GGBall: Graph Generative Model on Poincaré Ball
Authors: Tianci Bu
Chuanrui Wang
Hao Ma
Haoren Zheng
Xin Lu
Tailin Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Hyperbolic Space; Graph Generation; Flow Matching
Primary area: generative models
TL;DR: The first graph generation framework built upon the Poincaré Ball model of hyperbolic space.

Abstract: Generating graphs with hierarchical structures remains a fundamental challenge due to the limitations of Euclidean geometry in capturing exponential complexity. 
        Here we introduce **GGBall**, a novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms. 
        GGBall combines a Hyperbolic Vector-Quantized Autoencoder (HVQVAE) with a Riemannian flow matching prior defined via closed-form geodesics. This design enables flow-based priors to model complex latent distributions, while vector quantization helps preserve the curvature-aware structure of the hyperbolic space.
        We further develop a suite of hyperbolic GNN and Transformer layers that operate entirely within the manifold, ensuring stability and scalability.
        Empirically, GGBall establishes a new state-of-the-art across diverse benchmarks. On hierarchical graph datasets, it reduces the average generation error by up to 18\% compared to the strongest baselines.
        These results highlight the potential of hyperbolic geometry as a powerful foundation for the generative modeling of complex, structured, and hierarchical data domains. 
        Code is available at: https://github.com/AI4Science-WestlakeU/GGBall.

Published as a conference paper at ICLR 2026
GGBALL: GRAPH GENERATIVE MODEL ON POINCARÉ BALL
Tianci Bu B2+ Chuanrui Wang Hao Ma' Westlake University National University of Defense Technology (butianci, wangchuanrui, mahao, zhenghaoren, wutailin @west
Haoren Zheng'
Xin Lu Tailin Wul
ABSTRACT
Generating graphs with hicrarchical structures remains fundamental challenge due to the limitations Euclidean geometry in capturing exponential complexity. Here introduce GGBall, novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms. GG- Ball combines Hyperbolic Vector- cestto-Quantized Autoencoder (HVQVAE) with a Riemannian flow matching prior defined via closed-form geodesics. This de- sign enables flow-based priors to model complex latent distributions, while vee- tor quantization helps preserve the curvature-aware structure of the hyperbolic space. We further develop suite of hyperbolic GNN and Transformer layers that operate entirely within the manifold, ensuring stability and scalability. Em- pirically, GGBall establishes a new state-of-the-art across diverse benchmarks. On hierarchical graph datasets, it reduces the average generation error by up to 18% compared to the strongest baselines. These results highlight the poten- tial of hyperbolic geometry a powerful foundation for the generative model- ing of complex, structured, and hierarchical data domains. Code is available at: https:/ htt/ggtttuuu.ugg/ggg///gg com AT 4Sci ence West 1 aake//GGRa
INTRODUCTION
Graph generation plays a central role in many sci- entific and engineering domains, including molec- ular design, material discovery, and social network modeling (Miller et al.] 2024. [Shi et al.) 20200] Reiser et 2022 Luo et al. 2021] Wang et al. 2021a). Recent advances in deep generative mod- els have enabled powerful data-driver approaches to this task. Most existing models operate di- rectly in the discrete graph space, where gener- ation proceeds by sequentially constructing re- fining nodes and edges. For instance, diffusion- based models such GDSS (Jo et al. 2022) and DiGress (Vignac et al. 2023) iteratively denoise graphs in discrete space, while autoregressive models like GraphRNN (You al., (018)) and GRAN (Liao al. 2019) build graphs one node edge time, modeling structural dependencies step-by-step. These approaches offer fine- are-grained control and explicitly model structural dependen- cies the discrete graph domain.
Degree vs. Edge Accuracy (Reversed X-axis)
100.0
Hyperbolic Model
HVQVAE
99.0
Better Region
Edge Accuracy
HAE
08.5
98.0
VQVAE
Euclidean Model
AE
0.008
0.007
3.006
0.005
0,004
0.003
D.002
0.001
0.000
Degree (MMD)
4
Figure 1: Degree similarity and edge reconstruction accuracy on reconstructed dataset. Hyperbolic models consistently outperform Euclidean basclines.
While these models have shown promise, graph generation remains challenging due to the discrete, combinatorial, and often hierarchical nature graph data (Guo & Zhao (022). To address these issues, latent space generation has emerged scalable and flexible alternative. By encoding
"Equal contribution, listed in alphabetical order.
+Corresponding author.

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Published as a conference paper at ICLR 2026
Messaee
Patecare codeboot
Lotent fosture
Message passing edge
diccods
Pe (n Ib)
Alconia
NX
decods
GNN
Transformer
Transformer
Pa(e, h,.b))
min, da, (2,c)
Encoder
Decoder
quantized
G= E)
MP6)
Poincaré Flow Matching
Palisl
G= -X Reconstructed /Geneaaee Graph
Graph
Figure 2: Overview of our hyperbolic graph generation framework. We encode graphs into hyperbolic latent space using Poincaré GNN and geodesic-attention Transformer. The latent representations are quantized via a Poincaré codebook and modeled with a Poincaré flow prior. A hyperbolic Transformer then decodes the latent code to reconstruct or generate graphs, enabling structure-awan aware generation in non-Eucciiee annEuucidean geometry.
graphs into continuous latent representations and decoding from this space, methods like Graph- VAE (Simonovsky Komodakis 2018) and VQGAE (Boget et al. (023) emable efficient one-shot generation"] Nevertheless, these models typically rely on Euclidean latent spaces, which ill-suited for capturing the hierarchical and compositional nature of real-world graphs, such as community structures and power-law degree distributions (Krioukov et al., 2010).
This geometric mismatch motivates our shift to hyperbolic space, a natural framework for hierarchical representation learning (Mathieu et al. 2019). Unlike Euclidean embeddings that distort parent-child relationships, hyperbolic geometry intrinsically preserves graph hierarchies through its exponentially expanding volume (Chami et al. 2019) Ganea et al. 2018 Krioukov al. 2010; Sarkar (01T).
As demonstrated in Figure II hyperbolic latent models (e.g.. HAE, HVQVAE) achieve superior aligmment with power-law degree distributions (4x lower MMD) and higher edge reconstruction accuracy compared to Euclidean counterparts Community dataset.
Building on these insights, we present GGBall, the first Graph Generation framework fully built upon the Poincaré Ball model of hyperbolic space. Our core innovation is unified, node-only latent representation. This holistic design ensures that the entire graph topology, both nodes and their relational structure, is governed by single, consistent geometric framework, treating edge connectivity as an emergent property of the latent node geometry. This approach differs from discrete space hybrid models (Wen et al.) 2024) that assign nodes and edges to separate latent spaces. Specifically, we convert a standard Euchdean latent generative pipeline (Rombach et al. 2022) into fully hyperbolic one: For the encoder, we design Hyperbolic Vector-Quantized Autoencoder (HVQVAF) that captures graph structure via discrete tokens the Poincaré ball, initialized through geodesic clustering and optimized using Riemannian methods. For the latent generative process, we leverage flow matching in hyperbolic space to model expressive priors without relying on predefined noise. For architectural support, we develop a modular suite of hyperbolic GNN and Transformer layers that operate entirely within the manifold, ensuring numerical stability and scalability.
Our method achieves state-of off-the-ar art performance across diverse graph generation benchmarks. On SOMMUNITY-SMALL and EGO-SMALL, GGBall reduces the average generation error by up to 18% compared to the strongest competitors, highlighting its strength in modeling modular, tree-like structures. On molecular graphs (QM9), GGBall also delivers state-of- the-art results, achieving a best-in-class novelty of 93.77% while maintaining high chemical validity, culminating in the highest overall V.U.N. score. This demonstrates its robust ability to generate diverse and chemically plausible molecules.
hierarchical datasets like
For related work, please refer Appendix
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Published as a conference paper at ICLR 2026
2 PRELIMINARIES AND PROBLEM DEFINITION
To support our hyperbolic generative framework, we briefly review the Poincaré ball model as the underlying latent space in Section 2.1 followed by the formulation of graph generation non- Euclidean geometry in Section 2.2 For clarity, all hyperbolii-eeometty operators fully defined in the Appendix E.5
2.1
HYPERBOLIC GEOMETRY
2.1.1 RIEMANNIAN MANIFOLD
Riemannian manifold (M, g) a smooth manifold equipped with a Riemannian metric tensor field g. which smoothly assigns each point MM an inner product 9z T&M T&M R on its tangent space T.M (Gallot et al. 1990). For a n-ddimensional manifold, each tangent space T.M locally isomorphic IR",, providing linear approximation of the manifold at.
2.1.2 HYPERBOLIC SPACE AND POINCARÉ BALL MODEL
Hyperbolic space is a Riemannian manifold of constant negative curvature -c, offering a geometric framework for hierarchical data representation. Among isomorphic models, the Poincaré ball model (B ge) is widely adopted in machine learning due to its conformal structure and numerical stability. Here, B fæ E R" c||=:| 2 1} defines an open ball of radius 1/vC. and the metric tensor g. = (A:) II, scales Euclidean distances by the conformal factor X = 2 |3_ This induces a Riemannian inner product (u, v) = (x,)²(u, v) for u, V T (Ungar 2008 To enable algebraic operations on hyperbolic coordinates, the Möbius gyrovector framework extends vector space axioms to gyrovectors. The basic binary operation is denoted as the Möbius addition Be B x D - B", which is noncommutaive and nonassociative addition. We provide detailed introduction of basic operations in Appendix E.5 and E.6
2.1.3 GRAPHS IN HYPERBOLIC SPACE
Hyperbolic space is well-suited for embedding graphs with hierarchical tree-like structures, thanks to its exponential volume growth and negative curvature. Prior works have shown that hyperbolic embeddings better preserve hicrarchical relationships and long-range dependencies than their Euclidean counterparts (Nickel & Kicla 2017) Chami et al., 2019; Krioukov et al. 2010)). This further supported by Gromov's approximation theorem, which states tree-like structures admit approximate embeddings into hyperbolic space, formally linking hyperbolic geometry to tree-like structures (Appendix E.3
While these results motivate hyperbolic embeddings for representation learning, graph generation in hyperbolic space remains largely unexplored. Our work addresses this gap by introducing a generative framework that fully exploits curvature-awar geometric priors.
2.2 GRAPH GENERATION IN HYPERBOLIC SPACE
In this section, we formalise the task of graph generation in hyperbolic latent space and outline the core idea of our approach.
Problem definition. We consider undirected graph G (X. E). Node attributes represented as one-hot vectors, x (2......... )" E 1R and edges types are represented by e e m in such dense matrix representation. The absence of edges treated as additional edge type. We use m to denote the total number of nodes in single graph, h and ky as the number of classes for nodes and edges. The goal of graph generative model is learn distribution that can produce graphs whose structure and attributes match those observed in the training set.
Latent factorization. Motivated by the insight that hyperbolic space inherently captures edge infor- mation through geometry, our idea is to encode edge structure directly into node embeddings. There- fore, we assume that introduce set of hyperholic latent variables z (21, 22,... ... 2m), 2;
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Bğ and factorize the data likelihood
f BB x
pa( e) =
pe(=) =z
Two-stage paradigm. Inspired by the two-stage generation pipeline in image generation tasks (Rom) bach et al., 2022), we propose to generate graphs via two sequential steps: (i) Sample a set of node em- beddings that lie on n--dimensional Poincaré ball; (ii) Decode these embeddings into discrete node and edge attributes conditionally as Po( e x) = II pe x,) IT ITk=1 Po(E 2j. zk) where node labels depend solely on their own latent vectors and edge types depend only on hyperbolic pairwise relations. This conditional independence assumption reduces decoding complexity while fully leveraging hyperbolic distances when modelling edge likelihoods.
METHOD
We propose a fully hyperbolic framework operating within the Poincaré ball to address the geometric
mismatch between graph topology and Euclidean latent spaces mentioned before. First, we propose several basic Poincaré network architecture Section 3.1 then detail the training paradigm for encoder-decoder framework between graph space and hyperbolic space in Section Section B.3 further describes the modeling of latent prior with normalising flows. The overall architecture is illustrated in Figure
3.1
POINCARÉ NETWORK ARCHITECTURE
3.1.1 POINCARÉ GRAPH NEURAL NETWORK
As illustrated in the GNN module of Figure 2 goal is to construct hyperbolic message passing that encodes edge and node information directly into node representation. The key challenge lies in preserving hyperbolic structure during neighborhood aggregation. Our solution leverages two principles: (1) Adaptive modulation of features based on hyperbolic distances to maintain curvature- aware scaling, and (2) Projection-free operations using closed-form Möbius additions.
Given node embeddings and edge embeddings h;, hij E B each layer computes updated features follows.
Tangent Space Aggregation. To enable stable aggregation in curved space, we project neighboring nodes and edge embeddings to the tangent space at the origin using (og perform Fuclidean-style operations, and map the result back via exp8(-). This yields the following update for node i:
1+1
Σ JEN(G)
We [logā (h.), (h.)), logō logō(h!))]
m
4 1+1 expo(logo + W. (l))) logû (M(m{+1)))
where W., Wú, are learned weight functions. This aggregation scheme preserves hyperbolic geometry throughout the message-passing process.
Distance-Modulated Message Function. The message function M(-) integrates node and edge information by modulating the aggregated messages using parameters derived from hyperbolic distances. Specifically, for each edge j), we compute scale and shift coefficients Vij Bij as functions of (h,, h2), and apply them to the message: M(m,y) = Yi mij Biz-
This curvature-aware modulation allows the model to encode edge strength and structural hierarchy di- rectly into node representations. By design, this mechanism inherently preserves tree-like topologies, unlike Euclidean GNNs that often distort long-range relationships due to flat-space aggregation.
3.1.2 POINCARÉ DIFFUSION TRANSFORMER
After obtaining node representations from the hyperbolic GNN, to further model global graph structure, adapt diffusion transformers hyperholic space by aligning self-attention mechanisms
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with geometric priors. The core innovation lies in replacing dot-product attention with geodesic distance scoring and Möbius gyromidpoints to aggregate value, which respects the exponential growth of relational capacity in hyperbolic space. Each layer computes:
Geodesic Attention.
Score interactions uses hyperbolic distances rather than dot products:
Oij exp(-rde(q: ky));
(4)
where qi, j is projected by input features using Poincaré linear layers (Eq. 16.. Values fy are
aggregated using Möbius gyromidpoints to maintain geometric consistency:
Σ, E,og Σ, la,g(
Z, =
p auyle = 0.
(5)
j=1
Our Möbius gyromidpoint aggregation can be interpreted as a weighted Fréchet mean on the Poincaré ball, ensuring geometric consistency while being computationally efficient.
Cine-Conditioned Modulation. We extend the Poincaré transformer with adaptive time- conditioned modulation. Each layer injects timestep embeddings temb Rn through Euclidean affine transformations of normalized features.
The transformer block maintains hyperbolic consistency via: (1) Multi-head attention splits features using 3-scaling followed by hyperbolic concatenation (Appendix E.6.3). (2) Residual connections employ Möbius addition t instead of standard summation. (3) Layer normalization and feed-forward networks operate in tangent space via logo expĚ projections (Eq. 17
3.2 REPRESENTATION LEARNING IN HYPERBOLIC LATENT SPACE
Building upon the hyperbolic GNN and Transformer layers, we introduce hyperbolic autoencoding framework for learning graph representations in non-Euclidean latent spaces. As illustrated in Figure 2 this framework employs an encoder-decoder architecture operating entirely within the Poincaré ball model of hyperbolic space.
We explore three architectural variants: (1) a continuous Hyperbolic Graph Autoencoder (HGAE), (2) probabilistic Hyperbolic Variational Autoencoder (HVAE), and (3) discrete Hyperbolic Vector- Quantized Autoencoder (HVQVAE). While HVAE a natural extension of its Euclidean counterpart, found its Kullback-Leibh (KL) divergence term to be numerically unstable curved manifolds, often leading to degraded reconstruction quality (see Appendix F.I). In contrast, HVQVAE provides stable and expressive alternative through latent space discretization, making our primary model for the full generative framework.
3.2. HYPERBOLIC AUTOENCODER LEARNING
The Hyperbolic Graph Autoencoder (HGAE) maps graph nodes into hyperbolic space using a Poincaré ball encoder and reconstructs graphs via geometry-aware decoding.
Encoder & Decoder Design. The encoder enriches node features with spectral graph properties
capture global topology (Beaini et al., [202]] Viinac et al. 2023 Xu al. (019). These features are processed by Euclidean MLPs and projected onto the Poincaré ball via exponential mapping. Subsequent hyperbolic GNN layers aggregate local structural patterns, while stacked hyperbolic transformers propagate global dependencies, finally obtaining node-level representations 2,.
The decoder reconstructs node and edge attributes using intrinsic hyperbolic geometry, while edge connectivity and features are conditionally dependent node pairs. This factorization yields the joint reconstruction probability: Po Pa(ar, h) = I (az; h.) IT II Po (eçk h,j h2)- Node attributes are predicted by projecting hyperbolic embeddings to the tangent space and applying an MLP. For edge reconstruction, we compute pairwise geometric features:
fij [logo (h+), logo (h;), logh. (hz), ) de (h; hj), cos 0zy].
(6)
where d, measures hierarchical distance, 0, captures angular relationships, and logarithmic maps encode directional dependencies. These features decoded into edge probabilities via MIPP.
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The model is trained by minimizing composite loss function designed balance reconstruction fidelity with the generation of structurally valid graphs. The overall objective CAE combines three key components:
Optimization.
Lroon + Cdegres Xreg
(7)
where each term serves distinct purpose: (1) The reconstruction loss Lreen serves as the primary objective and consists of standard cross-entropy terms for predicting node and edge types from the hyperbolic latent embeddings. (2) A central component of our method is the degree- edge consistency loss Ldegree, which ensures that the expected degree implied by the edge probabilities with both the ground-truth degree and the degree predicted at the node level. This encourages the decoder to generate edge patterns that do not contradict local structural roles. The full formulation is provided in Appendix [...2 (3) Finally, to regularize the latent space, include an norm penalty Leg over latent representations.
consistent
3.2.2 HYPERBOLIC VECTOR-QUANTER VARIATIONAL AUTOENCODER LEARNING
To enhance latent space interpretability and expressiveness, extend HGAE with hyperholic vector quantization (HVQVAE), which discretizes embeddings into learnable codebook C E TBC and uses Riemannian optimizer for optimization.
Codebook Quantization. Codebook vectors are first initialized via hyperbolic k-means clustering (Alg. [6]. Each node-level latent representation z; is then quantized its nearest codebook entry via 2q = arg minezec de(2, Cj); where de denotes the geodesic distance in the Poincaré ball.
Training Objectives. The training objective combines three geometrically consistent components:
LuvoVAE = AI + X2 E. (dl (sg((ss)))) (z)), =)] X E. ((((
Commitment
Consistency
where Pa is the encoder and reconstruction loss is same as HAE; commitment loss anchors latent codes to quantized vectors, and consistency loss updates embeddings via straight-through gradient estimation sg(-).
Stability Mechanisms. To prevent codebook collapse, inactive entries are periodically replaced using expiration threshold. Codebook updates employ weighted Einstein midpoints in the Poincaré ball. More details can be found in Appendix E.7.4
3.3 LATENT DISTRIBUTION MODELING
After training the encoder and decoder, we aim to model prior over the latent space for generation. Unlike Euclidean spaces, defining stochastic generative processes in hyperbolic geometry challeng- ing due to the absence of canonical noise and well-defined stochastic dynamics (Fu et al. 2022). To address this, we adopt flow-based models, which offer flexible and deterministic mappings without relying stochastic processes. While autoregressive model is another choice, it underperforms FM on preliminary experiments, and we leave the exploration for future work.
Poincaré Flow Matching. Flow-based generative models (Lipman et al. 2023 define time- varying vector field uz(zz) that generates probability path P: (zz), transitioning between the base distribution po (zo) and the target distribution p1 (z1). In order to learn the vector field which lies in the tangent space M of z, E B" efficiently, following Chen & Lipman (202)), we minimize the Riemmanian conditional flow matching objective:
CRCFM E (((.))))) (.. (lo [1,, t) u, (z. 20)
2019) as the prior in hyperbolic space, and # as We define p(20) = NB I (Mathieu et al.
the hyperbolic latent encoding of graph data. The intepolation path is computed via deterministic geodesic interpolation 2, exp=, (s((t logz, (zo)). with x(t)
We parameterize the vector field up B [0, 1] TB using Poincaré DiT backbone, followed by logmap projection to the tangent space. For generation, integrate the manifold di vo (2, t) from an initial sample 20 ~ Po to obtain ε1, which then quantized via the VQ codebook and decoded to obtain generated graphs.

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Published as a conference paper at ICLR 2026
4 EXPERIMENT
We organize our experiments around four key questions to evaluate the advantages of hyperbolic latent spaces, focusing representation quality (QI, section 4.2), hierarchical structure modeling (Q2, section
4.3) diversity (Q3, section 4.4). and latent space smoothness (Q4, section 4.5
QI: How well can our model reconstruct input graphs from hyperbolic latent representations?
Q2: Does the hyperbolic latent facilitate the generation of complex hierarchical graph structures?
Q3: Does the hyperbolic space provide expressiveness to generate diverse structural molecules?
Q4: Does our method support smooth and chemically plausible interpolations between molecular?
4.1 EXPERIMENTAL SETUP
Baselines. We compare our methods (HAE: hyperbolic autoencoder; HVQVAE: vector quantized version; HVQVAE+Flow: latent space flow matching) with state-of-the-art models for both generic graph and molecular graph generation. Specifically, the baseline models include: VAE based models, such as StaphVAE (Simonovsky & Komodakis 2018) and VGAE (Boget et al. (023). autoregressive based model GraphRNN (You al. 22018) diffusion models, such as EDP-GNN (Niu et al. 2020). GDSS (Jo et al. (022). DiGress Vignac et al. 2023), and the recent HGDM (Wen et al. 2024) that incorporates Poincaré embeddings and edge-spes speeefeeeiiic denoising. Flow-based model including Graph Normalizing Flows (Liu et al., (019), Graph Autoregressive Flows (Shi et al. (020), and Categorical Flow matching (Eijkelboom al. 2024). Please refer to Appendix E. for training details.
4.2 STUDY ON GRAPH RECONSTRUCTION (Q1)
To answer Q1, we first evaluate the reconstruction fidelity of our autoencoders. This task assesses how effectively the hyperbolic latent space preserves graph information and establishes an upper bound on generative quality, reconstructions are decoded directly from the latent codes of ground-truth graphs.
As shown in Table D our models achieve near pear-perfect reconstruction. On abstract graphs, HVQVAE consistently improves upon HAE across all MMD metries and enhances edge accuracy. This trend continues the QM9 molecular dataset, where HVQVAE significantly boosts chemical validity from 95.18% to 99.14% and overall V.U.N. score from 94.97 to 97.34.
In intuition, compared to HAE, HVQVAE better captures discrete structural motifs (e.g., rings, branches). Compared to HVAE (Appendix F.J.. HVQVAE avoids unstable KL regularization in curved manifolds and yields more stable optimization.
Table I: Reconstruction performance of our baseline models HAE and HVQVAE on abstract graphs (Community-smen small, Ego-small) and molecular graphs (QM9).
Community-snan small
Ego-small
Model
Deg. 1 Clus.
Orb. 4
Edge Acc. f
Deg. J
Clus. 4
Orb.
Edge Acc. 1
HAB
0.0008
0.0310
0.0007
99.10
0.0019
0.0250
0.0048
92.40
HVQVAR
0.0004
0.0208
0.0005
99.39
0.0002
0.0194
0.0018
93.20
QM9
Model
Validity t
Unique t
Novelty 1
V.U.N 1
Edge Acc.
Node Acc. 1
HAE
95.18
99.79
100
94.97
99.48
100
HVQVAE
99.14
99.81
98.38
97.34
99.90
100
4.3 GENERIC GRAPH GENERATION (Q2)
Setup. Next, we address Q2 by evaluating generative performance on datasets known for their hierarchical and community based structures: Community-Small and Ego-Small. Following standard protocol (Vignac et al., 2023), we generate test-set-sized collection of graphs and measure the Maximum Mean Discrepancy (MMD) between the distributions of generated and real graphs across degree, clustering, and orbit statistics.
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Results. Table shows that our hyperbolic models excel at this task. Our one-shot HVQVAE drastically outperforms its Euclidean counterpart, GraphVAE, reducing the average MMD by 93.3% on Community-smate and 86.8% on Ego-small. This demonstrates the powerful inductive bias of the hyperbolic latent space for hierarchical data. When paired with a flow-based prior, our HVQVAE+Flow model sets a state-of-the-art, achieving the lowest average error both datasets. This superior performance directly answers Q2, confirming that hyperbolic geometry is highly effective for modeling complex, hierarchical graph structures.
Table 2: Abstract graph generation on Community-small and Ego-small dataset. We evaluate the difference in graph statistics (and their mean) between generated and ground truth graphs. Best results are in bold, second best are underlined.
Community-smal
Ego-small
Type
Space
Method
Deg.
Clus.
Orb.
Avg.
Deg. L
Clus.
Orb, 1
Avg.J
One-shot
GraphVAE
0.3500
0.9800
0.5400
0.6233
0.1300
0,1700
0.0500
0.1
Autoregressive
GraphRNN
0.0800
0,1200
0,0400
0.0800
0.0900
0.2200
0.0030
0.1043
VQGAE
0.0320
0.0046
0.0329
0.0210
0.0410
0.0230
G
EDP-GNN
0.0530
0.0260
0.0930
0.0507
Diffusion
GDSS
0.0450
0,0860
0.0070
0,0460
0.0210
0.0240
0.0070
0.0173
DiGress
0.0470
0.0410
0.0260
0,0380
X 3"
HGDM
0,0170
0.0500
0.0050
0,0240
0.0150
0.0230
0.0030
0.0137
G4
GNF
0.2000
0,2000
0,1100
0.1700
0.0300
0,1000
0.0010
0.0437
Flow
GraphAF
0.0600
0,1000
0.0150
0.0583
0.0400
0.0400
0.0080
0.0293
G4
CatFlow
0,0860
0.0070
0.0240
One-shot Flow
BC
HVQVAE (ours)
0.0085
0.0681
0.0488
0.0418
0.0071
0.0320
0.0070
0.0154
B
HVQVAE- Flow (ours)
0.0015
0.0589
0.0040
0.0215
0.0133
0.0182
0.0022
0.0112
4.4 MOLECULAR GRAPH GENERATION (Q3)
Setup. To answer Q3, assess model's ability to generate novel and diverse molecules the QM9 benchmark. This task challenges the model's expressiveness in a space that is less overtly hierarchical but rich in complex local structures rings, bonds). We follow standard protocols for molecular graph generation and reporting chemical validity, uniqueness, and novelty using RDKit.
Results. The results in Table highlight crucial challenge in molecular generation: balancing chemical validity with structural novelty. Many baselines excel at one while sacrificing the other. Our HVQVAE+Flow model resolves this trade-off, simultaneously achieving high validity while generating the most novel molecules (93.77%) of any method. This superior balance leads to state- of-the-art performance on the overall V.U.N. metric (85.34%) and the reliable novelty V.N. score (85.35%). This performance answers Q3 affirmatively, demonstrating that our hyperbolic framework highly expressive and effective for exploring complex chemical spaces.
Table Molecular graph generation on QM9. We report standard metrics and derived metrics V.N.
(Valid x Novel), N/U (Novelty rate). All values are reported percentages.
QM9
Type
Space
Method
Valid. f
Unique. 1
Novel.
V.N.
NUT
V.U.N.1 1
One-shot
E2
Gra
45.8
30.5
66.1
30.27
9.23
Autoregressive
Rd
VQGAE
88.6
EDP-GNN
47.52
99.25
86.58
41.14
87.2
40.83
Diffusion
G4
GDSS
95.72
98.46
86.27
82.58
87.6
81.31
Q
DiGress
99.0
96.2
33.4
33.07
34.7
31.81
C x 1
HGDM
98.04
97.27
69.63
68.27
71.6
66.40
Gd
GraphAF
67.00
94.51
88.83
$9.52
56.25
Flow
G
CatFlow
99.95
49.00
48.91
49.0
48.88
One-shot Flow
BC
HVQVAF (ours)
85.93
96.59
77.64
66.72
80.4
64,44
B.
HVQVAE+ (ours)
91.02
100.00
93.77
85.35
93.8
85.34
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4.5 INTERPOLATION BETWEEN MOLECULES (Q4)
Finally, to address Q4, we examine the smoothness and structure of the latent space via geodesic interpolation. Given two embeddings hg, h E B. we trace the geodesie path between them using h expho (t- logh. (h1)) for t E [0, 1] and decode the intermediate points. A well-structured latent space should yield chemically valid intermediates that blend the properties the endpoints.
As shown in Figure 3 our method generates smooth and valid molecular transitions. The decoded intermediates exhibit gradual changes structure and composition, such as ring formation and atom substitution. This high rate of validity along the geodesic path confirms the smoothness of the learned hyperbolic manifold, answering Q4 and suggesting its utility for tasks like property-guiee -uidded molecule design. We leave the integration of explicit node alignment a promising direction for future work.
1+02
LHO2
1-OS
EWER
Figure 3: Geodesic interpolation in hyperbolic latent space between molecular graphs. Top row shows 10-step latent transitions with 2D-projjeced node embeddings. Lower rows decode these into molecules, with source and target ends. Green labels mark valid intermediates; red indicates invalid ones. Chemical formulas and ring counts are annotated.
4.6 DISCUSSION
Our experiments validate a powerful, decoupled paradigm for graph generation. The proposed hyperbolic VQ-VAE learns a high-fidelity structural backbone (QI), enabling not only near-perfect reconstruction but also state-of-the-art generation of both hierarchical abstract graphs (Q2) and complex molecules (Q3), and a smooth latent space for manipulation (Q4), validating this approach as a robust and versatile alternative end-to-end models.
Furthermore, by operating on a compressed latent representation, our method presents path towards greater computational efficiency over models that work in the full, high-dimensional graph space (see Appendix G). For Limitations and Future Work, please refer to Appendix
5 CONCLUSION
In this work, we have introduced GGBall, novel framework for graph generation that leverages hyperbolic geometry to naturally encode the hierarchical structure of complex networks. By com- bining vector-quantized autoencoder with Riemannian flow matching in the Poincaré ball, our approach enables curvature-aware and one-shot graph generation. Extensive experiments on both abstract and molecular graph benchmarks validate the effectiveness of our method, showing consistent improvements in reconstruction accuracy and distributional fidelity over Euclidean and graph-space baselines. Our findings highlight the potential of non -on-Euclidean generative models to accelerate scientific discovery across chemistry, biology, and network science, including applications such as materials design, molecular modeling, and complex graph generation.

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ACKNOWLEDGEMENT
We thank Long Wei, Tao Zhang, Ruiqi Feng, and Tengfei Xu for helpful discussions and valuable feedback on the manuscript. We also gratefully acknowledge the support from the Westlake University Research Center for Industries of the Future and the Westlake University Center for High- Segh-Peeformance Computing. The content of this work is solely the responsibility of the authors and does not necessarily represent the official views of the funding entities.
ETHICS STATEMENT
This work focuses on developing a generative framework for graph-structured data using synthetic and publicly available datasets, such as Community-Small, Ego-Small, and QM9. All datasets are non-sensitive and do not involve any personally identifiable, medical, or proprietary information. Our method is intended for foundational research in machine learning, materials science, and chemistry, and is not designed for deployment in safety-critical applications without extensive downstream validation. While the model can be applied to molecule generation and scientific discovery, caution against interpreting the generated outputs as actionable or experimentally validated without proper domain-specift. screening. We advocate for careful and responsible use of generative models, especially in sensitive domains such as drug design or social network synthesis.
REFERENCES
Ivana Balažević, Carl Allen, and Timothy M Hospedales. Multi-relational poincaré graph embeddings. Advances in Neural Information Processing Systems, 02:4465-4475, 2019.
Dominique Beaini, Saro Passaro, Vincent Létournaau, Will Hamilton, Gabriele Corso, and Pietro Liò. Directional graph networks. In International Conference Machine Learning, pp. 748-758 PMLR, 2021.
Andreas Bergmeister, Karolis Martinkus, Nathanaël Perraudin, and Roger Wattenhofer. Efficient and scalable graph generation through iterative local expansion. arXiv preprint arXiv:2312.11529 2023.
Yoann Boget, Magda Gregorova, and Alexandros Kalousis. Vector-quantized graph auto-encoder. CoRR, 2023.
Herbert Busemann. The geometry of geodesics. Academic Press, 1955.
Ines Chami, Zhitao Ying, Christopher Ré, and Jure Leskovec. Hyperbolic graph convolutional neural networks. Advances in neural information processing systems, 32, 2019.
Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. In The Twelfth International Conference on Learning Representations, 2024. URL btttss:/openzevieww
net f teet/Eoorm21dd472hD111
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
Xiaohui Chen, Yinkai Wang, Jiaxing He, Yuanqi Du Soha Hassoun, Xiaolin Xu, and Li-Ping Liu. Graph generative pre-trained transformer. arXiv preprint arXiv:2501.01073, 2025.
Michel Coomaert, Thomas Delzant, and Athanase Papadopoulos. Géométrie et théorie des groupes: les groupes hyperboliques de Gromov, volume 1441. Springer, 2006.
Kaize Ding, Albert Jiongqian Liang, Bryan Perozzi, Ting Chen, Ruoxi Wang, Lichan Hong, Ed H Chi, Huan Liu, and Derek Zhiyuan Cheng. Hyperformer: Learning expressive sparse feature representations via hypergraph transformer. In Proceedings of the 46th international ACM SIGIR conference on research and development in information retrieval, pp. 2062-2066, 2023.
Floor Eijkelboom, Grigory Bartosh, Christian Andersson Naesseth, Max Welling, and Jan- Willem van de Meent. Variational flow matching for graph generation. Advances in Neural Information Processing Systems, 37:1 1775-177--774,, 2024.
10

---

Published as a conference paper at ICLR 2026
David B. A. Epstein and Robert C. Penner. Euclidean decompositions of non-compact hyperbolic manifolds. Journal of Differential Geometry, 27(1):67 80, 1987.
Aleksandr Ermolov, Leyla Mirvakhabova, Valentin Khrulkov, Nicu Sebe, and Ivan Oseledets. Hy- perbolic vision transformers: Combining improvements in metric learning. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp- 7409-7419, 2022.
Xingcheng Fu, Yisen Gao, Yuecen Wei, Qingyun Sun, Hao Peng, Jianxin Li, and Xianxian Li. Hyperbolic geometric latent diffusion model for graph generation. In International Conference Machine Learning, pp. 14102-14124. PMLR, 2024.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, al. Riemannian geomelry, volume 2. Springer, 1990.
Octavian Ganea, Gary Bécigneul, and Thomas Hofmann. Hyperbolic neural networks. Advances in neural information processing systems, 31, 2018.
Daniele Grattarola, Lorenzo Livi, and Cesare Alippi. Adversarial autoencoders with constant- curvature latent manifolds. Applied Soft Computing, 81:105511, 2019.
Caglar Gulcehre, Misha Denil, Mateusz Malinowski, Ali Razavi, Razvan Pascanu, Karl Moritz Hermann, Peter Battaglia, Victor Bapst, David Raposo, Adam Santoro, and Nando de Freitas. Hyperbolic attention networks. International Conference on Learning Representations, 2019. URL
net Eorum? ii- RcFF
Xiaojie Guo and Liang Zhao. A systematic survey on deep generative models for graph generation. IEEE Transactions Pattern Analysis and Machine Intelligence, 45(5):5370-5390 2022.
Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. Masked autoencoders are scalable vision learners. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 16000-16009, 2022.
Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. International conference learning representations, 2017.
Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction tree variational autoencoder for molecular graph generation. In International conference on machine learning, pp. 2323-2332. PMLR, 2018.
Jachyeong Jo. Seul Lee, and Sung Ju Hwang. Score-based generative modeling of graphs via the system of stochastic differential equations. In International conference on machine learning, pp. 10362-10383 10383. PMLR, 2022.
Diederik P Kingma, Max Welling, et al. Auto-encoding variational bayes, 2013.
Max Kochurov, Rasul Karimov, and Serge Kozlukov. Geoopt: Riemannian optimization in pytorch. arXiv preprint urXiv:2005.02819, 2020.
Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E E-Statistical,, Nonlinear, and Soft Malter Physics, 82(3):036106 2010.
Keegan Lensink, Bas Peters, and Eldad Haber. Fully hyperbolic convolutional neural networks. Research in the Mathematical Sciences, 944)600, 2022.
Tianhong Li. Yonglong Tian, He Li, Mingyang Deng, and Kaiming He. Autoregressive image generation without vector quantization. Advances in Neural Information Processing Systems, 37: 56424-56445. 56445, 2024.
Qian Liang, Wen Wang, Feng Bao, and Ge Gao. Fully hyperbolic rotation for knowledge graph embedding, arXiv preprint arXiv:241 (tt:::2/11.08000 1.03622, 2024.

---

Published as a conference paper at ICLR 2026
Renjie Liao, Yujia Li, Yang Song, Shenlong Wang, Will Hamilton, David K Duvenaud, Raquel Urtasun, and Richard Zemel. Efficient graph generation with graph recurrent attention networks. Advances in neural information processing systems, 32, 2019.
Yaron Lipman, Ricky Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference Learning Repre- sentations, 2023. URL https: http://pppeevvveeeeeeeeee net/ (o
Jenny Liu, Aviral Kumar, Jimmy Ba, Jamie Kiros, and Kevin Swersky. Graph normalizing flows. Advances in Neural Information Processing Systems, 32, 2019.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint 17110:1711.05101, 2017.
Youzhi Luo, Keqiang Yan, and Shuiwang Ji. Graphdf: A discrete flow model for molecular graph generation. In International conference machine learning, pp. 7192-7203, PMLR, 2021.
Emile Mathieu, Charline Le Lan, Chris Maddison, Ryota Tomioka, and Yee Whye Teh. Contin- uous hierarchical representations with poincaré variational auto-encoders. Advances in neural information processing systems, 32, 2019.
Benjamin Kurt Miller, Ricky TQ Chen, Anuroop Sriram, and Brandon M Wood. Flowmm: Generating materials with riemannian flow matching. In Forty-first International Conference on Machine Learning, 2024.
Joshua Mitton, Hans M. Senn, Klaas Wynne, and Roderick Murray-Smith A graph vae and graph transformer approach to generating molecular graphs. arXiv preprint arXiv:2104.04345, 2021. URL httth:://azxiii org/ http///b./2........ 04345
Yoshihiro Nagano, Shoichiro Yamaguchi, Yasuhiro Fujita, and Masanori Koyama. A differen- tiable gaussian-like distribution on hyperbolic space for gradient-based learning, arXiv preprint arXiv:1902.02992, pp. 41, 2019.
Maximilian Nickel and Douwe Kiela. Poincaré embeddings for learning hierarchical representations. In Advances Neural Information Processing Systems, volume 30, pp. 6338-6388860 0288-6347,1 2017.
Maximilian Nickel and Douwe Kiela. Learning continuous hierarchies the lorentz model of hyperbolic geometry. In Proceedings of the 35th International Conference Machine Learning, pp. 3779-3788, 2018.
Chenhao Niu, Yang Song, Jiaming Song, Shengjia Zhao, Aditya Grover, and Stefano Ermon. Permu- tation invariant graph generation via score bosre-based generative modeling. In International conference artificial intelligence and statistics, pp. 4474 4484. PMLR, 2020.
William Peebles and Saining Xie. Scalable diffusion models with transformers. Proceedings of the IEEE/CVF international conference on computer vision, pp. 4195 4205, 2023.
Kristina Preuer, Philipp Renz, Thomas Unterthiner, Sepp Hochreiter, and Gunter Klambauer. Fréchet chemnet distance: metric for generative models for molecules in drug discovery. Journal of chemical information and modeling, 58(9):1736-) 1741, 2018
Yiming QIN, Manuel Madeira, Dorina Thanou, and Pascal Frossard. Defog: Discrete flow matching for graph generation. In Forty-second International Conference Machine Learning, 2025. URL httpp::// htttps://epenrevleee net./ forum? (ateeuummmdkKK www.dd=KPRIWWI...
John G. Rateliffe. Foundations of Hyperbolic Manifolds. Springer, 2nd edition, 2006.
Ali Razavi, Aaron Van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. Advances in neural information processing systems, 32, 2019.
Patrick Reiser, Marlen Neubert, André Eberhard, Luca Toressi, Chen Zhou, Chen Shao, Houssam Metni, Clint van Hoesel, Henrik Schopmans, Timo Sommer, et al. Graph neural networks for materials science and chemistry. Communications Materials, 3(1):93, 2022.
12

---

Published as a conference paper at ICLR 2026
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF confer- on computer vision and pattern recognition, pp. 10684-10695, 2022.
Rik Sarkar. Low distortion delaunay embedding of trees in hyperbolic plane. In International symposium on graph drawing, pp. 355-366. Springer, 2011.
Chence Shi, Minkai Xu, Zhaocheng Zhu, Weinan Zhang, Ming Zhang, and Jian Tang. Graphaf: a flow- based autoregressive model for molecular graph generation. In International Conference on Learn- ing Representations, 2020. URLhttts::/// http::::s///peeeviee..... net wtte//ooummmm
Ryohei Shimizu, Yusuke Mukuta, and Tatsuya Harada. Hyperbolic neural networks++, arXiv preprint arXiv:2006. 0000000000.000, 2020.
Martin Simonovsky and Nikos Komodakis. Graphvae: Towards generation of small graphs using variational autoencoders. In Artificial Neural Networks and Machine Learning-CCANN 2018: 27th International Conference on Artificial Neural Networks, Rhodes, Greece, October 2018, Proceedings, Part I 27, pp. 412-422. Springer, 2018.
Abraham Albert Ungar. Analytic hyperbolic geometry and Albert Einstein's 's special theory of relativity. World Scientific, 2008
Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.
Max Van Spengler, Erwin Berkhout, and Pascal Mettes. Poincaré resnet. In Proceedings of the IEEF/CVF International Conference on Computer Vision, pp. 5419-5428, 2023.
Clement Vignac, Igor Krawczuk, Antoine Siraudin, Bohan Wang, Volkan Cevher, and Pascal Frossard.
Digress: Discrete denoising diffusion for graph generation. In The Eleventh International Confer- ence on Learning Representations, 2023. URLhttps:: petttps://openeeeee wwweeeiiwww.... net. firee/fforre forum? i d=
UaAD-Nu8 6WX
James Walker al. Text-conditioned graph generation using discrete In International Confer- ence on Learning Representations, 2021. URL http:ss///peeeeeeee...etett wet.ieewwwweeteee...e formmt/ffooummm
4UbhxQIjeSH
Chaokun Wang, Binbin Wang, Bingyang Huang, Shaoxu Song, and Zai Li. Fastsgg: Efficient social graph generation using a degree distribution generation model. In 2021 IEEE 37th International Conference Data Engineering (ICDE), pp. 564-575. IEEE, 2021a.
Shen Wang. Xiaokai Wei, Cicero Nogueira Nogueira dos Santos, Zhiguo Wang, Ramesh Nallapati, Andrew Arnold, Bing Xiang, Philip S Yu, and Isabel F Cruz. Mixed-curvature nulti-relational graph neural network for knowledge graph completion. In Proceedings of the web conference 2021, pp. 1761-1771, 2021b.
Lingfeng Wen, Xuan Tang, Mingjie Ouyang, Xiangxiang Shen, Jian Yang, Daxin Zhu, Mingsong Chen, and Xian Wei. Hyperbolic graph diffusion model. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 15823-15833 215331,,0221 2024.
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference Learning Representations, 2019. URL https 7/oeeneeview. net foret//foum?11 i ry5s
Zhe Xu, Ruizhong Qiu, Yuzhong Chen, Huiyuan Chen, Xiran Fan, Menghai Pan, Zhichen Zeng. Ma- hashweta Das, and Hanghang Tong. Discrete-state continuous -ime diffusion for graph generation. arXiv preprint urXiv:2405. 1416, 2024.
Menglin Yang, Harshit Verma, Delvin Ce Zhang, Jiahong Liu, Irwin King, and Rex Ying. Hypformer: Exploring efficient transformer fully in hyperbolic space. In Proceedings of the 30th ACM SIGKDD Conference Knowledge Discovery and Data Mining, pp. 3770-3781, 2024.
13

---

Published as a conference paper at ICLR 2026
Jiaxuan You, Rex Ying, Xiang Ren, William Hamilton, and Jure Leskovec. Graphrnn: Generating realistic graphs with deep auto-regressive models. In International conference on machine learning. pp. 100-5778-57 PMLR, 2018.
Viacheslav Yusupov, Maxim Rakhuba, and Evgeny Frolov. Knowledge graph completion with mixed geometry tensor factorization. arXiv preprint arXiv:2504.02588, 2025.
Yiding Zhang, Xinyi Wang, Chuan Shi, Ning Liu, and Guojie Song. Lorentzian graph convolutional networks. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 1333-1343, 2021.
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CONTENTS
Appendices
17
A Notation in this paper
17
The Use of Large Language Models (LLMs)
17
Related Works
17
Hyperbolic Graph Representation
17
Graph Generation
18
A Algorithms
19
Additional Technical Details
21
Remannian Manifold
21
Hyperbolic Geometry
E,3
Graphs in Hyperbolic Space
22
Why We Choose the Poincaré Ball Model
22
E.5
Basic Operations on Poincaré Ball
23
E.6
Hyperbolic Network Backbone
23
E.7
Hyperbolic AutoEncoder for Graph
26
Autoregssive Model for Latent Distribution
29
E,
Flow Matching
29
10
Training Details
30
Extra Experiments
31
Variational Autoencoder
31
Additional Experiment: Zinc250K
Additional Experiment: Sice-Strutture Graphs
32
Additional Experiment: Euclidean Ablation
................................
33
F.5
Stability Analysis via Mean/Std Reporting
33
G Efficiency Analysis
34
Additional Discussion: Higher-Order Structural Dependencies
34
Additional Discussion: Head-to-Head Comparison with HGDM
35
J Sampling Strategy
35
Statistical Metrics
35
K.1 Generic graph generation and Reconstruction
35
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K.2 2
Molecular graph generation
36
3 Evaluation Protocols
36
Limitations & Future Work
37
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A
NOTATION IN THIS PAPER
Table Notations used in this paper.
Notation
Description
M
Manifold
Metric
R
Real number field
a:
Point on manifold
Absolute value of curvature
B
Poincaré ball
Point on latent space
Remannian inner product graph
G
X
Set of graph nodes
Set of graph edges
T
Tangent space of manifold
m
Number of nodes
Number of dimension for specific space
Möbius Addition
exPo
Logarithm map
log.
Exponential map
B THE USE OF LARGE LANGUAGE MODELS (LLMS)
In this paper, we employ LLMs as general-purpose assist tools for refinement and language polishing. All core research ideas, datasets, and scientific conclusions presented in this paper are our own original contributions.
C RELATED WORKS
C.1 HYPERBOLIC GRAPH REPRESENTATION
Lorentz Model. The Lorentz model represents hyperbolic space as part of a two-sheeted hyper- boloid in Minkowski space, enabling efficient geodesic computation and optimization (Busemann 1955 Ratcliffe, 2006 Epstein & Penner 1987). It effectively captures hierarchical structures in graphs through continuous embeddings Nickel Kiela 2018). and Lorentz transformations im- prove knowledge graph embeddings with fewer parameters (Liang et al. 2024). Hyperbolic Graph Convolutional Networks (HGCNs) have shown great potential in capturing hierarchical structures in graphs (Chami et al. 2019). Lorentzian Graph Convolutional Networks (LGGNNs) enforce hyperbolic constraints, reducing distortion in tree-like graph representations (Zhang et al. (2021). Further ad-
vancements in hyperbolic geometry have enhanced neural architectures for better expressivity and
efficiency (Ermolov et al. 2022 Ding et al. 2023 Fu et al. 2024 Yang et al. 2024).
Poincaré Ball Model. The Poincaré ball models hyperbolic space within unit ball, supporting compact embeddings for hierarchical data (Nickel & Kiela (017). Geometry-awan neural layers improve performance in tasks like link prediction and classification (Ganea al.) 2018). MuRP applies Möbius transformations for relation-spectie knowledge graph embeddings (Balažević et al. 2019), and fully convolutional networks capture multi-scale graph features (Lensink 2022). Core components like attention and convolution have been extended to the Poincaré ball for improved efficiency and stability (Shimizu et al.) 2020), followed by a fully hyperbolic residual network that enhances robustness distributional shifts and adversarial perturbations (Van Spengler et al.) (023). Unlike prior work that focuses on discriminative tasks, our framework leverages the Poincaré ball for generative modeling speccficcllllll... learning and sampling discrete graph structures via a hyperbolic latent space. This extends the utility hyperbolic geometry from representation learning to structured data generation.
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C.2 GRAPH GENERATION
Autoencoder-Basse Models. Autoencoder-based models use latent representations to generate graphs. VAEs (Kingma et al., 2013) and VQVAEs (Van Den Oord et al. 2017, Razavi et al. 2019) are foundational this area. JT VAE builds molecular graphs by first generating a tree of substructures, then assembling them with message passing (Jin et al. 2018). Combining VA with graph transformers improves scalability and interpretability (Mitton et al. (02T). VQVAEs like VQ-T2G support rext-conditioned graph generation by learning discrete latent codes aligned with textual prompts (Walker et al., 2021). Our model extends prior autoencoder based approaches by using a hyperbolie latent space to better capture hierarchy and improve generative quality.
Autoregressive Models. Autoregressive models generate graphs by sequentially adding nodes and edges. GraphRNN models the conditional distribution of graph structures using an RNN, producing diverse graphs (You et al. 2018). GRAN improves efficiency by incorporating attention and generating graph blocks (Liao et al., 2019). G2PT adopts transformer- based next-token prediction for goal-oriented graph generation and property prediction (Chen et al. 2025). contrast, our method avoids sequential generation in graph space, which requires explicit ordering, and instead enables one-shot decoding with stronger inductive bias for hierarchical structures.
Diffusion Models. Diffusion models are effective for graph generation by transforming noise into discrete denoising diffusion model with graph transformer to handle categorical node and edge attributes, achieving strong results (Vignac) al. 2023). DisCo extends this by modeling diffusion discrete-state continuous-time setting, balancing quality and efficiency while preserving graph discreteness (Xu et al. 2024). Concurrent to our work, [Wen et (2024) proposed the Hyperbolic Graph Diffusion Model (HGDM), which performs diffusion in a brid latent space: node embeddings evolve in hyperbolic space while edge embeddings are modeled in Euclidean space, updated by two independent score networks. In contrast, our GGBBal framework embeds the entire graph into compact node-only hyperbolic latent space, implicitly encoding edge structure in the geometry of node embeddings. During generation, we only evolve these node embeddings, with edges reconstructed by the decoder. This unified design eliminates the need for separate latent processes and leads to improved scalability and generative quality.
structured graphs through diffusion process. DiGress
Flow Matching. Flow matching methods learn deterministic transport maps to transform simple distributions into complex graph structures. The Flow Matching framework trains continuous normalizing flows by regressing vector fields along fixed probability paths without simulation (Lipman) et al. 2023). DeFoG adapts this to discrete graphs, separating training and sampling for improved flexibility (QIN et al. 2025). CatFlow frames flow matching as variational inference, enabling efficient generation of categorical data (Dijkelboom et al. 2024). In materials science, FlowMM applies Riemannian flow matching to generate high-fidelity materials, showing the method's 's broad applicability (Miller et al.] (024)). Building these advances. extend flow matching to hyperbolic latent space, enabling geometry-awe aware graph generation with improved alignment to hierarchical structures.
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D ALGORIHTMS
Algorithm Hyperbolic k-Means Clustering
Require: Samples {z; IB N i I. cluster count K, iterations T
Ensure: Codebook (cj E [ = cluster sizes {ny}
1: Initialize C3 < s--plleevecttooee
K)
Random initial centers
2: for t = to T do
Compute pairwise distances: d; -del coj (1-11) 4: Assign clusters: by +- argmax,dij Vi
3:
(Bq. 14
5: Count clusters: n2 < Hi : b. j}}, vj
6:
for j = to K do if n then
7:
8:
(t-1)
b Retain dead clusters
9:
else
(0) WeighedMMidponn((1)) 5))
10:
11:
projag (c))
Manifold projection
end if end for
13:
(1) if max, de(e)) cj ((-11) e then break
15: end if
16: end for
Algorithm 2 Flow Matching for Graph Generation
Require: Pretrained VQVAE with hyperbolic latent space, flow-based model
Ensure: Generated graph Ggen
1: Sample latent codes from the standard normal distribution in hyperbolic space:
2: 2n ~ (0,
3: Apply flow transformation to move from the source distribution to the target distribution in the
latent space:
4: 21 Flowe((o, for t [0, 1]
5: Quantize the latent code to the nearest codebook vector:
6: 2q arg min, de, c2),c)
7: Decode the quantized latent code back to a graph:
8: Ggen (2g)
9: Return the generated graph Ggen gen
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Algorithm 3 Hyperbolic Graph Autoencoder (HGAE)
Require: Graph G (X, E), hyperbolic curvature c, layers L, margins Tmins Tmax
Encoding Stage
1: Compute structural and spectral features S (e.g., graph Laplacian eigenfeatures).
2: Preprocess attributes using Euclidean MLP:
X₁, = expo (MLP(XX,S))
E, expi (ML((E))
3: Initialize hyperbolic embeddings: (0) & GNNg (X,, E₁).
4: for -= ...... L do
5: Perform hyperbolic rreetion-based message passing:
Attentiona (2, (
6: end for
7: Obtain final embeddings: Z (2/")..
Decoding Stage
8: for each node do
9:
Project hyperbolic embedding to tangent space:
(z()) Tu
10:
Predict node attributes:
())
(4
11: end for
12:
for each edge (i.j) do
13:
Compute hyperbolic geometric features:
Juj = ( (T),, (zg (2)))) Jog (1)")) d<(, , cos 0zg]
14: Predict edge attributes:
(e(ei) }
(L) (L) Softmax(MLP(f;;))
end for
16:
return Final embeddings Z, trained encoder-decoder
Algorithm 4 Hyperbolic Vector Quantized Variational AutoEncoder (HVQVAE)
Require: Dataset D, codebook size K, hyperbolic curvature c, EMA parameter 8, margins
Tmin, Tiaa
Initialization
1:
Initialize hyperbolic codebook C = {4 C Be using hyperbolic k-means or random sam- pling.
Encoding and Quantization
2: for each data point æ D do
3: Obtain latent embedding = (ncoder,(x) E Bd
4: Quantize embedding nearest codebook vector using hyperbolic distance:
2g arg mine,eo Ec de (z, c;)
5: end for
Decoding
6: Reconstruct input from quantized embedding: è Decodera (2yy)
EMA Codebook Update
7: for each codebook vector Cj do
8: Compute centroid μj from assigned latent vectors
P. 2 O. Σ. Wr [on Tj E. TenEsT
9:
EMA update codebook vector using Einstein midpoint:
(++1) projud (lc))) Aee B. Iny, -8e)
10: end for
Dead Codes Revival
11: if a codebook vector cj has not been updated for long periods then
12:
Re-initialize codebook vector ej to new vector sampled from Bd.
13: end if
14: return Trained encoder, decoder, and hyperbolic codebook
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Algorithm 5 Training HVQVAE
Require: Dataset {G₂} of graphs
Ensure: Trained HVQVAE model on hyperbolic
1: Initialize hyperbolic encoder Egg, codebook C, and decoder Dep
2: for each graph G; in the dataset do
Map the graph's node and edge features to the hyperbolic space via the Poincaré ball model: > Exponential map to hyperbolic space Apply message passing and attention encode information in latent space
4:
2; = Exp, (Epp)()))))))
5:
6:
Quantize the latent code via the hyperbolic codebook C:
7:
2g = arg mine dR, (z,c) c)
8:
Decode the quantized latent codes back to the original graph structure:
9:
G (2g)
10:
Update model parameters using the loss Eq. 40
11:
end for
E ADDITIONAL TECHNICAL DETAILS
E. REMANNIAN MANIFOLD
Riemannian manifold (M.g) is smooth manifold M equipped with an inner product Sp TrMM Tp-M + R on the tangent space M at each point p E M. which varies smoothly with The inner product y is called the Riemannian metric and induces a notion of distance and angle on the manifold.
Given point p E M and a tangent vector e M,.M, the exponential map exp, TT.M + M maps to the point reached by traveling along the geodesic starting in the direction of for unit time. Conversely, the logarithmic map logp .M- - T.MMM when defined, maps point & E .M back to the tangent vector corresponding to the initial velocity of the geodesic from q. These allow the manifold geometry to be locally approximated using linear operations in the tangent space.
maps
E.2 HYPERBOLIC GEOMETRY
Let (M,g) g) be complete, simply connected Riemannian manifold of constant sectional curvature R < 0. Then (M, g) is called the n-dimensional hyperbolic space with curvature バi denoted H Up isometry, there exists a unique such space for each dimension and curvature k. making hyperbolic space a canonical example of a negatively curved space form in Riemannian geometry.
In practice, several coordinate models commonly used to represent this manifold in different ambient geometries. These include the Lorentz model, the Poincaré ball model, and the Beltrami- Klein model. Although their coordinate expressions differ. these models are all isometric one another and describe the same underlying hyperbolic space. Closed-form mappings exist between the models, allowing flexibility in geometric computation.
We summarize below the Riemannian metrics associated with each of these models.
Lorentz Model.
This model is realized as hypersurface embedded in dim=)s-iieasi Minkowski space 1R equipped with the Lorentzian inner product:
(x,y)c T scy, where Oc diagg-11 1,,,,,,,,).
Poincaré Ball Model. Defined on the open ball B = {x E 18" |||x|| < 1} and here - Poincaré model uses conformal metrie:
2
xxx
XLI, with Ax 1- c||x||2
Beltrami-Klein Model. Also defined the open ball, the Beltrami-Klein model uses non-conformal metric. Unlike the Poincaré model, it does not preserve angles, but has the advantage that geodesics are represented straight lines within the Euclidean ball:
cx (1 -
c -x=x Txxx
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These models provide flexible choices for representing hyperbolic geometry depending the task hand, including optimization, inference, and geometric embedding in machine learning applications. Poincaré ball here is preferred in graph embedding because it preserves local angles (conformal), naturally encodes hierarchy near the boundary, and supports efficient optimization in a bounded Euclidean space.
E.3 GRAPHS HYPERBOLIC SPACE
Previous works Sarkar 2011 Fu et al. 2024 have suggested that graph features are well-suited for embedding in hyperbolic space due to tree-like structure Coornaert al. (2006).
Theorem (de Gromov's 's approximation theorem) For any 8- 5-hyperbolic space, there exists contin- wous mupping from the spuce to an R-Tree such that the distances between points are approximately preserved with small error term. More formally, the mapping T X - ' satisfies the following distance preservation property:
S yl. [x - yl - 228 S (2(x) 2(v)1
where X is the ô-hyperbolic space, is the R-Tree, and is the number of sampled points in the mapping.
E.4
WHY WE CHOOSE THE POINCARÉ BALL MODEL.
Theoretical Motivation We selected the Poincaré Ball model over the Lorentz model for two primary reasons:
Numerical Stability: The Poincaré model is defined bounded domain (B2), which prevents latent variable norms from diverging and is inherently more stable for floating-pon point arithmetic. In contrast, the Lorentz model is unbounded and prone numerical precision issues in high dimensions, especially for points far from the origin.
Geometric Properties: The Poincaré model is conformal, meaning it preserves local angles. This property benefits learning tasks where local structure is important.
Practical Comparison Furthermore, we conducted a geometric round-trip experiment to compare the Poincaré and Lorentz models. Two points To, were sampled from wrapped normal distribution. We then computed the recovered point
2 = = (log.n (x1))
and measured both the Euclidean L2 error |||| x ll and the manifold distance do( æ1), which ideally should be zero. Results are reported as mean standard deviation over 10 runs.
Table 5: Round-trip numerical stability comparison between the Poincaré Ball and Lorentz models.
Model
Dim
L2 Error (Mean Std)
Manifold Error (Mean Std)
Poincaré Ball
5.59
(4.83 + 3.33)c 15
Lorentz
8
3.624
(1.4le-4 + 1.53e-11) 11) (4.66 e-59)e-15
Poincaré Ball
32
(8.25 + 4.01je-1
Lorentz
32
128.28
0.72 = 0.70 (4.01 2.82)e-15
Poincaré Ball
64
(8.86 + 5.90)e-16
Lorentz
64
2958.63 1546.06
5.95 + 0.84
Table 5 shows that the Poincaré Ball model consistently achieves round-trip errors on the order of machine precision (~ 10-15), demonstrating near-perfect numerical reversibility across all dimensions. In contrast, the Lorentz model's error explodes catastrophically higher dimensions. For example, at 64 dimensions, the mean L.2 error exceeds 2900 with large variance, indicating complete numerical breakdown.
This provides empirical evidence that the Lorentz model is unsuitable for high-dimensional deep learning applications, validating our choice of the Poincaré Ball model for its robustness and numerical stability.
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E.5
BASIC OPERATIONS ON POINCARÉ BALL
E.5.1 MÓBIUS ADDITION AND TRANSFORMATIONS
Möbius addition is defined on Poincaré ball as follows:
x Be y (1 + 2c(xx, y) + c x + (1 c||=]1
(10)
1 1 20 3) +
which reverse calculation is defined by
O - x 0e (-
(11)
This operation is closed on D and defines non-associative algebraic structure known gy- rogroup.
Unlike Euclidean addition, Möbius addition is not associative. Instead, it satisfies a weaker form of associativity known as gyroassociativity:
(v O₂ w) (u Be v) Be gyr(u, v)w),
Accordingly, when considering transformations on the Poincaré disk, it is important to account for the underlying aarosssociaiive gyrogroup structure. Throughout this work, all Möbius additions consistently defined as left additions, which naturally gives rise to operations where composing with the inverse yields the identity transformation.
E.5.2 LOGARITHM MAP, EXPONENTIAL MAP, AND GEODESIC DISTANCE
The exponential map expx. T. B. is described in as follows:
1
(elilla) [w], Vo E B E T.
expe (v) O
tanh
(12)
Ve
2
The logarithmic map log = (exp:) B TBB. provides the inverse operation:
2
log (y)
tanh-
(Vell On x C. ull) (Dar me v), Va,, y E Bg
(13)
Vc
Geodesic distance between E B is defined as:
2 de (x. y) tanh (Voll Dx al) log (4)
(14)
Despite the noncommutative aspect of the Möbius addition @c, this distance function becomes commutative thanks the commutative aspect of the Euclidean norm of the Möbius addition.
E.6 HYPERBOLIC NETWORK BACKBONE
Prior efforts to construct hyperbolic neural networks predominantly leveraged the Lorentz model, yet its numerical instability (particularly the non-invertibility of exponential and logarithmic maps in high dimensions) limits scalability. In contrast, the Poincaré ball model offers superior numerical robustness and conformal structure, making a pragmatic foundation for building deep architectures. Building these insights, we unify and extend hyperbolic machine learning by developing comprehensive suite of foundational components and mainstream network modules within the Poincaré framework.
E. POINCARÉ MULTINOMIAL LOGISTIC REGRESSION
Hyperbolic hyperplanes generalize Euclidean decision boundaries by constructing sets of geodesics orthogonal to a tangent vector a E T, : {0} at a base point p. However, traditional formulations suffer from over wer-parameterizzatoo of the hyperplane. [Shimizu et al. (2020) address this by proposing unidirectional hyperplanes, where the base point qı: is constrained as qı = expα(ra (ax]) using a
scalar bias r IR. This reduces redundancy and stabilizes optimization. Building this, hyperbolic
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multinomial logistic regression (MLR) generalizes Euclidean MLR by measuring distances margin hyperplanes. For K classes, the unidirectional hyperbolic MLR formulation for all a E B. is defined as:
Ui - ply = æ) x sign ((-qs O. æ, az)) ||ax|de (xx, Hag.r))
(15)
where de (x, H Ha.. denotes the hyperbolic distance from æ to the k-th unidirectional hyperplane. aş and r3 are learnable parameters. This formulation preserves the interpretability of Euclidean margins while leveraging the exponential growth of hyperbolic space for hierarchical classification.
Poincaré Fully Connected layer. The Poincaré fully connected (FC) layer generalizes Euclidean affine transformations to hyperbolic space while preserving geometric consistency. Following adopt reparameterized formulation that circumvents the parameter
Shimizu et al. (2020), redundancy inherent in earlier hyperbolic linear layers (Ganea et al. 2018). For an input x E B the layer computes the output y |B as:
y y= F (x;; r,r) == w01 / + +|//l)))) where w :- (c sinh (Vev (x))
(16)
where (x) denotes the signed hyperbolic distance from æ to the unidirectional hyperplane H
which is described in Eq. [5| derived via parallel transport and Möbius addition (Shimizu et al. (020).
This formulation ensures that each output dimension corresponds a geometrically interpretable distance metric in B aligning with the intrinsic curvature of the manifold.
Poincaré Layernorm. We introduce Poincaré LayerNorm, novel normalization module that stabilizes training in hyperbolic networks while preserving geometric integrity. Unlike Euclidean LayerNorm, which operates directly on manifold coordinates, our method leverages the logarithmic map to project hyperbolic features x E IB to the tangent space at the origin To Here, standard LayerNorm is applied to the paclidcan-flattenn the normalized result back to Br. Formally, for input æ::
features, followed by an exponential map to reproject
Enorm = (Layeraorno (z))),
(17)
where weights and biases in the LorerNorm submodule are initialized to minimize distortion. This approach decouples normalization from manifold curvature, ensuring stable gradient propagation without violating hyperbolic geometry.
Poincaré ResNet. For deep hyperbolic architectures, we adapt residual connections mitigate vanishing gradients, inspired by Chami et al. (2019). Given hyperbolic feature E IB and nonlinear transformation F(x)), the Poncaré ResNet block computes:
x DD F(x),
(18)
E.6.2 POINCARÉ GRAPH NEURAL NETWORKS
We introduce a Poincaré ball-based graph neural network (GNN) that generalizes message passing to hierarchical graph structures. Unlike Lorentz-based approaches that operate on hyperboloid coordinates, our architecture leverages the conformal structure of the Poincaré ball enable stable gradient propagation and efficient hyperbolic-lineat operations. Let Hx {h; E B J denote node features and Hg = [hy E B (t.))ee edge features. Our architecture cooperates via two geometrically consistent phases:
Hyperbolic Message Passing and Edge Update. For each edge (i,j), compute adaptive shift and scale parameters from the hyperbolic distance (h!, ht):
Sl..Bt Linear(dech!,. h2))
(19)
where Linear d + R2ad ensures acale-awareness. Project hT, hy and heg to To via (o) concatenate, and modulate to update the message:
m ++
Σ
AdaLN (We [log] (h(), log,( log (h!)) 3/))
(20)
JEN(()
where W. E 3nxxdd is Euclidean linear layer. We can get new edge representation by pulling the message back up to the Poincaré manifold.
h
[+1 expo (ml)) 11)
(21)
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Hyperbolic Node Aggregation. For node i, we have aggregated messages m, 1+1 from neighbors (i) in Eq. 20 Similarly, compute adaptive shift and scale parameters v S from the message and update node features;
h expé (ho) + AdaLN (W₂ (kog)(h)) logō
++1 ,,,))))))))))))) ))
(22)
with We 2 as Euclidean linear layer.
E.6.3 POINCARÉ TRANSFORMERS
We present a hyperbolic transformer architecture that extends self-attention mechanisms to the Poincaré ball model while preserving geometric consistency. Given the hidden feature H = B" the transformer first injects positional information via Möbius addition of learnable hyper- bolic embeddings. The sequence then passes through L identical layers, which containing Poincaré Multi-Head Attention and Poincaré Feed-Forward Network. Both components employ residual connections with Möhius addition and Poincaré LayerNorm.
Poincaré Attention Mechanisms. The attention mechanism generalizes scaled dot- dot-product atten- tion hyperbolic space through three stages:
Projection & Splitting. Inputs are projected into queries Q, keys K, and values V via Poincaré FC layers (Eq. 15). As described in Shimizu et al. (020),, B-splitting to split each tensor into heads:
Qi K, Vi Split, (Q), Split (p)i where Split, scales dimensions via beta functions.
(23)
Geodesic Attention Score. Attention weights computed using hyperbolic distance maintain the hyperbolic geometry structure, which can be formulated as:
azj softmax Q; K.) 7)
(24)
where T is an inverse temperature and y is bias parameter, which was proposed by Gulcehre et al. (2019). After applying a similarity function and obtaining each weight, the values are aggregated by Möbius gyromidpoint:
( Ej
Z:
Σ V asl. @c E, Lang 11
(25)
the symbol
which is described 8. æ 1+ 1+V1-cle11
Ungar 20088
Multi-Head Composition. Head outputs {Z.} h combined via @-concatenation and out projection:
0 Concat, 1Z .... Zn.) @ Wo.
(26)
where Wo is a Poincaré linear projection and Concat, inversely scales beta ratios from splitting.
Time-aware Transformer Layers. To integrate temporal dynamics into hyperbolic diffusion processes, we extend the Poincaré transformer with adaptive time-conditioned modulation. Each layer injects timestep embeddings temb IR through Fuclidean affine transformations of normalized features. Following Euclidean Diffusion Transformer (Peebles & Xie 2023), we regress time- conditioned gate ayy, shift B.z. and scale Yh parameters from tembi
Lutinn Blattn Vaten (' f/ Ty3n. = Linear(temb)
(27)
where Linear Raxd - 6 is Euclidean linear layer. These parameters drive AdaLN modulation mechanism and modulated feature representation hnew is computed as:
knew o Aataa (h) 7,,10)
PoincareAttennioon
(28)
where AdaLN(.) is the similar function in Diffusion Transformer (Peebles & Xie 2023). Subse- quently, knew processed similarly in the Poincaré Feed-Forward Networks, forming complete Time-aware Transformer Layer, and we use L Time-aware Transformer Layers to construct Poincaré Diffusion Transformer used in Section 3.3
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E.7 HYPERBOLIC AUTOENCODER FOR GRAPH
E.7.1 HYPERBOLIC GRAPH AUTOENCODER (HGAB)
Graph autoencoders are unsupervised learning methods designed to learn low-dimensional representa- tions of graphs. In hyperbolic space, graph autoencoders can more effectively capture the hierarchical structure of graphs. Our Hyperbolic Graph Autoencoder (HGAE) consists of a hyperbolic encoder and a hyperbolic decoder. The encoder maps graph nodes to points hyperbolic space, while decoder reconstructs the adjacency matrix of the graph.
Graph Encoder. The encoder first incorporates extra structural and spectral graph fea- tures eigeeeiigessssors and eigenvalues of the graph Laplacian- enrich global topological awareness. Such features boost the expressive capacity of graph neural networks and are helpful for training powerful autoencoder structure (Beaini et al., 2021; Vignac al. 2023). Since these features not exposed to the graph generation process, we could theoretically construct any features that can increase the expressiveness of graphs. These features undergo initial Euclidean preprocessing via multilayer perceptrons (MLPs), Ps), ensuring effective fusion of multidimensional attributes. The result is mapped onto the Poincaré ball with an origin- exponential map:
X expo(MLP((X) SJ)),
E, exPo(MMP(O)
(29)
Subsequent hyperbolic graph neural network (GNN) layers aggregate localized structural patterns, while L hyperbolic Transformer layers propagate information globally, yielding final hyperbolic embeddings z, = h E HD that preserve multiscale structural semantics.
h 4+1 = Attentiong (h*),
h GNNG EA))
(30)
Probabilistic Graph Decoder. The decoder architecture operates under the conditional indepen- dence assumption, where node attributes depend solely on their respective embeddings, while edge connectivity and features conditionally dependent node pairs. This factorization yields the joint reconstruction probability:
Pe X,
E
H
Pa [x:
h;)
Po b h2)
(31)
We first feed the latent repsentations z; into a 2 layer Poincaré self attention model, to allow full interaction between nodes, and obtain node-level representations h, For attribute reconstruction, we first project the hyperbolic feature on to TolBy, and then apply Euclidean MI.P-based prediction head followed by Softmax distribution.
(x h.) SoftmaxMLP, (log)(h:))
(32)
Edge probabilities are derived from a comprehensive set of hyperbolic geometric features:
loog (b,), (h;), logí, dhy),d(h,, h;), cos 0.y)
(33)
where the log-map vectors represents the directional relationship in the tangent space, d. (h; h2) com- putes the explicit hyperbolic distance encoding hierarchical relationships, and 0gy = (0, h4, hy) captures relative angular positions The resulting R3D+2 feature vector is processed through an MLP with softmax output to predict:
Ps (e h, h,) = Softmax (MLP (fr))
(34)
By jointly optimizing feature and topological reconstruction under hyper- bolic geometric constraints, the model ensures latent embeddings retain both structural hierarchy and coributte-ddde relationships, outperforming conventional Euclidean counterparts in preserving complex graph semantics.
Reconstruction Loss.
LAE Anode Loaite
Aedge Ledge + Arey Lreg:
(35)
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n
d1
mode (X, (bix.
Tie log logPo(Tic hj).
(36)
16
Cdge
B)xD En>po(b|x.F)
Cije log h hy).
(37)
==1
With edge reconstruction loss, node reconstruction loss, and regularization loss, and lambda being hyperparameters..
Training instability remains a fundamental challenge in hyperbolic neural networks, primarily due to the exponential growth of hyperbolic distance points away from the origin toward the Poincaré ball boundary. Empirical observations reveal that embeddings with small norms (e.g., below 0.1) exhibit Euclidann behavior, while those approaching the boundary (norms 0.8) frequently suffer from gradient instability during optimization. To mitigate this issue, we adopt an adaptive regularization approach inspired by recent advances in clipped hyperbolic representations. Unlike hard constraints that strictly enforce norm boundaries, we introduce band loss composed of two hinge loss terms, which softly encourages hyperbolic embeddings h; remain within a stable intermediate region.
1 n
Lband
Σ 7 i=1
(max(0, ||z,|| - Tmax})) (max((, Tmin ||]
(38)
Where we set the inner margin Imin 0.2, and the outer margin Imma 0.8
E.7.2 DEGREE-EDGE CONSISTENCY Loss
To ensure structural coherence between node-level predictions and edge-level decoding, we introduce a degree-edge consistency loss. This loss does not attempt to match the global degree histogram; instead, it enforces local consistency for each node between the expected degree implied by the edge decoder, (ii) the ground-truie degree, and (iii) the degree predicted by the node-level classifier.
For node let
N
de
edge
Σ
Poleis hh, h1)
denote the expected degree induced by the edge probabilities, and let
K
d
pred
Elde
h:]
Σ
pooeee = k ha)
A=O
denote the expected degree predicted by the node-degree classifier. The full degree-edge consistency loss is then defined as:
Laggre
112
Ilddd (prodd1)).
This term encourages the local comnectivity implied by the edge decoder to agree with both the true degree and the internal node-level prediction. Such consistency is particularly beneficial for abstract graph datasets, where degree patterns reflect structural roles and strongly correlate with motif and orbit statistics.
E.7.3 HYPERBOLIC VARIATIONAL AUTOENCODER (HVAE)
Variational Auto-Encoders augment a classic auto-encoder with a probabilistic KL penalty, helping to reduce the high-variance in latent space and enable sampling. In hyperbolic space H the Euclidean Gaussian is no longer appropriate, so we adopt the Wrapped Normal distribution (Mathieu et al. 2019 Grattarola al. 2019. Nagano al. 2019). for both prior and posterior:
p(z,) (O,
(s(z; X, F)) Nu((ss(x), Ex(x))
(39)
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where PoincareLinear (h,) E Ba, Σ = Softplus (Linear (logō (h:)) E RR. We choose the prior standard deviation be isotropic and renormalized by the square root of the dimension of hyperbolic space, Σ dimensionality increases.
Vd I, to ensure sampled points don't get pushed to the boundary as
Reconstruction Loss. minimising LVAB 1o
HVAE is trained by maximising the evidence lower bound, or equivalently log (x e
+/3 [T] x, e) p(z)), where the lambda term
Reconstruction
KL.
controls the eeconstructionngeeeaatee trade-off (Higgins et al. for these distributions are unavailable, we compute the KI. term via Monte Carlo estimation.
2017). Since closed-form solutions
We have also implemented a hyperbolic variational autoencoder (HVAE), using the wrapped normal distribution as the prior in latent space, and tested the reconstruction performance. However, we observe that sampling in high-dimensional hyperbolic space is highly unstable, and Monte Carlo esti- mation of the KL divergence often leads to numerical instability. As result, the HVAE significantly underperforms in both reconstruction and generation tasks compared to HAE and HVQVAE.
E.7.4 HYPERBOLIC VECTOR QUANTIZED VARIATIONAL AUTOENCODER (HVQVAE)
We introduce hyperbolic vector quantized variational autoencoder (HVQVAE) that discretizes latent representations while preserving hierarchical structure.
Hyperbolic Codebook. We propose novel hyperbolic codebook that resides entirely within the Poincaré disk, ensuring all operations remain consistent with the hyperbolic geometry.
Codebook Initialization. The hyperbolic codebook C B. is initialized via manifold-awars random sampling optimized through hyperbolic k-means clustering. This algorithm uses initial vectors are drawn from the Poincaré neural network, ensuring uniform coverage across hierarchical scales and minimizes the sum of squared geodesic distance, which is defined as Eq. [4] The algorithm terminates when the centroid movement after 10 iterations (Alg. 16). We choose random initialization which achieves slightly better results, while K-means initialization method often offers faster convergence,
Quantization Method. In the Poincaré VQVAE, quantization maps an input embedding Z B% to the nearest codebook vector using the hyperbolic geodesic distance. Formally, the quantized embedding By obtained by selecting the codebook vector c; that minimizes the geodesic distance.
Loss Function.
The training objective combines three geometrically consistent components:
LHvOVAE A Ep log pe (x, xn)]
+\2 EEg (sg( (,,) =]] + [g[d sg(z)))
(40)
Reconstruction
Commitncot
Consistency
where reconstruction loss is same VAE; commitment loss anchors latent codes to quantized vectors, and Codebook loss updates embeddings via straight-through gradient estimation sg(-). It is worth noting that since the hyperbolic codebook in hyperbolic space and the model parameters are in Euclidean space, we use both Riemannian optimizer and traditional optimizer, and the same is true for the positional encoding in Transformer.
Hyperbolic Exponential Moving Average. To stabilize training and avoid mode collapse, incorporate Hyperbolic Exponential Moving Average (EMA) for codebook updates.
EMA Update. Codebook vectors Cj E 1 update via Riemannian EMA to balance historical states and new features. We compute a new centroid μj from assigned samples z; with weights Wij = I(2, - c,), which is an indicator function that assigns weight of 1 when the sample z; is mapped to the codebook vector cj and otherwise:
N Wij X
1 8c
(41)
i=
Wey
(X - 1)
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Similarly, update codebook vectors via Einstein midpoint (Ungar, hypernetworks++)::
(t+1) q
= projs ([o8 Be [mg. - -
(42)
where the projag is projection in Poincaré ball, 3 is custom decay, and the weighted midpoint operation is defined as:
w
[x, w/e
(43)
ow
To avoid the problem of inactive codebook entries, we implement expira- tion mechanism. If a codebook entry is not frequently updated, it is replaced with a new codebook vector. This mechanism ensures that the codebook remains active and capable of efficiently represent- ing the latent space, thereby preventing collapse during training.
Dead Codes Revival.
This hyperbolic codebook design, paired with tailored loss function and updating method, enables the Poincaré VQVAE to effectively model complex hierarchical data while preserving the manifold's intrinsic properties, offering a robust foundation for discrete representation learning in hyperbolic spaces.
E.8 AUTOREGSSIVE MODEL FOR LATENT DISTRIBUTION
Autoregressive modeling in the latent space has become common approach, especially for generating high-fidelity images (Li et al. 2024 He et al. 2022 Razavi et al. 2019). Since we have already trained a VQ model, we perform autoregressive learning over the discrete latent tokens. During training, causal mask is applied to ensure that the model only attends past tokens, with Poincare Transformer backbone model. During generation, we compute the geodesic distance between the predicted next token embedding and all entries in the codebook, and apply softmax sampling select the next token.
(44)
E.9 FLOW MATCHING
Let IRd denote the data space where the data samples x, E IRd, The variable t E [0. 1] represents the inference time, where P1 (11) is the target distribution we aim generate, and (o( a base distribution that is easy to sample from.
Flow-based generative models, as introduced by Chen al. 20188 and Lipman et al. (2023, define a time-van varying vector field ny (2z) that generates probability path (2t), transitioning between the base distribution Pa (aro and the target distribution p1 (1r). By first sampling sy from po (rro) and then solving the ordinary differential equation: dt Tr (xx)). Specifically, we define loss function
that matches the model's vector field 2 (xr) to the target vector field u, (z)):
E E,~U(0 B.-0T(..) [mv;( 0) ue 2xx)))
However, in practice, we cannot explicitly obtain the probability path (x())) or the vector field vz(z),, and thus, used approach in practice Conditional Flow Matching (CFM), which provides a practical way to approximate these elements and facilitate the generation process. CFM defines the conditional vector field Uzjz (z+l2) to obtain the conditional probability path p(>(01). By marginalizing this conditional vector field and conditional probability path, we obtain the marginal vector field and marginal probability path. Specifically, the marginal vector field uy (xxx) is defined as:
unable directly compute these quantities. Instead, a more commonly
vz(xx)) : f map (1le(ard))))
where p(z(2z)
pل(x|)))p(a) which generates the marginal probability path Pa(xa) n(x))
J pr (x)12))(()dd Thus, the marginal vector field governs the generation the marginal prob- ability path.
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The CFM loss is given by the following formula:
Be..it0))..-p(h....aaaaal [1 () -ej (Ir [=]l=
It can be shown that this loss is equivalent to the FM loss (Lppman et al. 2023
Chen & Lipman 2024) extends this principle to Riemannian manifolds by ensuring that all vector fields lie in the tangent space Mz, and distances are measured using the Riemannian metric g. The
loss becomes:
t) - us (21 zo)1l
E.10 TRAINING DETAILS
We primarily selected the hyperbolic VQVAE the base autoencoder model. Subsequently, conducted two-stage training for the autoregressive model and the flow matching model to enhance the generation performance. Similar to the StableDiffusion (Rombach et al. 2022). chose to combine the quantizer and the decoder, which applies on the latent representation. It is worth noting that for our model, we employed the optimizer Adam @Loshchilov Hutter (2017) for the Euclidean parameters, and the Riemannian optimizer RiemannianAdam (Kochurov et al. 20020) for all Riemannian parameters, which include the codebook and the positional embedding in the transformer. Most of the other operations are carried out in the Euclidean space, and they mapped to different spaces through the exponential map and the log map.
We provide the number of layers and latent space dimensions used in models in Table 6 All models are trained on a single A6000 GPU. On the QM9 dataset, the vector-quantized autoencoder trained for approximately 6 hours, followed by 12 hours of flow training. For the COMM20 and Ego-Small datasets, the autoencoder training takes about 2 hours, and flow training takes around hours. Training details provided in Table 7 The loss weights
follows: the coefficients for node and edge reconstruction are both set to 5.0, the 1.2 regularization loss coefficient set to le-4, the commitment loss coefficient
to 0.25, and the VQ loss coefficient is set 1.0.
Table 6: Model Parameters across datasets.
Dataset
Comm-Small/Fgo-Smal
QM9
Hyperbolic Channels
64
128
Codebook Size
32
512
Encoder GNN Layers
2
Encoder Transformer Layers
8
8
Encoder Dropout
0.1
0.1
Decoder Transformer Layers
2
2
Decoder Dropout
0.1
0.1
Decoder Layers
2
2
Flow DiT Transformer Layers
Hyperparameter tuning. Most hyperparameters in implementation are fixed across all datasets; we do not perform dataset-specific grid search or tuning. Specifically:
Loss weights: coefficients (reconstruction loss, degree loss, VQ commitmen//consistencyy loss, regularization loss) are set to constant values for different dataset, chosen to balance magnitudes at initialization and ensure stable optimization.
Codebook size: selected via codebook perplexity monitoring during VQ-VAE training, to avoid codeword collapse or extreme fragmentation, which is a standard practice in VQ-based methods (Van Den Oord al. 2017).
Curvature of hyperbolic latent space: fixed to 0.1. We do not tune curvature per dataset, because a constant curvature offers stable geometry and avoids potential gradient instability or embedding degeneration.
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Empirically, this exed-hyperparamet design yields robust performance across diverse datasets (Comm20, Ego-Small, QM9) without per-dataset hyperparameter search.
Table 7: Optimizer settings across geometric parameter spaces and training stages.
Parameter Space
Parameter
Autoencoder
Flow
Optimizer
Riemannian Adam
Riemannian Adam
Learning Rate
0.005
0.005
Poincaré
Weight Decay
0
Betas
(0.9, 0.999)
(0.9, 0.999)
Grad Clipping
1.0
1.0
Optimizer
A
Adam W
Learning Rate
0.0005
0.0005
Euclidean
Weight Decay
lc-12 (0.9, 0.999)
1c-12 (0.9, 0.999)
Betas
Grad Clipping
1.0
1.0
F EXTRA EXPERIMENTS
F.1 VARIATIONAL AUTOENCODER
As shown Table & HVQVAE consistently outperforms both HAE and HVAE across all recon- struction metrics. While HVAE suffers from instability introduced by KI. regularization on curved manifolds, HVQVAE achieves both lower distributional discrepancics (e.g., degree and clustering MMD) and higher edge reconstruction accuracy. Notably, on Community-small HVQVAE reduces degree error by over 95% compared to HVAE, and improves edge accuracy to 99.39%. These results confirm the benefit of discrete latent representations in stabilizing learning and preserving topological structures in hyperbolic space.
Table 8: Reconstruction performance of our baseline models HAE, HVAE, and HVQVAE abstract graphs (Community-small Ego-small).
Community-small
Ego-small
Model
Deg. J
Clus.
Orb. 4
Edge Acc. ?
Deg. +
Clus. 4
Orb.
Edge Acc. 1
HAE
0.0008
0.0310
0.0007
99.10
0.0019
0,0250
0.0048
92.40
HVAE
0.0175
0.0910
0.2347
97.36
0.0050
0.0320
0.0390
89.82
HVQVAR
0.0004
0.0208
0.0005
99.39
0.0002
0.0194
0.0018
93.20
While variational autoencoders (VAEs) are natural choice for probabilistic modeling, we found that the KL divergence term becomes numerically unstable when optimized in hyperbolic space. As shown in Fig. 4 the KI. divergence fluctuates dramatically during training. This instability stems from the mismatch between the prior and posterior distributions in non-Fuclidean geometry, which lacks a tractable and behaved KL formulation. As result, the VAE often fails to converge to a coherent latent structure.
Moreover, despite moderate improvements reconstruction accuracy, the edge prediction per- formance of HVAE remains significantly inferior our proposed HVQVAE. These observations motivate our choice to adopt a discrete latent representation, which provides greater numerical stability and structural expressiveness.
In future work, we plan to explore alternative variational formulations and prior structures that may stabilize training and fully leverage the geometric inductive bias of hyperbolic space.
F.2 ADDITIONAL EXPERIMENT: ZINC250K
To further evaluate the scalability and generalization capability of our proposed framework on large- scale sparse graphs, extended our experiments to the ZINC250k dataset. ZINC250k consists of
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KL Divergence (train/kl_og loss)
Edge Reconstruction Accuracy
22.5
0.9
20.0
0.8
12.5
0.7
0.0
7.3
0.5
2000
4000
5000
B00G
10000
0
2000
4000
6000
8000
10000
Step Index
Step Index
Figure 4: Training dynamics of the hyperbolic VAE (HVAB). Left: KI. divergence exhibits large oscillations, indicating numerical instability. Right: Edge reconstruction accuracy improves slowly and remains suboptimal. These results highlight the limitations of using continuous probabilisticc models in curved latent spaces.
approximately 250,000 drug-like molecular graphs, representing a significantly larger and structurally more complex domain compared QM9 and Community-Sman This experiment serves verify the model's performance in modeling realistic, sparse graph structures at scale.
Table 9: Generation performance on the ZINC250K dataset. We report the mean and standard deviation for GGBall over 3 independent runs. indicates higher is better. The best results are highlighted in bold.
Model
Validity (%) 1
Uniqueness (%) f
Novelty (%) t
V.U.N 1
GraphAF MoFlow
68.00
99.10
100.00
67.38
81.76
99.99
100.00
81.75
GraphDF
89.03
99.20
100.00
88.32
EDP-GNN
82.97
100.00
100.00
82.97
GGBall (Ours)
83.50 = 0.98
100.00
0.00
98.88 + 1.01
83.48 + 1.58
Discussion. The quantitative results on ZINC250K are summarized Table GGBall achieves robust performance with Validity of 83.50% and V.U.N. score of 83.48, outperforming flow- based baselines such as GraphAF (67.38) and MoFlow (81.75), and performing comparably to the energy-based EDP-GNN (82.97). While GraphDF achieves higher validity rate, GGBall maintains 100% Uniqueness with high Novelty, indicating that our model can generate diverse and chemically valid structures without collapsing into repetitive patterns. These results confirm that the proposed hyperbolic framework generalizes effectively to large-scale real-world datasets.
F.3 ADDITIONAL EXPERIMENT: TREE-STRRR STRUCTURED GRAPHS
Tree graphs constitute prototypical example of hierarchical structure and well aligned with the inductive bias of hyperbolic geometry. To assess how well our latent hyperbolic model captures such structure, we trained HVQVAE on the synthetic Tree dataset (Bergmeister al. 2023 following the same protocol as in the main experiments.
Results.
The hyperbolic autoencoder reconstructs trees with high accuracy across all structure-
sensitive metrics.
Model
Degree-MMD +
Clustering-MMA 1
Orbit-MMD +
HVQVAE (ours)
7.9 10-4
2.6 1 2
1.0 -0-3
Discussion. These results indicate that the hyperbolic latent space provides the correct geometric bias for modeling hierarchical connectivity. The extremely low Degree- and OrD values
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show that HVQVAE recovers the branching patterns and local motif statisties of trees with near- perfect accuracy. This supports our claim that hyperbolic latent structure is especially well suited for hierarchical graphs.
F.4 ADDITIONAL EXPERIMENT: EUCLIDEAN ABLATION
We performed an ablation by replacing our full HVQVAE + hyperbolic-latent framework with a fully Euclidean counterpart (i.e. Euclidean GCN Euclidean VQ- VA-VAE: + Euclidean Transformer). This isolates the effect of hyperbolic geometry while keeping network architecture, codebook, and training procedure unchanged. The results (on Community-small and Ego-Small) are summarized below:
Table 10: Reconstruction Performance: Hyperbolic vs. Euclidean Parametrization
Model
Comm20
Ego-Small
Deg. |
Clus.
Orb. L
Deg. +
Clus. 4
Orb.
Hyperbolic AE
0.0008
0.0310
0.0007
0.0019
0.025
0.0048
Hyperbolic VQVAE
0.0004
0.0208
0.0005
0.0002
0.0194
0.0018
Euclidean VQVAE
0.0457
0.1344
0.7555
0.006
0.0815
0.0205
On Comm20, Fuclidean VQ-VAE yields larger Orbit-MMD (0.76) and lower structural fidelity, whereas our hyperbolic HVQVAE obtains near-zero Orbit-MMD and extremely low degrec/clustering discrepancy. This dramatic gap suggests that the gains are due network architecture or codebook quantization per se, but indeed arise from the inductive bias of hyperbolic geometry.
When replacing the Poincaré parametrization with Euclidean space, the model largely loses the ability to compactly and faithfully represent hierarchical or the hyperbolic latent space is the main driver of performance improvements.
graphs, confirming that
F.5 STABILITY ANALYSIS VIA MEAN/STD REPORTING
To assess the robustness of our method beyond single-run performance, we report the mean and standard deviation of reconstruction metrics across 3 random seeds. As shown in Table [1] the hyperbolic VQVAE consistently achieves low reconstruction error and high edge accuracy with only minor fluctuations across runs, indicating that the model behaves stably under different initializations.
On the abstract graph datasets (Ego-Smmal and Community-Smal), degree, clustering, and orbit MMD values remain tightly concentrated, and reconstruction accuracy remains above 98% across seeds. For the molecular datasets (QM9 and ZINC250K), validity. uniqueness, and novelty also show narrow confidence ranges, demonstrating stable behavior in both reconstruction and generation.
Overall, the small variance observed across all datasets confirms that the performance gains reported in the main text are consistent and reproducible, rather than the result of favorable random seed or training instability.
Table 11: Reconstruction and generation stability across multiple seeds (Mean + Std).
Dataset
Deg. |
Clus. |
Orb. +
Edge Acc.
Ego-Small
0.0012 + 0.0007
0.0237 + 0.0006
0.0017 0.0003
99.03+0.33%
Community-Small
0.0008 + 0.0012
0.0215 + 0.0012
0.0009
0.0010
98.49+1.12%
Validity
Unique 1
Novelty 1
Edge Acc.
QM9
97.42 + 1.00
98.96 + 0.46
93.55 + 2.70
99.21 +0.30%
ZINC
90.01 + 2.36
100.00 + 0.00
99.47 + 0.32
99.39
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G EFFICIENCY ANALYSIS
We conduct a targeted efficiency analysis to quantify the practical benefits of our architectural design. Unlike dual-space diffusion models such as HGDM, which require iterative denoising over both nodes and edges with separate networks, GGBall employs unified hyperbolic latent space for nodes, from which the entire graph is decoded. This design avoids the computational overhead and complexity of explicit, iterative edge modeling. We compare inference performance community graphs and batch size 128, with results summarized Table 12
Table 12: Inference efficiency comparison on Community-smme (N=18) with a batch size of 128. Models marked with * are the primary configurations used in the main generation benchmarks. Lower better for memory and time.
Model (Inference Steps)
Params (M)
GPU Memory (MB) 4
Time/Batch (s)
HGDM (100 steps)
0.18
342.3
13.49
HGDM (10 steps)
0.18
342.3
9.03
GGBall (100 steps)*
1.64
151.8
12.45
GGBall (10 steps)
1.64
151.8
1.40
(1 steps)
0.64
110.8
0.04
Table 12 highlights GGBall's significant advantages in two key areas. First, is remarkably memory- efficient. Despite having approximately 9 times more parameters in its diffusion configuration, our model uses less than half the GPU memory of HGDM (151.8 MB vs. 342.3 MB). This stems directly from unified architecture, which eliminates the need for separate, nemory-intensiie edge-denoision network and its associated intermediate activations.
Second, GGBall offers superior inference speed and flexibility. Our benchmark model (GGBall*) already faster than a comparable HGDM configuration with the same number of steps. More importantly, framework uniquely supports one-shot generation mode (GGBall-Oneshol) caaaeoteccsatabbele of decoding an entire batch of graphs in mere 0.04 seconds. This near-instantaneous generation capability stands in stark contrast the iterative nature of diffusion models, unlocking possibilities for applications requiring real-time or large-scale graph deployment.
H ADDITIONAL DISCUSSION: HIGHER-ORDER STRUCTURAL DEPENDENCIES
Although our decoder factorizes the likelihood into node-wise and pairwise edge terms for computa- tional efficiency, this design does not prevent the model from capturing rich higher-order dependencies. The latent hyperbolic Transformer aggregates global information through attention layers, producing latent embeddings that already encode hierarchical, long-range, and motif-level patterns prior to decoding.
Furthermore, the framework naturally supports extensions to explicit higher-order decoding, For example, one may incorporate a triangle- consssseens objective by matching the expected triangle count of each node:
pred Σ
Pij Pik PjR
to
gt
3.k
where Piy denotes the predicted edge probability. Alternatively, a triadic potential module can be added:
(A) MLP((lo((za) log(2,)) log(#:))),
whose outputs can be integrated into the pairwise logits Siy to provide motif-aware corrections.
Importantly, these extensions require change the encoder, hyperbolic latent geometry, or the VQ representation. The architecture is therefore fully compatible with k-wiise hyperbolic potentials and higher-order structural modeling. We view explicit motif-aware hyperbolic decoders as promising direction for future work, and our current design is intentionally structured to accommodate such extensions seamlessly.
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I ADDITIONAL DISCUSSION: HEAD-TO-HEAD COMPARISON WITH HGDM
is helpful to explicitly contrast our latent-space design with that of HGDM (Wen al. 2024 to clarify where the novelty and trade-off's lie with other hyperbolic graph generation methods.
HGDM performs diffusion in a hybrid latent space. Specifically, node features X₃ evolve in a hyperbolic space Ig, edge features A, in Euclidean space. Both are separately updated during generation using two independent score networks:
dX.- (X+, g.. V xx, Pe
(egPe(x,,) (x,, A.)] dt + 91,t dui
dA, - (A., 92,r V A, log Ax)] 92, dx02
In contrast, GGBall maps the entire graph G (X, A) into unified, node-only representa- tion in hyperbolic latent space. Formally, we embed the whole graph into node embeddings Z (BB)), where N is the number of nodes and B is the Poincaré ball.
This representation is both compact and information-rich: edge connectivity is instead implicitly encoded node embeddings.
During generation, we only sample and evolve these node embeddings inside By and decoder reconstructs the full graph from the node-only representation. This makes GGBall a truly hyperbolic latent-space generation model.
J SAMPLING STRATEGY
In large-scale graphs, naïvely decoding all pairwise edges leads to (nn²) complexity, and without constraints, the number of sampled edges can grow exponentially with the number address this challenge, we introduce principled strategy to control edge sparsity by aligning the predicted number of edges with that of real graphs.
of nodes. To
Temperature-Contten Sampling. At generation time, avoid thee-threshold binarization uncalibrated Multinomial sampling, which can lead to dense or overly sparse graphs. Instead, we introduce learnable temperature parameter to scale the edge logits:
Pij = (s:s///)
(45)
where 8ty is the raw logits for edge (). The temperature is adjusted gradually on the validation set, such that the expected number of sampled edges, Eie Pij. approximates the desired edge count. This provides dynamic mechanism modulate edge density function of graph size and latent structure.
Together, this strategy ensures that the number of generated edges respects the dataset feature, rather than quadratically with the number nodes, preserving both scalability and fidelity to real-world graph sparsity.
K
STATISTICAL METRICS
K.1 GENERIC GRAPH CENERATION AND RECONSTRUCTION
To assess the structural fidelity generated graphs, we compare the distributions of key graph statistics with those from the reference dataset, including:
Degree distribution, reflecting node connectivity,
Clustering coefficient, measuring local triangle density,
Orbit counts, capturing subgraph motif frequencies.
We compute the Maximum Mean Discrepancy (MMD) between the empirical distributions of these statistics for generated and ground truth graphs. The MMD is computed with a Gaussian kernel:
MMD E[k(G, G')] + (H, H') 2E[K(G, H)),
where G, G are generated graphs and H, H' are reference graphs.
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K.2 MOLECULAR GRAPH GENERATION
We evaluate the quality of generated graphs using four standard metrics: Validity, Uniqueness, Novelty, and Distributional Fidelity.
Validity measures the proportion of generated graphs that are structurally valid, i.e., they satisfy predefined syntactic or domain-specific constraints (e.g., chemical valency rules in molecular graphs). Formally, if n, graphs are generated and V of them are valid, then Validity = |V//ng.
Uniqueness evaluates the diversity of valid samples by computing the fraction of non- isomorphic graphs among the valid set. If C' is the set of valid graphs and set(C") the set of unique ones, then Uniqueness (s)
Novelty quantifies the generative model's ability produce new, unseen structures. It is defined as the proportion of valid unique graphs that are not present the training dataset.
Let Drain denote the training graph set, then Novelty 1 [set(C) n Drein // [ct(())).
Distributional Fidelity evaluates how closely the distribution of generated samples matches that of the real data. We use the Fréchet ChemNet Distance (FCD) (Preuer et al. 2018) as the metric, which computes the Fréchet distance between the feature distributions of generated and real molecules. Specfflill,,e each molecule is encoded using the penultimate layer of a pretrained ChemNet model. The resulting embeddings are assumed follow multivariate Gaussian distributions. Let (m, C*) and (ma,, Cw.) denote the mean and covariance of embeddings from generated and real molecules, respectively. The FCD defined as:
de
m mw 12 + Tr(C + Cw 2(CC)
which corresponds to the 2- 2assseesteiin distance. Lower FCD indicates that the generated samples are closer distribution to the real data.
K.3 EVALUATION PROTOCOLS
We follow the standard evaluation practices established in prior graph generation literatures (Vignac et al. 2023) Wen et al. 2024; |Boget et al.) 2023). For all datasets used in this work (COMM20, Ego-Small, and QM9), adopt the official train/validationntest splits released by these benchmarks to ensure comparability and reproducibility.
At time, we generate batch of graphs equal in size the set and compute the corresponding MMD-based structural metrics (degree-, clustering- and orbit-MMD) or chemical metrics (validity, uniqueness, novelty) depending on the dataset. All metrics are computed exactly following the implementations used in prior works to avoid discrepancies due to evaluation code.
Dataset Splits. For each abstract dataset, with the same seed from the original paper, we construct the vain//vlidation//ess partitions in three steps. First, we reserve 20% of all graphs as the test set. From the remaining graphs, we take 80% as the training set, and the remainder becomes the validation set. This results in approximate proportions of 64% training, 16% validation, and 20% testing. For the QM9 molecular dataset, we assign 100k molecules to the training set, 10% of the full dataset to the test set, and use the remaining molecules as the validation split.
Checkpoint Selection. To guarantee full reproducibility and prevent test leakage, all model selec- tion is performed using validation data:
VQ- stage:
We select the checkpoint with the lowest validation reconstruction structural divergence, measured as the sum MMD scores across degree, clustering, and orbit distributions, or the product of molecular validity, uniqueness, and novelty.
Flow Matching stage: We select the checkpoint achieving the lowest validation generative structural divergence.
This two-stage evaluation protocol follows standard practice and ensures that test metrics reflect the generative model's 's true performance rather than hyperparameter search or checkpoint selection on the test split.
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L LIMITATIONS & FUTURE WORK
While our current experiments focus on small to medium-scale graphs such as Community-Small Ego-Small, and QM9, key direction is to scale our method much larger graph domains (e.g. social networks, citation graphs, molecular datasets with thousands of atoms). It is not yet clear whether the inductive biases of hyperbolic latent geometry will continue to offer advantages at scale.
Another exciting direction is to explore mixettcurvature latent spaces. Recent work in knowledge graphs and representation learning has shown that embedding a single curvature space (pure hyperbolic or Euclidean) can be limiting when the data exhibits heterogeneous structural patterns.
For example, M2GNN embeds KGs product of different curvature spaces and shows performance gains in multi-relational reasoning tasks (Wang et al. (2021b). Also, more recent studies examine learning mixed geometry latent models (e-g. Yusupov et al. (2025) discuss expressive capacity of mixed geometry for embeddings). Such hybrid geometry paradigm may allow our model to adaptively allocate curvature to different substructures, providing both flexibility and structural fidelity.
Finally, we plan to validate our model in more application-oricnted downstream tasks, such as conditional molecule design (e.g. generating molecules with target properties), knowledge graph augmentation, or graph- based program synthesis. Extending our hyperbolic latent framework to such tasks would enhance its utility and demonstrate applicability beyond benchmarking.
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Reviews:
[{"review_id": "c0JreixLyD", "reviewer": "Reviewer_FJcZ", "summary": "The main goal of the paper is to develop a graph generative model that leverages hyperbolic geometry to naturally capture hierarchical and tree-like structures. Specifically, it aims to show that embedding graphs in the Poincaré ball and generating them through manifold-aware neural components (hyperbolic GNN, geodesic attention, HVQVAE, and Riemannian flow matching) can more effectively represent the relational geometry of graphs compared to traditional Euclidean models.", "questions": "Look in the Weaknesses Section. I am open to considering answers for concerns mentioned in the weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The work presents a coherent framework that combines hyperbolic graph neural networks, geodesic attention, vector quantization, and Riemannian flow matching in a single end-to-end model. This integration is technically nontrivial and demonstrates careful engineering of both discrete and continuous latent components.\n\n2. Paper is well written and maintains consistent notation throughout to follow.", "weaknesses": "1. Authors assume node labels depend only on their own latent and edges depend only on pairwise hyperbolic relations which I agree cuts decoding complexity, but it also forbids higher-order dependencies (motifs, triads) and makes long-range constraints modelling not accounted. It can miss global combinatorial constraints that are not pairwise-decomposable.\n\n2. Authors add a degree-edge consistency term aligning predicted degree to ground-truth degrees and it improved MMD on degree but might risk over-regularizing towards degree histograms at the expense of other structures (e.g., motif diversity), if authors can comment on it?\n\n3. Authors shared the anonymous repo but all files on the anonymous link are not accessible. It compromises the reproducibility.\n\n4. Hyperbolic geometry’s expressive power critically depends on curvature c. However, methodology fixes c a priori. Curvature effectively controls the “branching factor” of the embedding manifold. Without adaptive curvature learning: 1) Graphs of different hierarchy depths collapse into a single scale OR The model may under- or over-stretch distances, biasing flow priors and reconstruction.\n\n5. No error bounds are shown for the tangent-space linearization (how much curvature is lost per layer). Similarly, is there any theoretical grounding that repeated log–exp projections preserve manifold consistency?\n\n6. The results mainly demonstrate performance on small synthetic or molecular settings, leaving open whether the proposed hyperbolic framework generalizes to large-scale or sparse real-world graphs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "15bPc8RWf0", "reviewer": "Reviewer_tqE4", "summary": "The paper presents a framework for generating graphs with hierarchical structure termed “GGBall”, consisting of\n\n1. hyperbolic message passing and DiT layers which use\n    \n    1. message aggregation in the tangent space via log/exp maps (GNN)\n        \n    2. scale and shift values used for FiLM like message modulation derived from hyperbolic distances  (GNN)\n        \n    3. value aggregation using Möbius gyromidpoints (DiT)\n        \n    4. relevancy score calculation leveraging hyperbolic distances on poincare linear layer proejcted q,k values (DiT)\n        \n    5. hyperbolic auxillary operations (diT): layernorm aggregation in the tangent sapce like the GNN message aggregation, residual connections using möbius addition instead of standard addition and multi-head splitting and concatenation in a hyperbolic geometry perserving fashion\n        \n2. a novel graph auto encoder leveraging this hyperbolic latent space parametrization, trained to reconstruct node and edge types, regularized by matching degrees consistently + an l2 norm (evaluated in continuous, standard AE,variational AE and quantized VAE form, termed HGAE, HVAE and HVQVAE respectively, where quantized version was motivated by numerically unstable KL divergence for the HVAE)\n    \n3. a manifold flow matching method taken from [https://openreview.net/pdf?id=g7ohDlTITL](https://openreview.net/pdf?id=g7ohDlTITL)\n    \n\nThe method is evaluated on community-small, ego-small and qm9, with the interpolation properties of the latent state being studied in particular", "questions": "1. would it make sense to do do ablations over mechanisms in the poincare parametrization since a lot of them are introduced, or are they an all or nothing operation (I assume the lattern)?\n    \n2. why does HVQVAE change from e.g. 0.002 in table  1 to 0.0071 in table 2? should report mean/std across multiple rounds (at least for their method/closest baseline) for CIs\n    \n3. why no experiment on generating trees? this would seem like the clear “hello world” example for this?\n\n+ address as many weaknesses as possible please", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. seemingly strong performance\n2. overall very clear exposition of a complex, but theoretically well motivated approach\n\nhitting the guides dimensions:\n\n- originality: hyperbolic embeddings are well established, but creating end to end VQVAE+ flow models I haven't seen yet\n- quality: well written, decent evaluation, proofs appear to be correct after a single close read\n- clarity: fully understandable, with some small nits\n- significance: solid incremental advance, evaluation on larger graphs/trees required to say more", "weaknesses": "1. The density of exposition and hyperbolic-geometry terms can make the paper somewhat difficult to follow (einstein midpoint, möbius gyromidpoint etc). I’d suggest the following two tweaks\n    \n    1. state direction around around 107 that all terms not immediately defined are defined in the appendix for space constraint reasons (to warn the reader some will be just mentioned)\n        \n    2. make use of latex’ glossary feature [https://www.overleaf.com/learn/latex/Glossaries](https://www.overleaf.com/learn/latex/Glossaries) and a reasonable link color/style, to allow hovering over the term to see the definition in modern browsers (I think this will help the definition quite a lot) + enabling backlinks (if readers click through)\n        \n2. should detail how  hyperparameter choices/tuning were performed\n    \n3. nice to have: consider adding a  test on larger graphs (guacamol,moses), as noted in appendix K\n    \n4. nit: shouldn’t it be $\\frac{2}{\\sqrt{c}\\lambda_x^c}$ , see e.g. [https://arxiv.org/pdf/1805.09112](https://arxiv.org/pdf/1805.09112)  eq 12?\n    \n5. interpolation experiment needs reporting of a baseline (in the appendix)\n    \n6. not: I think $\\lambda_{valid}$ is meant to be $\\lambda_{degree}$ on eq 7?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x0RzLPsPOa", "reviewer": "Reviewer_LTqy", "summary": "This paper introduces a hyperbolic framework for graph generation to address the limitations of Euclidean geometry in capturing hierarchical graph structures. It integrates a Hyperbolic Vector-Quantized Autoencoder with Riemannian flow matching based on closed-form geodesics in the Poincaré Ball model. Empirically, GGBall achieves state-of-the-art performance across benchmarks.", "questions": "Please refer to the weakness.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The proposed model is a fully hyperbolic graph generation framework using the Poincaré Ball, leveraging its exponential volume growth to naturally preserve hierarchical structures.\n\n2. Combining HVQVAE with Riemannian flow matching (for flexible prior modeling) resolves stability issues of continuous hyperbolic VAEs and enhances generative capacity.\n\n3. It outperforms SOTA baselines across abstract graph generation and molecular graph generation.", "weaknesses": "1. Experiments focus on small/medium graphs (e.g., QM9, small community graphs); performance on large-scale graphs is unproven.\n\n2. While HVQVAE avoids HVAE’s KL issues, the paper does not explore alternative variational formulations to fully leverage hyperbolic probabilistic modeling. In addition, what is a L_degree in Line 273?\n\n3. The Poincaré Ball’s fixed negative curvature may struggle with heterogeneous graphs (mixing hierarchical and non-hierarchical structures), unlike mixed-curvature alternatives not explored here.\n\n4. The loss function has many hyperparameters. How are their values ​​determined? What are their values ​​for different datasets? What are the sensitivity experiments? Won't so many parameters increase the difficulty of hyperparameter tuning, thereby reducing the practical usability of the method? The inability to answer these questions may cause concern for the reader.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OqRj1rQec7", "reviewer": "Reviewer_BT1X", "summary": "This paper proposes a new generative framework for graph data based on a hyperbolic latent space. The authors motivate the use of hyperbolic geometry by highlighting its theoretical advantages for representing hierarchical and complex graph structures compared to standard Euclidean latent spaces.\n\nThe approach embeds graphs into a hyperbolic space through a Vector-Quantized Autoencoder (VQ-VAE) operating within the Poincaré ball model. The paper introduces dedicated architectural components, including a Poincaré Graph Neural Network (GNN), a Poincaré Transformer, and corresponding hyperbolic encoders and decoders that allow mapping graphs to and from the hyperbolic latent space.\n\nThe model is evaluated on three datasets containing relatively small graphs.", "questions": "The theoretical contribution of the paper is strong and well-founded, but the empirical evaluation does not yet convincingly demonstrate the claimed benefits of hyperbolic latent spaces.\n\nI would encourage the authors to consider the following improvements:\n\n1. Include a systematic ablation comparing hyperbolic and Euclidean latent spaces across all tasks and datasets.\n2. Describe the experimental protocol in detail, including data splits, training configurations, and evaluation procedures (possibly in the supplementary material).\n3. Expand the empirical evaluation by including larger datasets (e.g. Zinc250, Planar, SBM) and additional performance metrics (e.g., FCD, NSPDK).\n\nSuch additions would substantially reinforce the paper’s claims and make the overall contribution more compelling.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "The paper is well-written and easy to follow, even though it introduces non-trivial concepts from hyperbolic geometry. The theoretical development is solid, clearly presented, and supported by extensive supplementary material.\n\nThe idea of leveraging a hyperbolic latent space for graph generation is both neat and original, and the proposed framework represents a meaningful conceptual step forward in the design of geometry-aware generative models.\n\nI would like to particularly emphasize that the theoretical contribution and its presentation are of very high quality. The mathematical rigor and clarity are exemplary.", "weaknesses": "**Empirical Evaluation and Euclidean Baseline**\n\nThe paper’s main claim is the superiority of hyperbolic over Euclidean latent spaces for graph generation. Consequently, it is essential to provide clear empirical evidence supporting this claim.\nIncluding an Euclidean VQ-VAE baseline in Table 1 would greatly strengthen the experimental section, as it would directly demonstrate the empirical advantages of the hyperbolic space. Similarly, presenting detailed results and experimental settings for Figure 1 would help assess the validity of the reported improvements.\nSince the proposed model is theoretically and empirically more complex, it is important to show that it provides a clear and consistent performance advantage over its Euclidean counterpart, ideally across all reported datasets and experiments, including those in Tables 2 and 3.\n\n**Evaluation Metrics and Protocol**\n\nThe evaluation protocol requires clarification.\n\n* *Community-Small*: The results for the HVQVAE+Flow model appear to outperform the training set itself as given in SPECTRE, which should not occur in a properly calibrated evaluation. Moreover, the absence of standard deviations prevents assessment of variability or statistical significance. The paper refers to results from DiGress (which relies on SPECTRE), but the evaluation chain is not made explicit.\n\n* *Ego-Small*: Similar concerns arise here, as the model’s results also systematically outperform the training set as given in DGAE (which seems to be the journal paper of VQGAE) , which appears inconsistent. Clarifying the evaluation setup is crucial to ensure reproducibility and interpretability of the reported findings.\n\n**Evaluation on QM9**\nThe use of the *novelty* metric on QM9 is problematic. QM9 is an exhaustive enumeration of small organic molecules satisfying specific constraints; thus, generating molecules outside this set does not necessarily indicate successful generalization. This metric is therefore not commonly used for evaluation. It would be beneficial for the authors to justify its inclusion or reconsider its use.\nFor the same reason, achieving 100% *uniqueness* is not necessarily desirable. \n\nTherefore, the only meaningful metric in this setting is *validity*, which remains substantially below various baselines.\nTo provide a more comprehensive evaluation, including additional metrics such as FCD (Fréchet ChemNet Distance) or NSPDK similarity would be highly valuable.\n\n**Graph Size and Dataset Diversity**\nThe experiments are limited to datasets with small graphs, with a maximum of around 20 nodes (e.g., Community-Small). Although this limitation is acknowledged in the paper, it remains a major limitation, as many contemporary models handle graphs with up to 200 nodes and report results on larger benchmarks such as Planar, SBM, Zinc250K, or Moses.\n\nIncluding experiments on at least one larger-scale dataset would considerably strengthen the empirical evidence and demonstrate scalability.\nThis is particularly important because the Community-Small and Ego-Small datasets each contain only around 200 instances, making overfitting likely and reducing the statistical robustness of the results.\n\n\n--------\n\n**SPECTRE**: Karolis Martinkus, et al.. SPECTRE:\nSpectral conditioning helps to overcome the expressivity limits of one-shot graph generators. In Proceedings of the 39th International Conference on Machine Learning. PMLR, 17–23 Jul 2022.\nhttps://proceedings.mlr.press/v162/martinkus22a.html.\n\n**DGAE**: Yoann Boget, et al.. Discrete graph auto-encoder. Transactions on Machine Learning Research, 2024. https://openreview.\nnet/forum?id=bZ80b0wb9d.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)