard deviations for baseline methods in the Appendix of the final paper.\n\n---\n**W-E2) Important evaluation metrics are missing.**\n\n**A)** Please refer to W4 of Reviewer Z9nJ.\n\n---\n**W-E3) Lack baselines for one-shot graph generation, such as [2].**\n\n**A)** To ensure a fair comparison, our current set of baselines includes four well-known one-shot graph generation methods. Following your suggestion, we attempted to run [2] using its public code. Unfortunately, the substantial computational hurdles (as noted on its public GitHub) hindered us from reporting results within the deadline. However, DiGress [1] has already demonstrated superior results over [2] on both general and molecular graph benchmarks (SBM, Planar, Community-small, and QM9). Since TAGG consistently showed superior or comparable performance to DiGress (Table 1; W1, W4 of Reviewer Z9nJ), we can reasonably infer TAGG would also outperform [2].\n\n---\n**W-E4) More datasets like molecular graphs will be beneficial.**\n\n**A)** To address the general capabilities of our method across diverse graph types, we conducted additional experiments on standard datasets (SBM and Planar, as detailed in W1 of Reviewer Z9nJ) and on a molecular graph dataset (QM9). Below, we report the results for QM9 dataset:\n\n|QM9|Val.|Uniq.|Nov.|\n|---|---|---|---|\n|ConGress|98.9|97|39.8|\n|DiGress|99.5|100|33.7|\n|TAGG|99.0|100|50.6|\n\nTAGG demonstrates superior or comparable performance across all metrics, notably achieving significantly higher novelty (+16.9%p) compared to DiGress. One possible explanation for this superior novelty is that, unlike standard discrete diffusion models that often overfit to frequent local motifs, our PDM loss encourages the exploration of diverse yet valid topologies. This helps the model avoid mode collapse and generate structurally novel graphs that still preserve key global properties, leading to the observed performance gains.\n\n---\n**W-M1) Better accounting for homological features may not account for diverse structural patterns.**\n\n**A)** We acknowledge that our fixed filtration and vectorization choices might not capture *all* graph structural diversity. However, graph topology is a critical structural aspect largely overlooked in prior graph generation works, and persistent homology offers a principled way to encode these global structures.\nAs noted in Sec. 6 (Limitations), our specific choices are widely adopted and well-established within the TDA community. Our results demonstrate that even with this straightforward design, our approach leads to meaningful and consistent improvements in both quantitative metrics (Tables 1–4) and qualitative analyses (Figures 3–4). Furthermore, our ablation studies confirm that each PH-based component significantly contributes to the observed performance gains.\n\n---\n**W-M2) It is unclear whether a simple averaging of $\\mu$ would work across different graph distributions.**\n\n**A)** Using the average $\\mu$ is a deliberate choice that provides a representative topological summary of the dataset's general distribution. This simple yet effective strategy works by capturing the dominant topological patterns, guiding our model to produce graphs aligning with the dataset's characteristics. Our strong experimental performance across diverse datasets empirically validates its sufficiency.\nCrucially, this average $\\mu$ encodes meaningful, learned topological information, which is fundamentally different from sampling from random noise (as detailed in Q4).\n\n---\n**Q1) Distributions of betti_0 and betti_1.**\n\n**A)** We agree that the distribution of Betti numbers is crucial for understanding homological similarity, thus we report the averaged Betti numbers (mean of 50 graphs) below:\n\n|ADNI|Test|EDPGNN|GDSS|DiGress|LocalPPGN|TAGG|\n|---|---|---|---|---|---|---|\n|$\\bar\\beta_0$|**2.98**|1|6.54|5.76|6|**2.92**|\n|$\\bar\\beta_1$|**225.64**|1120.64|90.98|238.45|148.5|**229.96**|\n\nGiven that the real brain network exhibits $\\bar\\beta_0=2.98$, our proposed TAGG model closely approximates these real topological features, yielding $\\bar\\beta_0=2.92$, demonstrating strong homological similarity. In contrast, most other models generally yield higher $\\bar\\beta_0$ (around 5-6), deviating from the real data.\n\n---\n**Q2) Impact of the choice of vectorization and filtration function.**\n\n**A)** Below, we report the generation performance using different filtration functions: degree, clustering coefficient, and betweenness centrality. As can be seen, the choice of filtration function does not have a crucial impact on the overall performance, as well as the vectorization method (refer to W1 of Reviewer zh2S).\n\n||ADNI|||ENZY.|||Com-s|||Ego-s|||\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n|   |Deg|Clus|Orb|Deg|Clus|Orb|Deg|Clus|Orb|Deg|Clus|Orb|\n|degree|0.160|0.886|0.237|0.003|0.045|0.015|0.048|0.068|0.004|0.012|0.024|0.010|\n|clust coef|0.242|0.802|0.198|-|-|-|0.061|0.057|0.008|0.016|0.030|0.015|\n|btw_cent|0.252|0.847|0.236|0.009|0.047|0.012|0.051|0.059|0.017|0.015|0.036|0.010|\n\n---\n**Q3) Does the incorporation of in the attention layer lead to a constant additive term in Eq. (9) across all diffusion steps?**\n\n**A)** Yes, the homological embedding is injected to the attention computation as a global bias term, which is employed across all diffusion steps.\n\n---\n**Q4) Random $\\mu_{G_0}$ sensitivity.**\n\n**A)** We report results using random $\\mu_{G_0}​$. Performance degrades notably across all datasets and metrics, which aligns with our design intuition: the averaged $\\mu_{G_0}​$ serves as a proxy for general topological characteristics of the original distribution and guides the network to generate topologically faithful samples. Random $\\mu_{G_0}​$ lacks such semantic structure, leading to poor topological alignment. Degradation is more severe for larger graphs (ADNI), likely due to the increased importance of embedded topological information.\n\n||Com-s|||Ego-s|||ADNI|||\n|---|---|---|---|---|---|---|---|---|---|\n||Deg|Clu|Orb|Deg|Clu|Orb|Deg|Clu|Orb|\n|avg|**0.048**|**0.068**|**0.004**|**0.012**|**0.024**|**0.010**|**0.160**|**0.886**|**0.237**|\n|rand|0.091|0.069|0.026|0.024|0.030|0.015|1.126|1.880|0.693|\n\n---\n**Q5) How does the proposed model compare with Digress on their evaluation setup?**\n\n**A)** All comparisons strictly follow the evaluation setups suggested in the manuscript or using the public code, including DiGress. Additional hyperparameter details for TAGG are provided in Appendix A.3.\n\n---\n**Q6) In line 112, what if the homological feature does not disappear?**\n\n**A)** When a homological feature does not disappear during the filtration, it is referred to as an essential feature, represented as barcodes of the form $(b, \\infty)$. In the context of graphs, essential 0- and 1-dimensional features typically correspond to connected components and loops, respectively.\nIn practice, since $\\infty$ cannot be directly represented in computation, it is approximated using a sufficiently large constant or the maximum filtration value."}, {"review_id": "PuZ6NlqwzD", "reviewer": "Reviewer_zh2S", "summary": "This manuscript introduces a method to integrate persistent homology into graph diffusion models. The approach comprises three steps: (1) compute the persistence landscape from input graphs; (2) embed the landscape into the diffusion backbone via an attention module; and (3) augment the base diffusion loss (as in [38]) with an additional term comparing persistence diagrams.", "questions": "1. The loss in Equation (11) involves an optimization subproblem. How do you compute gradients through this inner optimization during backpropagation?\n2. In Line 103, you describe “adding the node $v_1$,” but multiple nodes may be added simultaneously. Could you clarify this step in the algorithm?\n3. In Section 5.4, TAGG’s superior homology preservation is expected by design. Can you perform an ablation study that adds only the persistence-loss term to other baseline models to disentangle the contributions of topology loss and attention fusion?", "limitations": "Besides the limitations pointed out by the authors, I'd like to bring up one another possible limitation: the computation of persistence diagrams seem to limited to graphs with small sizes and maybe this is worth pointing out in the paper.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The overall idea of incorporating global information of a graph into the generative model is nice. The proposed method of using the persistence landscape might be a good way to do so. The presentation is clear and easy to follow and the problem is important in the context of graph generative models.\n\nHowever, the idea of using persistence diagrams in graph models is not new. For example, this work [1] already explored the idea of using persistence diagrams in message passing neural networks. I understand that the authors are using a different approach by combining the persistence landscape with an attention module, but there is no justification of why this is better than using persistence image + message passing as done in [1]. I think at least some empirical comparison is needed to justify the proposed method. \n\n[1] Zhao et al. 2020, Persistence Enhanced Graph Neural Network, AISTATS", "was_revised": true, "final_justification": "I think this work is solid after the authors addressed my and other reviewers' concerns by providing more ablative studies. These help with clarifying what is effective in the proposed method.", "rebuttal": "We sincerely appreciate your detailed review. We will address the reviewer's concerns for further improvements.\n\n[1] Zhao et al., \"Persistence Enhanced Graph Neural Network.\" AISTATS, 2020\n\n[2] Maria, Clément, et al. \"The gudhi library: Simplicial complexes and persistent homology.\" Mathematical Software–ICMS, 2014\n\n---\n**W1) Why is TAGG better than [1]?**\n\n**A)** A) We appreciate the reviewer’s reference to [1] and agree that persistent homology has been previously explored in graph learning. However, our work departs significantly from prior approaches such as [1], which focused on graph classification and incorporated persistence images as input features into message passing neural networks. In contrast, our method targets graph generation and leverages topological signals at a deeper architectural level; by introducing TAM that dynamically modulates the denoising process, and by enforcing topological alignment via PDM loss. These components are embedded directly into the generative model, rather than simple feature-level augmentation, offering a principled and structure-aware approach to topology-driven generation.\nWe also agree that empirical comparison of vectorization methods is valuable. We therefore evaluated both persistence landscapes (PL) and persistence images (PI) on Community-small and Ego-small datasets (larger graphs will be reported in the Appendix):\n\n|   |Com-s||||Ego-s||||\n|---|---|---|---|---|---|---|---|---|\n|   |Deg.|Clus.|Orb.|Avg.|Deg.|Clus.|Orb.|Avg.|\n|PL|0.048|0.068|0.004|0.040|0.012|0.024|0.010|0.015|\n|PI|0.050|0.079|0.004|0.044|0.011|0.025|0.016|0.017|\n\nResults indicate minimal performance differences, suggesting our method is robust to different vectorization choices. While we do not claim PL or any specific filter function to be universally superior, our novelty lies in recasting graph generation from a topological perspective and effectively employing TDA tools (TAM and PDM) to ensure topology-aware generation, rather than in the selection of a single vectorization strategy.\n\n---\n**Q1) How do you compute gradients through this inner optimization (Eq. 11) during backpropagation?**\n\n**A)** Addressing the concern about the backpropagation, the 1-Wasserstein distance used in Equation (11) is implemented based on a differentiable optimal transport method by leveraging Gudhi library [2]. Also, the homological feature vector is fed to the MLP layer before incorporating it to the node and edge features in the attention module, which ensures the end-to-end training.\n\n---\n**Q2) In Line 103, you describe “adding the node v_1,” but multiple nodes may be added simultaneously. Could you clarify this step in the algorithm?**\n\n**A)** Thank you for the great insight and question. As you mentioned, multiple nodes with the same filter value can be added simultaneously in the filtration process. In this case, following the sublevel set filtration, their incident edges are also included in the evolving subgraph at that filtration step. While Line 103 offers a simplified exposition, the underlying process correctly handles such additions. Notably, this does not limit the homological information; persistence barcodes are inherently recorded as a multiset of intervals, which perfectly captures all births, deaths, and persistence of homological features occurring at any given filtration value, regardless of simultaneous additions. For the readers' better understanding, we will add additional explanation in the Appendix.\n\n---\n**Q3) Can you perform an ablation study that adds only the persistence-loss term to other baseline models?**\n\n**A)** Thank you for pointing out an important analysis. As you correctly pointed out, disentangling the effects of topology preservation and attention design is essential for a thorough understanding of TAGG’s performance.\nTo address this, we conducted the exact ablation study you suggested, i.e., adding only the persistence-loss term to existing baseline models, and present the results in Table 3. As shown, incorporating the PDM loss alone leads to consistent improvements across not only baselines but also all datasets, demonstrating the effectiveness of PDM loss, independent of our attention design.\n\n---\n**L1) Persistence diagram computation may not scale to large graphs.**\n\n**A)** We appreciate your insightful suggestion regarding this potential limitation. We agree that the computational complexity of persistent homology generally increases with graph size. However, as discussed in Appendix B, the datasets used in our work span a range of graph sizes (average node counts from 6.41 to 160), and the computational burden for generating persistence diagrams proved manageable. Our experiments show that a complete TAGG training (1000 epochs) is feasible within a single day. Thus, we demonstrate that incorporating persistent homology is practical for real-world applications across various typical graph scales, even if not maximally efficient. We agree that this aspect warrants explicit mention and will include it in the limitations section of the manuscript."}]

Author rebuttal:

Meta-review: The paper was reviewed by four expert reviewers. 
The reviewers generally agree that the paper is well-written and mostly appreciate the novelty of incorporating topological features into a diffusion graph generative model. Compelling experimental results are supported by both real and synthetic datasets, with quantitative and qualitative evaluations demonstrating the performance of TAGG in comparison to multiple baselines. 
The authors provided a rebuttal that addressed the concerns raised in the reviews. After the rebuttal and disucssion, the reviewers were satisfied with the justifications and details provided in the rebuttal and recommended acceptance. 

The reviews provided valuable feedback to the authors, which the authors should carefully consider to improve the final version of their paper, in addition to experiments, additional evaluation measures, run-time, and ablations conducted during the discussion period, which should be added to the final version of the paper.
Decision: Accept (poster)