HOG-Diff: Higher-Order Guided Diffusion for Graph Generation
Authors: Yiming Huang
Tolga Birdal
Venue: ICLR 2026 Poster (2026)
Keywords: Topology
Topological Deep Learning
Graph Generation
Higher order
Guidance
Primary area: generative models
TL;DR: We introduce, HOG-Diff, a coarse-to-fine graph generation framework that explicitly exploits higher-order topological cues.

Abstract: Graph generation is a critical yet challenging task, as empirical analyses require a deep understanding of complex, non-Euclidean structures. Diffusion models have recently made significant advances in graph generation, but these models are typically adapted from image generation frameworks and overlook inherent higher-order topology, limiting their ability to capture graph topology.
In this work, we propose Higher-order Guided Diffusion (HOG-Diff), a principled framework that progressively generates plausible graphs with inherent topological structures. HOG-Diff follows a coarse-to-fine generation curriculum, guided by higher-order topology and implemented via diffusion bridges. We further prove that our model admits stronger theoretical guarantees than classical diffusion frameworks. Extensive experiments across eight graph generation benchmarks, spanning diverse domains and including large-scale settings, demonstrate the scalability of our method and its superior performance on both pairwise and higher-order topological metrics. Our project page is available [here](https://circle-group.github.io/research/hog-diff/).

Published as a conference paper at ICLR 2026
HOG-DIFF: HIGHER-ORDER GUIDED DIFFUSION FOR GRAPH GENERATION
Yiming Huang, Tolga Birdal
Department of Computing, Imperial College London, UK (y huang24, t.bi sdirdall ) imperial iirliall .ac..c..kk
ABSTRACT
Graph generation is a critical yet challenging task, as empirical analyses require a deep understanding of complex, non-Euclidean structures. Diffusion models have recently made significant advances in graph generation, but these models are typi- cally adapted from image generation frameworks and overlook inherent higher- order topology, limiting their ability to capture graph topology. In this work, we propose Higher-order Guided Diffusion (HOG-Diff), a principled framework that progressively generates plausible graphs with inherent topological structures. HOG-Diff follows coarse-to-fine generation curriculum, guided by higher-order topology and implemented via diffusion bridges. We further prove that our model admits stronger theoretical guarantees than classical diffusion frameworks. Exten- sive experiments across eight graph generation benchmarks, spanning diverse do- mains and including large-scale settings, demonstrate the scalability of our method and its superior performance on both pairwise and higher-order topological met- rics. Our project page is available here.
INTRODUCTION
Graphs provide an elegant abstraction for representing complex systems by encoding entities as ver- tices and their relationships as pairwise edges. As such, they have played key role in generative modeling in unstructured domains, enabling the synthesis of novel graphs faithful to the data dis- tribution. This representational power has positioned graph generative models as crucial tool for discovering new molecules, materials, and biostructures (Jumper et al., 2021).
Despite these advances, graph generation still lags behind its Euclidean counterparts. Classically, oversmoothing and oversquashing limit expressive capacity (Alon Yahav, 2021), but an even deeper obstacle persists: most generative frameworks treat graphs purely as collections of pairwise edges, overlooking the higher-order structures that govern the organization of real-world systems, such triangles, cliques, rings, and motifs (Battiston et al., 2020). These higher-order structures play a decisive role in fields ranging from chemistry to neuroscience: molecules function through coordinated multi-atom assemblies; collaborative teams operate effectively only specific group sizes; and neural populations exhibit coherent activity patterns that fundamentally rely on multi- unit interactions (Gardner et al., 2022: Papamarkou and theoretical work underscores that such higher-order topology is not an optional structure, but defining characteristic of real data (Segler et al., 2018: Ertl et al., 2025).
al., 2024). growing body of empirical
Recent studies make this picture even sharper. Approved drug molecules, for instance, inhabit only few hundred distinct ring systems (Ertl al., 2025), far fewer than the astronomical chemical space of plausible compounds (1023_1060) (Segler et al., 2018). Meanwhile, advances in topological deep learning (TDL.) show that explicitly modeling complex structures boosts the expressivity and stability of graph and molecular representation learning (Hajij et al., 2022: Papamarkou et al., 2024; Liu et al., 2024; Wang et al., 2025; Hajij et al., 2025). Nevertheless, no existing graph generative model integrates higher-order structure
explicit guiding signal.
In this work, motivated by this gap, we propose the Higher-order Guided Diffusson (HOG-Dit) framework, principled, topology- aware framework that places higher-order topology at the core of the generative process. As illustrated in Fig. 1, HOG-Diff implements a coarse-to-fin generation curriculum: it first synthesizes the higher-order skeleton of a graph, its key 2-cells, faces, other

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Published as a conference paper at ICLR 2026
Enzymes (Avg)
Enzymes (rr)
HOG-Diff
Ego-small (Avg)
GuacaMol (FCD)
Ega-small
GuacaMal
Community
MOSES
Go
Grg
G
small (uvg)
Community small (Kol
MOSES (Kre)
SBM (Avg)
ZINC250K (FCD)
Tk
TR+1
Tx
SBM
ZINC250k
(Kee
(Koal
Authentic
Higher-order Structures
Random
(NSPDK)
(FCD)
QMS
(NSPDK)
QMB
(Krx)
Figure I: Overview of the HOG-Diff framework and its performance. (Left) The dashed trajectory above illustrates the classical generation process, where graphs rapidly degrade into structureless randomness. In contrast, as shown in the coloured region below, HOG-Diff adopts coarse-to- fine generation curriculum via a diffusion bridge, explicitly learning higher-order structures during intermediate steps. (Middle and right) Relative improvements over prior sate--ff-theart on pairwise and higher-order topological metrics.
motifs, and then refines these coarse structures into full pairwise connectivity. To realize this, combine cell-complex filtering with a novel generalized Orustein-Uhlenbbect (OU) diffiusion bridge in the spectral domain, enabling the diffusion process transition smoothly between increasingly detailed topological states while preserving the global structure implied by higher-order interactions.
This coordinated two-stage design brings several benefits. It aligns the generative trajectory with the intrinsic hierarchical organization of real-world graphs; it avoids the collapse of intermediate states into meaningless noisy adjacency matrices; and it allows dynamics for stable, simulation-free training. Theoretically, we prove that HOG-Diff enjoys faster convergence in score matching and tighter reconstruction error bounds than classical diffusion mod- els. Practically, the explicit presence of topological guides enables interpretability: by varying the guide structures, we can directly prohe which topological motifs are most influential in determin- ing the generative process. Across molecular and generic graph benchmarks, HOG-Diff consistently achieves sate-off-thear performance, sharply reducing both statistical and topological discrepancies relative to real data. Our findings underscore central message: higher-order topology is power- ful generative signal and incorporating it transforms graph diffusion from an edge-level denoising procedure into truly structure-awant generative paradigm. Our concrete contributions
leverage closed-form bridge
We introduce cell fiieeixfiltering to extract higher-order skeletons from graphs as valuable gen-
cration guides.
We propose principled, coarse-to-fing graph generation framework guided by higher-order topol- ogy and implemented via the generalized OU diffusion bridge.
We theoretically show that HOG-Diff achieves faster convergence during score-matching and sharper reconstruction error bound compared to classical diffusion models.
Comprehensive evaluations across eight benchmarks show that topology-informed HOG-Diff achieves state-of-the- performance on both pairwise and higher-order metrics, highlighting the value of topological guidance.
2 PRELIMINARIES
Higher-order Networks. Graphs are elegant and useful abstractions for various empirical objects. Formally, a graph can be represented as G A (V, E, X), where V denotes the node set, E C V x V the edges, and X the node feature matrix. However, many empirical systems exhibit group interactions that go beyond simple pairwise relationships (Battiston et al., 2020). To capture these complex interactions, higher-order networks such as hypergraphs, simplicial complexes (SCs), and cell complexes (CCs)-ofe) more expressive alternatives by capturing higher-order interactions among multiple entities (Papamarkou al., 2024). Among these, cell complexes fundamental in algebraic topology, offering a flexible generalization of pairwise graphs (Hatcher, 2001).
Definition (Regular cell complex). A regular cell complex is a topological space S with a partition into subspaces (cells) (æ Joe Pa. where Ps is index set, satisfying the following conditions:
For any x E S. every sufficiently small neighborhood of x intersects finitely many cells. 2. Each cell Fa is homeomorphic to R where na dim(sr.) denotes the dimension of Fa-
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3. For each cell Fa- the boundary az is finite union of cells of dimension less than di
4. For every a E Ps, there exists a homeomorphism de of a closed ball B* C R to the closure
Ta such that the restriction of Oa to the interior of the ball is homeomorphism onto De-
Lifting: From Graphs to Cell Complexes. A cell complex can be constructed hierarchically through a gluing procedure, which is known as lifting. It begins with a set of vertices (O-cells), to which edges (I-cells) are attached by gluing the endpoinns of closed line segments, thereby forming a graph. This process can be extended by taking a two-dimensional closed disk and gluing its boundary to a simple eycle in the graph, see Fig. for illustration. While we typically focus on dimensions up to two, this framework can be further generalized by gluing the boundary of n-- dimensional balls to specific (n ce-ellls in the complex.
From the definition, we can derive that the cell complex S is the union of the interiors of all cells. In this work, we also consider simplicial complexes (SCs), class of topological spaces represented by finite sets of elements that are closed under the inclusion of subsets. Intuitively, SCs can be viewed as a more constrained subclass of cell complexes, where 2-cells are limited to triangle shapes. A comprehensive introduction to higher-order networks can be found in App. B.1.
Score-based Diffusion Models. A fundamental goal of generative models is to produce plausible samples from an unknown target data distribution p(xo). Score-based diffusion models (Song & Er- mon, 2019; Song et al., 2021) achieve this by progressively corrupting the authentic data with noise and subsequently training a neural network reverse this corruption process, thereby generating meaningful data from a tractable prior distribution, i.e., Xgenerated p(Xo).
Specifically, the time-dependent forward process of the diffusion model can be described by the following stochastic differential equation (SDE):
dxę (xx) dt + g.dw
where f, : I" + R" is a vector-valued drift function, [0, + R is a scalar diffusion coefficient, and w; represents a Wiener process. Typically, p(xo) evolves over time t from 0 sufficiently large T into p(xx)) through the SDE, such that p(xr) will approximate a tractable prior distribution, for example, standard Gaussian distribution.
Starting from time T. p(x)) can be progressively transformed back to p(xo) by following the tra- jectory of the reverse-time SDE dx; = (x,,) - of Vx log pt (x:) dī + g:dw, (Anderson, 1982), where (-) denotes the probability density function of x, and is reverse-time Wiener pro- cess. The score function Vx, log (xx) is typically parameterized by a neural network se s0(x,, t)
and trained using the conditional score-matching loss function (Vincent, 2011):
((0) A Exx,, (w(() [|s((x,,)) t)- Vx, log)((x))) Xx,x
(w(() (xx, f) Xx, logre(x; xo)
where (t) is a weighting function. The second expression is more commonly used since the condi- tional probability P. (x, X2) is generally accessible. Ultimately, the generation process is complete by first sampling Xy from a tractable prior distribution p(xr) R3 Pprior and then generating Xo by numerically solving the reverse-time SDD.
Doob's h-transform. Doob's h-transform is mathematical framework widely used to modify stochastic processes, enabling the process to satisfy specific terminal conditions. By introducing an h-function into the drift term of an SDE, this technique ensures that the process transitions to predefined endpoint while preserving the underlying probabilistic structure. Specifically, given the SDE in Eq. (I), Doob's h-transform alters the SDE to include an additional drift term, ensuring that the process reaches a fixed terminal state at = T. The modified SDE is expressed as:
dx,
[ft (xx) + 9; hxxx t, xT, T)]dt g,dw
where 1o(xe, f, XT, T) = xx log logp(>1 (x)) Crucially, the construction drives the diffusion process towards a Dirac distribution at xr, i.e., lim;-7 p(xe Xx, xT) = 8(x; xT).
HIGHER-ORDER GUIDED DIFFUSION MODEL
We now present our Higher-order Guided Diffusion (HOG-Diff) model, which enhances graph gen- cration by exploiting higher-order structures. We first describe our coarse-to-fing generation frame- work, followed by an introduction the supporting diffusion bridge technique. Finally, we provide theoretical evidence to validate the efficacy of HOf-Diff.
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3.1 COARSE-TO-FINE FRAMEWORK WITH TOPOLOGICAL FILTERING
We draw inspiration from curriculum learning. a paradigm that mimics human learning by system- atically organizing data in a progression from simple (coarse) to complex (fine) (Abbe et al., 2021; Soviany et al., 2022). Specifically, we model coarse intermediary structures as higher-order cells, which encapsulate rich structural properties beyond pairwise interactions (Battiston et al., 2020). These cells can be obtained by lifting the original graph and retaining associated 2-faces as the higher-order skeleton. Our generative processes then follow a curriculum to progressively generate graphs, starting with higher-order cells and gradually refining them into the full complex graph.
To implement our coarse-to-fine generation, we first introduce a key operation termed cell complex filtering (CCF).
(a)
(b)
Proposition (Cell complex filtering). Given graph G (V, and its associated cell complex S = UaTa (obtained via lifting). The p-cell complex filtering operation defines a filtered graph Gipl (Vp), E(p)), where ViP) {u E V
Ia with (x) = : e E JoJ. and Epp = {(u,v) E
(c)
(d)
To with (x) = : (u,o) S Ta}. Here, Ta denotes the closure of aa.
dz
As illustrated in Fig. we first lift the graph to a cell complex and then apply CCF to generate intermediate states by pruning nodes and edges that do not belong to higher-order cells. In prac- tice, CCF offers substantial speedup, as it avoids the expensive enumeration of all cells required by lifting (see App. D for de- tailed complexity analysis). The filtering operation decomposes the graph generation task into hierarchically structured and man- ageable subtasks.
do
Figure Cell Complex transfor- mations. (a) An example graph. (b) Lifting: closed 2D disks are glued the boundary of the rings to form the 2-cell complex. (c) The resulting cell complex and the corresponding homeo- morphisms the closed balls for three representative cells of dif- ferent dimensions in the com- plex. (d) Black elements repre- sent higher-order structures ex- tracted through 2-cell filtering, while grey elements denote cor- responding peripheral structures pruned by the filtering operation.
Based on the filtering results, the diffusion process is structured into K hierarchical time windows, denoted as {[Tk-1, 1, The Tx]) k== where 0 = To < ... < Th-1 < Tk <... < -K = T, with the fil- tered results serving as natural intermediaries in the hierarchical generation process. The overall framework of HOG-Diff is de- picted in Fig. (lefft). In general, we first generate coarse-grained higher-order order skeletons and subsequently refine them into finer pairwise relationships, thereby simplifying the task of capturing complex graph distributions. Formally, our generation process factorizes the joint distribution of the final graph Go into prod- uct of conditional distributions across these time windows:
p(Go) = P(Gol G., )p(G), (...G/.....GGG....
P(G
Here, the intermediate states G Gr,,, Gr represent progressively finer cell complex fil- tered graph representations, aligning intermediate diffusion stages with realistic hierarchical graph structures. This coarse-to-fine approach enables model to first focus on fundamental topolog- ical structures and then add finer connectivity, inherently aligning with the hierarchical nature of many empirical systems. Consequently, our model benefits from smoother training and improved sampling performance (see Sec. 3.3 for theoretical analysis).
To ensure smooth transitions between intermediate states within each interval [Tk-1. nx]. the graph evolves according to the general form of a diffusion bridge process (see Sec. 3.2 for details):
aG( (k) = fr. (G )t + dW, E [Tk-1,
(4)
The forward diffusion process introduces noise in stepwise manner while preserving intermedi- ate structural information. Reversing this process enables the model to generate authentic samples with desirable higher-order information. Moreover, integrating higher-order structures into graph generative models improves interpretability by allowing analysis of their significance in shaping the graph's properties. Rather than directly conditioning on higher-order information, HOG-Diff em- ploys it incrementally as a guiding structure. This strategy allows the model to build complex graph structures progressively, while maintaining meaningful structural integrity at each stage.
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3.2 GUIDED GENERATION VIA DIFFUSION BRIDGE PROCESS
As building block of our generative framework, we leverage a class of diffusion processes with fixed terminal states, namely the generalized Ornstein-Uhlenbeer (GOU) bridge, to realize the pro- posed guided diffusion process in Eq. (4), while enabling simulation-free training.
The generalized OrnstennUUhlebbed (GOU) process (Ahmad, 1988; Luo al., 2023b), also known as the time-varying OU process, is stationary, Coussian-MMarko process characterized by its mean- reverting property. Specifically, the marginal distribution of the GOU process asymptotically ap- proaches fixed mean and variance. The GOU process is governed by the following SDE:
dG: 0e(u Gt)dt + g.
(5)
where 4 is the target terminal state, 0g denotes a scalar drift coefficient and 9r represents the diffusion coefficient. For analytical tractability, Oe and Ii constrained by ge /0, 20 (Luo et al., 2023b), where x² is a fixed constant, yielding a closed-form transition probability:
pGG G.)) = (mmma of =N (1= (G,+ (G, e , e
Here, 0yt - S- 0_dz. For notional simplicity, for replaced by 0g when s =
Diffusion Bridge. Applying Doob's h-transform (Doob & Doob, 1984) to the GOU process un- der the terminal condition μ G we can derive the GOU bridge process as follows (detailed derivation of the bridge process provided in App. A.I):
2
dG; = 0e 1+
G + gw.
(7)
20
The conditional transition probability admits an analytical form p(G. Gre-s Goa) = ((n,, of
m, = G + +Grr.1 G. -00 1
o
v2 The 1st
Viire /02
(8)
1376"
Here, Ba:b - 10eh 0,ds, and Va:b = 0² e- 28...))
The GOU bridge process eliminates variance in the terminal state by directing the diffusion toward a Dirac distribution centered at Gr,, making it well-suited for stochastic modelling with terminal constraints (Heng et al., 2021; Yue et al., 2024). Moreover, we can directly use the closed-form solution for one-step forward sampling without expensive SDE simulation. Note that the Brownian bridge process used in previous works (Wu et al., 2022) is special case of the GOU bridge process when 0, + (see App. A.
Training and Sampling. Classical graph diffusion approaches typically inject isotropic Gaussian noise directly into the adjacency matrices A. leading to various fundamental challenges, such as permutation ambiguity, sparsity-induced signal degradation, and poor scalability (see App. C.2 for detailed discussion). To address these challenges, inspired by Luo et al. (2023a), we introduce noise in the spectra of the graph Laplacian = D A, instead of the adjacency matrix A, where D denotes the diagonal degree matrix. As symmetric positive semi-definite matrix, the graph Laplacian can be diagonalized as = UAU T. Here, the orthogonal matrix U = [ui. u,) comprises the eigenvectors, and the diagonal matrix A diag(A .A,) holds the corresponding eigenvalues. Therefore, the target graph distribution p( (Go) represents joint distribution X, and Ag. exploiting the permutation invariance and structural robustness of the Laplacian spectrum.
Consequently, the GOU bridge process in Eq. (7), along with its time-reversed counterpart, can be formulated as the following system of SDEs for graph G::
[xx, (X, + d.!
[xx; = (X,, oi. Vx log (G. di + gw!d dA, = (A.) oh. VA log ( (G, G.,,)] di + dW?
AA, (A,) W??
Here, the reverse-time dynamics of the bridge process are derived using the theory of SDEs, the superscript of x!()) and (a) dropped for simplicity, and fg.t is determined according to Eq. (7).
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To approximate the score functions Vx, (og and VA, log Pa GT. employ neural network so (Gr, Graz t), which outputs predictions for both node-level (s ($ (Gr, Gz.,t)
(R)
(k) and spectrum ($ (G,, T t)) components. The network is optimized by minimizing:
(lh) (0) = Br, Cryy [t(t - Vx log (G. +all+ VA log )(å))
where (tt) is a positive weighting function, and et, C2 control the relative importance of vertices and spectrum. The training procedure is detailed in Alg. in App. C.
In the inference procedure, we sample (X Arx) from the prior distribution and select Uo as an eigenbasis drawn from the training set. Reverse diffusion is then applied across multiple stages to sequentially generate (X A A (X.,, An (Xo, (Xo.Ao), Ao), where each stage is imple- mented via the diffusion bridge and initialized from the output the previous step. Finally, plau- sible samples with higher-order structures can be reconstructed as G. = (Xo, Lo = Ug ). Further details of the spectral diffusion process and the complete sampling procedure are provided in App. C, while ablation studies comparing diffusion in the spectral domain versus the adjacency matrix presented in App. F.3.
The score network plays critical role in estimating the score func- tions required to reverse the diffusion process. Standard graph neural networks designed for classi- cal tasks such as graph classification and link prediction may be inappropriate for graph distribution learning due to the complicated requirements. For example, an effective model for molecular graph generation should capture local node-edge dependence for chemical valency rules and attempt to cover global graph patterns like edge sparsity, frequent ring subgraphs, and atom-type distribution.
Score Network Architecture.
To achieve this, we introduce unified score network that explicitly integrates node and spectral representations. As illustrated in Fig. 7 in the Appendix, the network comprises two different graph processing modules: standard graph convolution network (GCN) (Kipf & Welling, 2017) for local feature aggregation and a graph transformer network (ATTN) (Dwivedi & Bresson, 2021; Vignac al., 2023) for global information extraction. The outputs of these modules are fused with time information through Feature-wise Linear Modulation (FiLM) layer (Perez et al., 2018), and the resulting representations are concatenated form a unified hidden embedding. This hidden em- bedding is further processed by separate multilayer perceptrons (MLPs) to produce predictions for Vx log (G, and VA A logp( G. (..),, respectively. It is worth noting that our score network is permutation equivalent, as each component of our model avoids any node ordering- operations. Our model detailed App. C.3.
3.3 THEORETICAL ANALYSIS
We now provide theoretical evidence for the efficacy of HOG-Diff, demonstrating that the proposed framework achieves faster convergence in mctre-matching and tighter reconstruction error bounds compared to standard graph diffusion. We experimentally verify our theories in Sec. 4.4.
Theorem 3 (Informal). Suppose the loss function (()) Eq. (9) is B-smooth and satisfies the Jt-PL condition in the ball B (0o, R). Then, the expected loss at the i-th training iteration satisfies:
by2 E pt)) (0.) S BN(3N2 + p(b ((bb-1))
((K) (0o)
where N denotes the size of the training dataset, and the mini-batch size. Furthermore, holds that BHOG-DiG S. Betassical, implying that the distribution learned by the proposed framework converges to the target distribution faster than classical generative models.
Following Luo et al. (2023a), we define the expected reconstruction error at each generation process as E(t E | G. - G where G, represents the data reconstructed with the ground truth score V Vlogpe and G. denotes the data reconstructed with the learned score function $g. Next, we establish that the reconstruction error in HOG-Diff is bounded more tightly than in classical graph generation models, thereby ensuring superior sample quality.

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Published as a conference paper at ICLR 2026
Table Comparison different methods on molecular datasets. The best results are highlighted in bold. The full version with more baselines, metrics, and standard deviations is provided in App. F.I.
QM9
ZINC250K
Method
Val. 1
Uni.
Nov. f
FCD
NSPDK +
Val. 1
Uni. f
Nov. 7
FCD +
NSPDK
GraphAF
74.43
88.64
86.59
5.625
0.021
68.47
98.64
99.99
16.023
0.044
GraphDF
93.88
98.58
98.54
10,928
0.064
90.61
99.63
100.00
33.546
0.177
GraphArm
90.25
95.62
70.39
1.220
0.002
88.23
99,46
100.00
16.260
0.055
MiCaM
99.93
93.89
83.25
1.045
0.001
100.00
88.48
99.98
31.495
0.166
SPECTRE
87.30
35.70
97.28
47.960
0.163
90.20
67.05
100.00
18.440
0.109
GSDM
99.90
2.650
0.003
92.70
12.956
0.017
EDP-GNN
47.52
99.25
86.58
2,680
0.005
82.97
99.79
100.00
16.737
0.049
GDSS
95.72
98.46
86.27
2.900
0.003
97.01
99,64
100.00
14.656
0.019
DiGress
99.00
96.66
33.40
0.360
0.0005
91.02
81.23
100.00
23.060
0.082
MoFlow
91.36
98.65
94.72
4.467
0.017
63.11
99.99
100.00
20.931
0.046
Cometh
99.59
96.75
72.06
0.248
0.0005
CatFlow
99.81
99.95
0,441
99.95
99.99
13.211
DeFoG
99.26
96.61
72.57
0.268
0.0005
94.97
99.98
100.00
2.030
0.002
HOG-Diff'
98.74
97.10
75.12
0.172
0.0003
98.56
99.96
99.53
1.633
0.001
HOG.00 HOG-DHT
Method
Val.t
Uni.
Nov.t
FCD
Filterst
DiGress
85.7
100.0
95.0
1.19
97.1
OMs
MOSES
DisCo
88.3
100.0
97.7
1,44
95.6
Cometh
90.5
99.9
92.6
1.27
99.1
DeFoG
92.8
99.9
HOG-DiiT
99.7
100.0
89.1
0.94
96.7
ZINC25NN
Method
Val.f
Uni.
Nov.t
FCD scoref
KL div.
DiGress
85.2
100.0
99.9
68.0
92.9
GuacaMal
DisCo
86.6
86.6
86.5
59.7
92.6
DiOres HOG
Cometh
98.9
97.6
96.7
MOBES
DeFoG
99.0
99.0
97.9
97.7
HOG-Diff
99.1
100.0
97.0
78.5
96.9
Figure 3: (left) Large molecule generation performance. Best results are highlighted in bold. Full versions presented in App. F.2. (right) Generative trajectories progressing from left to right,
illustrating that HOG-Diff preserves topological integrity across intermediate steps.
Theorem 4. Under standard Lipschitz and boundedness assumptions, the reconstruction error of HOG-Diff satisfies the following bound:
Enog S Anog Thos
(10)
where Anog = C² (0) 1Th At and Thog =
Ek= STh 7A (s)ds
+
CEKI K-1 1Th ds. Moreover, under the same assumptions, HOG-Diff admits a tighter recon- struction error bound than classical diffusion models.
The theorems above rely primarily on mild assumptions, such as smoothness and boundedness, without imposing strict conditions like the target distribution being log-concave or satisfying the log-Sobolev inequality. Their formal statements and detailed proofs are postponed to App. A. We experimentally verify these Theorems in Sec. 4.4.
4
EXPERIMENTS
We assess HOG-Diff against state-of-the-art baselines in both molecular and generic graph gener- ation. Ablation studies are further conducted to analyze the impact of different topological guides. Complexity analysis and experimental settings are deferred to Appendices and E, while App. reports additional results, including scalability large benchmarks, variance statistics, diffusion- domain analysis, the rationale for filtering choices, and visualizations.
4.1 MOLECULE GENERATION
Molecular design is a prominent application of graph generation. We evaluate our method on four widely used molecular benchmarks: QM9 (Ramakrishnan al., 2014), ZINC250K (Irwin et al.,

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Table 2: Generation performance on generic graph datasets. Best bold and second- best underlined.
Hyphen (-) indicates missing results in the original paper.
Community-small
Enzymes
Ego-small
Method
Deg.l
Clus.
Orbit
Avg.L
Deg.4
Clus.
Orbitl,
Avg.
Deg.4
Clus.
Orbit
Avg.
GraphRNN
0.080
0.120
0.040
0.080
0.017
0.062
0.046
0.042
0.090
0.220
0.003
0.104
GraphAF
0.180
0,200
0.020
0.133
1.669
1.283
0.266
1.073
0.030
0.110
0.001
0.047
GraphDF
0.060
0.120
0.030
0.070
1.503
1.061
0,202
0.922
0.040
0.130
0.010
0.060
Gra
0.350
0,980
0.540
0.623
1.369
0.629
0.191
0.730
0.130
0.170
0.050
0.117
GNF
0,200
0,200
0.110
0.170
0,030
0.100
0.001
0.044
EDP-GNN
0.053
0.144
0.026
0.074
0.023
0.268
0.082
0.124
0.052
0.093
0.007
0.051
GPrinFlowNet
0.021
0.068
0.021
0.037
0.021
0.088
0.009
0.039
SPECTRE
0.048
0.049
0.016
0.038
0.136
0.195
0.125
0.152
0.078
0.078
0.007
0.054
GDSS
0.045
0.086
0.007
0.046
0.026
0.061
0.009
0.032
0,021
0.024
0.007
0.017
DiGress
0.047
0.041
0.026
0.038
0.004
0.083
0.002
0.030
0.015
0.029
0.005
0.016
HOG-Diff
0.006
0.022
0.002
0.010
0.011
0.061
0.007
0.027
0.015
0.027
0.004
0.016
2012), and the large-scale MOSES (Polykovskiy et al., 2020) and GuacaMol (Brown et al., 2019) benchmarks. contrast to many prior works that report results on only a subset of these datasets, our evaluation provides comprehensive coverage. Intermediate higher-order skeletons are extracted using 2-cell complex filtering; the rationale for this choice is discussed in App. F.4. For QM9 and ZINC250K, we follow the dataset split and evaluation protocols of Jo et al. (2022). For MOSES and GuacaMol, we adopt their official splits and benchmark-speciffr metrics. We report Validity without correction (Val.), Uniqueness (Uni.), Novelty (Nov.). Fréchet ChemNet Distance (FCD) (Preuer et al., 2018), NSPDK (Costa & Grave, 2010), Filters, and KL divergence (Brown et al., 2019).
We benchmark against diverse set of representative molecular generators. Autoregressive ap- proaches include GraphAF (Shi et al., 2020), GraphDF (Luo et al., 2021), GraphArm (Kong et al., 2023), and the fragment-based MiCaM (Geng et al., 2023). The remaining methods adopt a one- shot paradigm: SPECTRE (Martinkus et al., 2022) incorporates spectral conditioning within a GAN framework, while GDSS (Jo al., 2022). DiGress (Vignac et al., 2023), and Cometh (Siraudin al., 2025) are diffusion-based models. We also compare against advanced flow-based models, namely MoFlow (Zang& & Wang, 2020), CatFlow (Fijkelboom et al., 2024), and DeFoG (Qin et al., 2025).
Sampling Quality. Tab. and Fig. 3(left) show that HOG-Diff generally outperforms both auto- regressive and one-shot models. Notably. the dramatic decrease in NSPDK and FCD implics that HOG-Diff is able to generate molecules with data distributions close to those of real molecules in both graph and chemical spaces. We further visualize the molecule generation process in Fig. (righht with more examples deferred to App. F.5. It can be observed that our model explicitly preserves higher-order structures during the generation process.
4.2 GENERIC GRAPH GENERATION
To display the topology distribution learning ability, we assess HOG-Diff over four common generic graph datasets: (1) Community-smmn small, containing 100 random community graphs; (2) Ego-small, comprising 200 small ego graphs derived from the Citeseer network dataset; (3) Enzymes, featuring 587 protein graphs representing tertiary structures of enzymes from the BRENDA database; and (4) SBM, larger-scale stochastic block model dataset. The SBM benchmark is presented separately in App. F.2 due to its distinct evaluation protocol. Intermediate higher-order skeletons are obtained via 3 3-simplicial complex filtering. We employ the same train/test split as Jo ct al. (2022) for a fair com- parison. Maximum mean discrepancy (MMD) is used to quantify the distribution differences across key graph statistics, including degree (Deg.), clustering coefficient (Clus.), and 4-node orbit counts (Orbit). A low MMD signifies close alignment between the generated and evaluation datasets, suggesting superior generative performance. We also report the average MMD across all metrics as an overall indicator.
We compare the following graph generative models: GraphRNN (You et al., 2018) is an autore- gressive model, while Graph VAE (Simonovsky & Komodakis, 2018), GNF (Liu al., 2019), and GPrinFlowNet (Mo et al., 2024) are one-shot models. GraphAF, GraphDF, EDP-GNN, SPEC- TRE, GDSS, and DiGress were introduced previously. The results in Tab. 2 show that HOG- Diff is not only suitable for molecular generation but also proficient in generic graph genera- tion, demonstrating its ability to capture the intricate topological interdependencies effectively.
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4.3 TOPOLOGICAL PRESERVATION ANALYSIS
Table Quantitative evaluation of higher- order topology preservation using Curvature Filtrations. The best results are bold.
To further verify the capability of HOG-Diff in pre- serving higher-order structures, conduct quanti- tative evaluation using Curvature Filtrations (South- al., 2023). Unlike simple graph statistics, curvature filtrations combine discrete curvature no- tions with topological data analysis (TDA) capture multi-scale topological features.
Dataset
Method
RFR
KOR
GDSS
0.925
0.601
DiGress
0.251
0.343
QM9
DeFoG
0.177
0.286
Cometh
0.216
0.314
HOG-Diff
0.077
0.098
Specifically, we measure the distance between the mean persistence landscapes of generated graphs and the test set, using two filtration functions: (I) Bal- anced Forman-Rice curvature KFR. which focuses on edge-based local clustering and cycles, and (2) Oliivier-Ricei curvature KOR, which captures global geometry and transport properties via Wasserstein distance. A lower distance indicates a generated dis- tribution that is topologically closer to the test set.
MiCaM
9,436
7.251
ZINC
GDSS
1.781
1,331
250k
DeFoG
0.728
0.498
HOG-Diff
0.190
0.098
DiGress
0,260
0.223
MOSES
HOG-Diff
0.159
0.183
DiGress
0.862
0.075
GuacaMol
HOG-Diff
0.083
0.057
DeFoG
3.998
5.206
Fig. (right) reports the KFR results, while Tab. 3 provides the full evaluation for both КPn and KOR across all molecular and generic datasets. HOG- Diff consistently achieves the lowest distance scores across both molecular and generic graph datasets. Notably, on complex molecular datasets, our method outperforms strong baselines by significant margin, suggesting that while baseline models may capture basic chemical validity, HOG-Diff is superior in re- constructing the intrinsic topological backbone and higher-order geometric relations. Even on generie graphs like SBM and Enzymes, which possess distinct community structures, HOG-Diff maintains better topological fidelity, verifying that the performance improvement stems from the successful preservation of higher-order structures.
SBM
Cometh
2.748
3.938
HOG-Diff
1.453
3.740
Comm.
GDSS
12.515
14,949
-small
HOG-Diff
6.734
6.522
GDSS
13.031
12.399
Enzymes
HOG-Diff
12.114
9.830
Ego -small
GDSS
2.311
1.301
HOG-Diff
1.679
1.042
4.4 ABLATIONS: TOPOLOGICAL GUIDE ANALYSIS
Coarse
Throughout the experiments, we observe that HOG-Diff ex- hibits superior performance on complex molecular datasets, but comparatively modest results on the Ego-small dataset. Statistics and visualizations in Apps. E and F.5 reveal that Ego contains the fewest higher-order structures among the
Fine
Spectrum Loss
Coarse+ Finc
Classical (Noise)
4
datasets analyzed, suggesting that the choice of guide plays
pivotal role in the effectiveness of generation. To validate this hypothesis, we conduct further ablations using different types of topological information as guides.
0.3 0.6 0.9 1.2 1.5 1.8 Training Steps (x10*)
We first compare two types of guides: structures derived from 2-cell filtering (Cell) and Gaussian random noise (Noise). Employing noise as a guide aligns with classical diffusion paradigms that generate samples by progressively denoising noisy data. Fig. 4 visualizes how the spectrum loss changes during the training process. It shows that our framework (red curve) converges faster than the classical method (blue curve), which is consistent with our thcoretical results in Theorem 3.
Figure 4: Training curves of the score-matching process. The entire process of HOG-Diff is divided into two stages, i.e., K as coarse and fine, respectively. The combined loss of these two stages labelled Coarse+Fine.
referred to
In the sampling procedure, we further evaluate peripheral structures (Periph.), obtained by remov- ing cell components, as guides. As shown Tab, 4, both peripheral and noise guides underperform cell-based guides, providing empirical support for the tighter reconstruction error bound established in Theorem 4. These results indicate that certain topologies, particularly cells, are more effective in guiding generation, highlighting the importance of selecting appropriate topological structures

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Published as a conference paper at ICLR 2026
steer toward meaningful outputs. Moreover, this finding suggests that guides could serve diagnos- tie tools for assessing whether specific topologies are essential to a given architecture. Systematic analysis of different guides promises to enhance interpretability, clarify how topologies affect gen- eration, and inform the design of more effective graph generative models.
5 RELATED WORK
We review graph and higher-order generation meth- ods, with a more detailed discussion in App. B.
Table 4: Sampling results of various topolog- ical guides.
Graph Generative Models. Early work on graph generative models dates back to random network models (Barabási & Albert, 1999). which offer foun- dational insights but are too simplistic for capturing real-world graph distributions. Recent advances in generative models have leveraged the power deep neural networks, significantly improving the ability to learn graph distributions. Notable approaches in-
Datasct
Guide
Val.1
PCD4
NSPDK]
Noise
91.52
0.829
0.0015
QM9
Peripheral
97.58
0,305
0.0009
Cell
98.74
0.172
0.0003
Noise
96.78
1.665
0.002
ZINC 250k
Peripheral
97.93
1.541
0.002
Cell
98.56
1.533
0.001
clude autoregressive or simultaneous models based techniques such variational autoencoders (VAE) (Jin et al., 2018; Simonovsky & Komodakis, 2018), recurrent neural networks (RNN) (You al., 2018), normalizing flows (Zang Wang, 2020; Shi et al., 2020; Luo et al., 2021), and gener- ative adversarial networks (GAN) (Martinkus et al., 2022).
Diffusion-based Graph Generation. A breakthrough in graph generative models has been marked by the recent progress in diffusion based generative models (Niu et al., 2020; Ho et al., 2020; Song et al., 2. Recent models employ various strategics to enhance the generation of complex graphs, including capturing node-edge dependency (Jo al., 2022), addressing discretization challenges (Vignac et al., 2023; Huang et al., 2023; Siraudin et al., 2025), exploiting low-to-high frequency generation curriculum (Mo al., 2024; Huang & Birdal, 2025), and improving computational effi- ciency through low-rank diffusion processes (Luo et al., 2023a). CatFlow (Eijkelboom et al., 2024) and DeFoG (Qin et al., 2025) adopt flow matching as an alternative diffusion, achieving more efficient generation. Recent studies have also enhanced diffusion based generative models by incor- porating diffusion bridge processes, i.e. processes conditioned on the endpoints (Wu et al., 2022: Jo et al., 2024; Boget et al., 2025). Despite these advances, existing methods either overlook or inad- vertently disrupt higher-order structures during graph generation, or struggle to model topological properties, denoising noisy samples does not explicitly preserve the intricate structural depen- dencies required to generate realistic graphs. This highlights the need for a graph-friendly diffusion framework that explicitly learns higher-order topology, preserves meaningful intermediate states and trajectories, and avoids inappropriate noise injection.
Higher-order Generative Models. Generative modeling uses higher-order information mostly in the form of motifs and hypergraphs. MiCaM (Geng et al., 2023) synthesizes molecules by iteratively merging motifs. HypeBoy (Kim et al., 2024) learns hypergraph representation through hyperedge filling, while Hygene (Gailhard et al., 2025) reduces hypergraph generation to bipartite graphs. To the best our knowledge, we are the first to consider higher-order guides for graph generation.
6 CONCLUSION
We introduce HOG-Diff, a coarse-to-fine generation framework that exploits higher-order topology as explicit guidance. By employing cell complex filtering and diffusion bridges, HOG-Diff de- composes graph generation into structured, casier-to-learm, topology-awes meaningful intermediate states. Our theoretical analysis supports its advantages over classical diffu- sion approaches, including improved training dynamics and a tighter reconstruction error bound. Ex- tensive experiments eight molecular and generic graph benchmarks further validate these gains. Overall, this work represents an important step toward topological diffusion models, highlighting the importance of higher-order features that prior methods overlook and leaving ample room for future work.
substeps and preserves
Limitations and future work. The performance of HOG-Diff depends on the presence of explicit higher- order structures, which certain graph types might lack. Nevertheless, higher-order motifs are often deducible in realistic graphs. Our future work will explore various filtering mechanisms beyond CCF We will adaptively determine the order (dimension of the higher-order skeleton).
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ACKNOWLEDGMENTS
T. Birdal acknowledges support from the Engineering and Physical Sciences Research Council [grant EP/X01 1364/1]. T. Birdal was supported by UKRI Future Leaders Fellowship Igrant num- ber MR/Y018818/11 as well as Royal Society Research Grant RG/RI/241402.
ETHICS AND REPRODUCIBILITY STATEMENT
This paper presents work whose goal is to advance the field of deep generative models. Positive applications include generating graph-structured data for scientific discovery and accelerating drug discovery by generating novel molecular structures. However, like other generative technologies, our work could potentially be misused to synthesize harmful molecules, counterfeit social interactions, or deceptive network structures.
Detailed thcoretical derivations are provided in App. A. The complete architecture of HOG-Diff, along with the training and sampling procedures, described in App. C. Details of the datasets, preprocessing steps, and experimental settings are provided in App. E. Additional experimental results, ablation studies, and visualizations can be found in App. F.
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Appendix
Organization. The appendix is structured as follows: We first present the derivations excluded from the main paper due to space limitations in Section A. Additional explanations of related work are provided in Section B. Section details the generation process, including the spectral diffusion framework, the architecture of the proposed score network, and the training and sampling proce- dures. Computational efficiency is discussed in Section D. Section E outlines the experimental setup, and Section I' reports additional experimental results, covering the impact of diffusion domain choice, scalability to large graphs, standard deviation analysis, and visualizations of the generated samples. Section G concludes with limitations.
A FORMAL STATEMENTS AND PROOFS
This section presents the formal statements of key theoretical results along with their detailed deriva- tions. We will recall and more precisely state the theoretical claims before presenting the proof.
A.1 DIFFUSION BRIDGE PROCESS
Recall that the generalized Ornstein-Uhlenbee (GOU) process, also known as the time-varying OU process, is a stationary Gaussian-Markov process whose marginal distribution gradually tends towards stable mean and variance over time. The GOU process is generally defined as follows (Ahmad, 1988; Luo et al., 2023b):
dG, = 0. ( Gi) dt + g:dW.
(1I)
where μ is given state vector, 0g denotes scalar drift coefficient, and gr represents the diffusion coefficient. Additionally, we assume the relation gi /0, = 2o²,, where o² a given constant scalar. As result, its transition probability possesses closed-form analytical solution:
p (G. G.) = =N(m 1)
mg: = l (G, p1) 6ese
(12)
22 δ² 1 20.in
S- Here, 0..: = f 0_dz. When we write 0, = Do for notation simplicity.
Doob's li-transform can modify an SDE to pass through specified endpoint (Doob & Doob, 1984). When applied to the GOU process, it eliminates variance in the terminal state by driving the diffusion process toward Dirac distribution centered G (Heng et al., 2021; Yue et al., 2024), making well-suited for stochastic modelling with terminal constraints.
In the following, we derive the generalized Ornstein-Ullenbeer (GOU) bridge process using Doob's ss h-transform (Doob Doob, 1984), and subsequently examine its relationship with the Brownian bridge process.
Generalized Uhhstein-Uhlenbeck (GOU) bridge. Let G, evolve according to the generalized OU process Eq. (5), subject to the terminal conditional μ G. Applying Doob's h-transform (Doob & Doob, 1984), we can derive the GOU bridge process follows:
dG. Oc (1
2
(G=, Gt)dt + dW,
(13)
201
The conditional transition probability p(G G G admits an analytical expression:
p(G. G G.,)) =N N(m,, T)
m, G + (GTz-1 G -re
(14)
2 Th-ii 1 2 "
Here, 0ash 1 0,ds, and Uos a2 -8 )
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Proof. Without loss of generality, consider one generation interval [Tk-1, Tk] and denote T X; (()) = Tk-1, and endpoints Xo = - XT Gia-
Tke
From Eq. (6), we can derive the following conditional distribution
p(xT x,) Nx (x; xr)e our Vur 1). Hence, the h--fnccion can be directly computed as:
(15)
h h( 1, XT, T) Xx, log p(gp( xx)
Xx
XT)
)=
-20er
xx
const
20
(16)
-0
(XT - X1)
(XT
x1)0
(6 1).
Following the approach in Yue et al. (2024), applying the Doob's h- h-traanformm yields the representa- tion of an endpoint xy conditioned process defined by the following SDE:
dx,
[x t) + gh xr, T)] dt g.dw,
01 + (2 - 1) -1)
(xT -x) xx + de
(17)
2
02 1+
(x- xt)dt + g.
220rr
Given that the joint distribution of (xo, Xx, XT] is multivariate normal, the conditional distribution p(x, Nb, xy) is also Gaussian:
p(x: Xy) N (m,, D
(18)
where the mean m, and variance os are determined using the conditional formulas for multivariate normal variables:
m, -[xx; Xo: xr] = E[x; xol + Cov(xx,x) xx) Var(xr xx) "''xx- - E[xr xol);
0f = VVr(xx XT) = Var(xe xn) - Cov(xr, XT Xo) Var(xr xxo) =oCo=XCC(XT,, X xx)
(19)
Notice that
Cov(x,, xx) Cov (x,, (x, xT)e d.r xx)
Tr Var(x: xx)).
(20)
By substituting this and the results in Eq. (6) into Eq. (19), we can obtain
m (XT (xo xT)e B.) + 2u: /v7 XT xT (xo xx)e Ar
=xT (xo xT) (e-8) Best up /17
(21)
-20
26.T (1 228))
=xr (xo xr)e 9.
1 - c 20
(Xo xr)e 0. 2 /
and
07
2
(e
(c- p3? /02
2
v
(vz
24 12)
(22)
u? 0² (1 (1-e
20 c- -0 (1- e-2)))
U
= /
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Finally, conclude the proof by reverting to the original notations.
Note that the GOU bridge process, also referred to as the conditional GOU process, has been studied theoretically in previous works (Salminen, 1984; Heng al., 2021; Yue al., 2024). However, are the first to demonstrate its effectiveness in explicitly learning higher-order structures within the graph generation process.
Brownian Bridge Process. In the following, demonstrate that the Brownian bridge process is particular case of the generalized OU bridge process when 0r approaches zero.
Assuming 0 is constant that tends to zero, obtain
[
0ab
02 ds 0(b a) a)
O.
(23)
Consider the term e 20 1. we approximate the exponential function using first-order Taylor expansion for small :E:T::
20
1 ~ 20LT. - 20( - ()-
(24)
Hence, the drift term in the generalized OU bridge simplifies to
2
2
9: 11
1+ 20( -
(25)
20tirx
Tk
Consequently, in the limit 0, + O, the GOU bridge process described in Eq. (13) can be modelled by the following SDE:
dG; G., Gi + dWW
(26)
Tk
This equation precisely corresponds to the SDE representation the classical Brownian bridge process.
In contrast to the GOU bridge process in Eq. (13), the evolution the Brownian bridge is fully determined by the noise schedule 9k.t, resulting in simpler SDE representation. However, this constraint in the Brownian bridge reduces the flexibility in designing the generative process.
Note that the Brownian bridge is endpoint--oondtionee process relative reference Brownian motion, which the SDE governs:
dG. WW
(27)
This equation describes a pure diffusion process without drift, making it a specific instance of the GOU process.
A.2 PROOF OF THEOREM 3
To establish the proof, we begin by introducing essential definitions and assumptions. Definition (3-smooth). A function f R R is said to be 3-smooth if and only if
llf(w) f(v) Vf(v)(ww-- v)ll ww vll YW, R.
(28)
Theorem (Formal), Let ((K(()) be a loss function that is 3-smooth and satisfies the p-PL (Polyak- Lojasiewicz) condition in the ball (0u. R) of radius R 2N /0k////ee 2/((() (0g) where O. Then, with probability 1 - 8 over the choice of mini-batch of size b. stochastic gradient descent uN (SGD) with learning rate 17" NN((N converges to global solution in the ball B with exponential convergence rate:
byr
(K) (0.) S
3N (3N2 (ubb-1) 1))) ((4) (0o)
(29)
Here, N denotes the size of the training dataset. Furthermore, the proposed generative model yields smaller smoothness constant SHoG-DaY compared that of the classical model Betassical i.e., SuoG-DiF S Betassicul, implying that the learned distribution in HOG-Diff converges to the target distribution faster than classical generative models.
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Proof. Assume that the loss function (0) in Eq. (9) is minimized using standard Stochastic Gradient Descent (SGD) on a training dataset S = fx' IN
1=1- At the i-th iteration, parameter 0, is
updated using a mini- of size b as follows:
0 0+11 0. 1 (0.).
(30)
where 77 is the learning rate.
Following Liu et al. (2020) and Luo al. (2023a), we assume that (0) 3-smooth and satisfies the μ-PL condition in the ball B(,, RR) with R = 2N (0) where > 0. Then, 8 over the choice of mini-batch of size SGD with learning rate 1)
with probabilityy RN ( 1)) converges to global solution in the ball B(0u, R) with exponential convergence rate (Liu al., 2020):
2 (Te()((O))) 1 (1- bumn" ) (w)) = -(1- 8N(8N2 + p(b (1bb)))) 1))
g(0) (())
(31)
N
Next, show that the proposed framework has smaller smoothness constant than the clas- sical one-step model. Therefore, we focus exclusively on the spectral component (*s VA log Ps G. GT. )l2k from the full loss function in Eq. (9), as the feature-related part of the loss function in HOG-Diff aligns with that of the classical framework. For simplicity, we use the notation
()) | V log (G.
(x+)(x+) - Vx log (xx))] as the feature-related part of
the loss.
Next, we verify that ()) is 8-smooth under the assumptions given. Notice that the gradient of the loss function is given by:
Vε(0) = 2E [xso(x) - V Vlogp(x)) [x)
(32)
Hence,
[[V[(0)) - -((0)) = |||l||2|||||/ [ss, (x) - logp(x)) Vsa, (x) - (80, v logp(x)) T V5a, [x)]
$ 2F[[ so, (x) so, (x)l 11 V/, (x)] + ||s0 (x) logp(x)) V (x) Vso, (x)]l)-
(33)
Suppose [x)]| < C and [x) v log ( - C22 then we can obtain
VE((,)) vE(02)l < 2E B0 00 - 021 + C2 |0 - 02ll]
(34)
2(3 C1 Bv |1 0zll-
To satisfy the 3-smooth of ())), we require that
2(Ci3ag + C2Bray) S 33
(35)
This implies that the distribution learned by the proposed framework can converge to the target distribution. Therefore, following Chung al. (2022), we further assume that sg a sufficiently expressive parameterized score function so that 3xp = B7 log and Br=sD Br2 log Pilva-1"
The loss function used in classical diffusion generative models given by E((p) = E. (x,) Vx, (x, (x)]22)) To demonstrate that the proposed framework converges faster to the target distribution compared to the classical one-step generation framework, it suffices to show that: @VPITt_ < & and ( Σ 3
Let 2rjo and x' ~ 1 Since inject topological information from into x', x' can be viewed as being obtained by adding noise e (0,0 I")). The variance of Gaussian noise σ² controls the information retained x'. Hence, its distribution can be expressed as p(x') J q(x' <) (e) de.
Hence, can model x' as x' + where
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Therefore, obtain
|| xxx) (= / (a(x)) 6) - q(x2 <)) a(()
/ q - c) - - ()) de
(36)
-x² / x(e)
g -xxx)xx))
(x))x(x))ll -x1-x()) x²).
Hence, v log p(px)(()))))) llip S || log qxx'(())))))))))))) (lupp
By setting k = 3 and ho 4, we can obtain S7 Po S Ar log ocjo and Pos 8 log 9ejo" Therefore SHOG-Diff I Belassical, implying that the training process of HOG-Diff (8g) will converge faster than the classical generative framework (8p).
A.3 PROOF OF THEOREM 4
Here, we denote the expected reconstruction error each generation process as E(t)
E G -
Before comparing the reconstruction error bounds of HOG-Diff and the classical diffusion model, we first relate their optimal score-matching losses at the function-class level.
Lemma 6. Let (((()) denote the score-matching loss of the k-th HOG-Diff stage [Tk-1, Te), and let (cta(()) be the classical single-stage loss on [0, TV. Define the corresponding optimal values
e inf g(k) (O(K()))) g(k)
Lela, inf Lele (4p).
(37)
40
Assume that, for each k, the score network used in HOG-Diff on [Tk-1, is instantiated from the same backbone architecture as the classical score network on [0, (with the coarse graph G provided as an additional input that can be ignored by suitable parameter choice). Under this construction, the classical score class is contained each stage-wise score class, and we have
K
(() Σ
< Lels...
(38)
Proof. Fix any parameter to of the classical model and consider the associated score network Sip (G,, t) on [0, 7]. By the expressivity assumption, for each k there exists a parameter vector (kk) such that the HOG-Diff score network on [Tk-1, 7x] can represent exactly the same score function while ignoring the coarse guide:
(k) 000 (Ge, B
t E
1, TM]-
(39)
By definition, the stage-wise loss g(k) (0(+)) is obtained by restricting the classical objective (xp) to the time window [T--1, Fk] and using the same score function that interval. Therefore, under the above parameter choice we have
((k) (0())
(K) cls (4p),
(40)
where (4 (sp) denotes the contribution of the interval [TK-1, to the classical loss. Summing over all disjoint windows yields the exact decomposition
K
K
(k)
(g(k))
Σ
(k) cls
(4p = lch((p)
(41)
k=
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Taking the infimum over (0(1),.. ((.., on the left-hand side and up the hight-hand side gives
K
inf (()) (O(R) S inf (cla(p) fele, (
(42)
L
Lemma 6 reflects the structural fact that the classical model is special case of the multi-stage archi- tecture when all stages share the same backbone and the coarse graph input is ignored. Intuitively, each HOG-Diff stage only needs approximate a smoother conditional score (given the coarse graph), whereas the classical model has to fit the full marginal score in a single stage. For fixed network capacity, decomposing the problem into these easier conditional subproblems is therefore not expected to deteriorate the overall score-matching accuracy.
Theorem 4. Under standard Lipschitz and boundedness assumptions, the reconstruction ervor of HOG-Diff satisfies the following bound:
Enog (0) S Anog e eTwox
(43)
where Anog -CE PEREEE000 S ds and Thog EE ST. ds. Moreover, under the same assumptions, HOG-Diff admits tighter recon- struction ervor bound than classical diffusion models.
Ek
(s)dds +
Proof. Recall that E(t) E G: GGi reflects the expected error between the data reconstructed
with the ground truth score V log pt(-) and the learned scores so(.). In particular, G is obtained by solving the following oracle reversed time SDE:
dG; (f Dh V log logp.((.)) df + Is.t dW,, E (rk-1:7k),
(44)
whereas G. is governed based on the corresponding estimated reverse time SDE:
so (G.,) Here, fiet is the drift function of the Ornstein Uhlinnbehhlkk bridge. For simplicity, denote the Lipschitz norm by I|ip and fe, (G.)) ha.s (C., Go), where hu.s = 0, (1+ 20arr
dG. = (G. (G.) so
di + Iu.t dW, [TK-1.
(45)
Step 1: Single stage error bound.
To bound the expected reconstruction error E G Gr. || at each generation process, we
begin by analyzing how E G. - Gi evolves as time / is reversed from T to Th-1- The recon- struction error goes follows
/'
E(t) E
fr (G.) fG.) IR. (so(G..s) Vologps ds
Th
(h... - ff., ds + CE I th. 1se (o((G,,)) - Vologp. (G.) ds
< CE
|1a, -E()dds CE(TT) {" f'R2,,ds 12
S0
["
+ c
A I (R((so(G..8)) so(Gs. th. (s0((G..))) Va loggo (G.))l
Tk.s
(0) (dds+CC(T))) '' 112 ds 1' (C + Cl (i E(s)ds
$ (2e(k)(
(s())
azl
at L (s)))ss))
(46)
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Let vtt) = ETT t) and s' T it can be derived that
(() ETT - t) s (T () / (u s')o(s') ds'.
(47)
Here, (1:(T - 1) is a non-decreasing function. By applying Grönwall's inequality, we can derive that
v(t) S (T exp / 7a(Ta s')ds'
exp ou - 1) exp (x())ss.
(48)
(49)
Tu-
Hence,
E(t) < (t) exp J. (s)dss..
(50)
Evaluating the bound at the beginning of the k-tth reverse stage, that is t Tk-1, yields
(L"h, (x())ds)
E(Tk-1) S ax (Tk-1)
(51)
-00011100 l" Is CE(TT)
Th f ds exp (1,h n((s)ds
For later convenience, introduce the nonnegative stage-wise coefficients
A; C=((( dhods, D₁ C
1 ds,
Ta 1 (x(d)s.
(52)
T6-1
TW-1
Then the error bound can be written compactly as
E(Te-1) (Ax + D₂E(())) eh
k 1....... K.
(53)
The classical diffusion model corresponds to a single reverse stage on [0, T), and its error bound can be formulated in the functional form as
(0) S Acta ci
(54)
where Act. := (o)) 1od 9, ds and Tels := =h= (s)ss + C So" Aāds.
Step Multi-stage accumulated error bound.
The reverse process starts from the same prior for both oracle and Iearned SDEs, hence E(TK) = 0. Unrolling the recursion Eq. (53) backward from k = yields
E(TK-1) A
(55)
and
E(TK-2) & (Ak-1+ + Dx- ( Tk-1 & (AK-1 + D D exx-1.
(56)
Continuing inductively, one obtains for the initial time
To the general bound
K
A-1
Ehog (0)
A exp
Σ m=l
Tm
Dj
(57)
1=
j=1
where the empty product is defined as I[, D := 1.
Since each Dy is nonnegative, the elementary inequality Dj S De for all x. 2 0 implies
K
k-1
K
K
K-1
Enog (0) S
Ag exp
Σ Tm + Dj Dj
<
Σ
A
Σ
T + Σ Dj
(58)
m=1
j=1
A=1
m=1
1=1
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The latter inequality holds as all quantities in the exponent are nonnegative, enlarging the sum- mation ranges can only increase the exponent. Therefore, obtain the compact multi-stage error bound
Ehog (0) S Anog Thog
(59)
where Anog == Σ Ag. and Thog Σ Em=1 Tm + Σド Dj. This bound admits the same functional form Eq. (54) but accumulates the contributions of all K reverse stages from T 7k to To-
Step Comparing the two error bounds.
We now compare (Acla, in Eq. (54) with (Anog Thog) in Eq. (59).
First, by construction each subinterval [Tk-1, 1, Tk] lies inside [0, T] and the OU bridge horizon of the k-th HOG-Diff stage satisfies S T. The corresponding drift and diffusion coefficients are therefore pointwise dominated by those of the classical bridge:
Gh.s S g.
h S h2
(s(s) S (s),
E [1
(60)
Since all these quantities are nonnegative, their integrals over a subinterval are bounded by the integrals over the full horizon:
K
ds s 1-' ds
K
1" (s(s) - /-" 7ot(s))ss
Th
Σ
Σ
(61)
k=1
and
Σ Ns S {" 1i ds.
K
Σ
(62)
k=1
For the coefficient Anog. using the definition in Eq. (52) we have
Anog C
K
(K) (0)
L" Ih.o ds C²
Σ (O)))O
1/"
ge ds.
(63)
k= Tk-1
At the level of optimal score-matching losses, Lemma shows that the coarse-to- curriculum of HOG-Diff cannot be intrinsically worse than the single-stage formulation: Σk=1 ((() S Lels, In the regime where the optimization error is negligible and the loss is dominated by approximation error, it is natural to focus on solutions for which the curriculum does not increase the total score- matching loss, namely
( (0) < (eta(p))
(64)
This condition is clearly satisfied in our experiments. For K = 2, the combined spectrum loss of the Coarse and Fine stages, denoted as Coarse+Fine in Fig. 4, is lower than that the classical model (Classical (Noise)), providing empirical support for the above comparison.
Hence, we can bound Anog as
Anog S (alaalpp)) (' Gdds Acts-
(65)
For the exponent Thog in Eq. (59), we use the same domination obtain
K K Σ
K
T Thog =
Σ Tk +
7a
(s)ds + +C
Σ
1
ds
k=1
A=1
T6-1
(66)
(al(s)ds + C ff n3ds
<
10
Tak.
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Together with the bounds in Eq. (54) and Eq. (59), we obtain
Anog e ThoR I C
(67)
Combining these inequalities. we can finally conclude (no < Ec)s(0), which shows that the reconstruction eтror bound of HOG-Diff over the full trajectory T - 0 is no worse, and can be strictly sharper, than the corresponding bound for the classical diffusion model.
B ADDITIONAL EXPLANATION ON RELATED WORKS
B.1 HIGHER-ORDER NETWORKS
Graphs provide elegant and practical abstractions for modeling irregular relationships in empirical systems, transforming unstructured data into analyzable representations. However, their inherent restriction to pairwise interactions limits their ability to represent group dynamies (Battiston et al., 2020). For example, cyclic structures like benzene rings and functional groups play holistic role in molecular networks; densely interconnected structures, like simplices, often have collective influ- ence on social networks; and functional brain networks exhibit higher-order dependencies. To ad- dress this, various topological formalisms have been employed to describe data in terms of its higher- order relations, including simplicial complexes. cell complexes, and combinatorial complexes (Hajij al., 2023). As such, the study of higher-order networks has gained increasing attention for their capacity to capture higher-order interactions, with broad applications across domains such as influ- ence maximization (Zeng et al., 2024), financial system analysis (Mai et al., 2024; Xiao al., 2024), graph signal processing (Sardellitti et al., 2021; Roddenberry et al., 2022), and graph representation learning (Bodnar et al., 2021; Huang et al., 2024; Papamarkou al., 2024).
Given their broad applicability and theoretical richness, we next focus on two core frameworks for modelling higher-order interactions: simplicial complexes and cell complexes.
Simplicial Complexes. Simplicial complexes (SCs) are fundamental concepts in algebraic topology that flexibly subsume pairwise graphs (Hatcher, 2001). Specifically, simplices generalize funda- mental geometric structures such as points, lines, triangles, and tetrahedra, enabling the modelling of higher-order interactions in networks. They offer a robust framework for capturing multi-way relationships that extend beyond pairwise connections typically represented in classical networks.
A simplicial complex A' consists of a set of simplices of varying dimensions, including vertices (dimension 0), edges (dimension 1), and triangles (dimension 2).
A d-dimensional simplex is formed by set of (d 1) interacting nodes and includes all the subsets of 8 + nodes (with d < d), which are called the 5-dimensional faces the simplex. simplicial complex of dimension d is formed by simplices of dimension at most d glued along their faces.
Definition 7 (Simplicial complexes). A simplicial complex X is finite collection of node subsets closed under the operation of taking nonempty subsets, and such a node subset E X is called simplex.
We can obtain a clique complex, a particular kind of SCs, by extracting all cliques from a given graph and regarding them simplices. This implics that an empty triangle (owning [11, U2]. [y]. (02, U3] but without [vi, U2, 0z)) cannot occur in clique complexes.
Cell Complexes. Cell complexes (CCs) generalize simplicial complexes by incorporating gener- alized building blocks called cells instead of relying solely on simplices (Hatcher, 2001). This broader approach allows for the representation of many-body interactions that do not adhere to the strict requirements of simplicial complexes. For example, square can be interpreted as cell of four-body interactions whose faces are just four links. This flexibility is advantageous in scenarios such social networks, where, for instance, discussion group might not involve all-to-all pairwise interactions, or in protein interaction networks, where proteins in a complex may not bind pairwise.
Formally, cell complex is termed regular if each attaching map is a homeomorphism onto the clo- sure of the associated cell's image. Regular cell complexes generalize graphs, simplicial complexes, and polyhedral complexes while retaining many desirable combinatorial and intuitive properties of
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(a)
(b)
(c)
(d)
Triangle
Tetrahedron
Sphere
Torus
Figure 5: Visual illustration of cell complexes. (a) Triangle. (b) Tetrahedron. (c) Sphere. (d) Torus.
these simpler structures, In this paper, all cell complexes will be regular and consist of finitely many cells.
As shown in Fig. 5 (a) and (b), triangles and tetrahedra are two particular types of cell complexes called simplicial complexes (SCs). The only 2-cells they allow are triangle-shaped. The sphere shown in Fig. 5 (c) is 2-dimensional cell complex. It is constructed using two O-cells (i.e., nodes), connected by two 1-cells (i.e., the edges forming the equator). The equator serves as the boundary for two 2-dimensional disks (the hemispheres), which are glued together along the equator to form the sphere. The torus in Fig. 5 (d) is 2-dimensional cell complex formed by attaching a single 1- cell to itself in two directions form the loops of the torus. The resulting structure is then completed by attaching a 2-dimensional disk, forming the surface of the torus. Note that this is just one way to represent the torus cell complex, and other decompositions might lead to different numbers of cells and faces.
These topological frameworks provide the mathematical foundation for capturing multi-way inter- actions beyond pairwise graphs. Building upon this background, we next review advances in graph generative models, highlighting how existing approaches attempt learn graph distributions and where they fall short in preserving such higher-order structures.
B.2 GRAPH GENERATIVE MODELS
The study of graph generation seeks to synthesize graphs that align with the observed distribution. Graph generation has been extensively studied, which dates back to the early works of the random network models, such as the Erdös-Rényi (ER) model (Erdös et al., 1960) and the Barabási-Alls (BA) model (Barabási & Albert, 1999). While these models offer foundational insights, they often too simplistic to capture the complexity of graph distributions we encounter in practice.
Recent graph generative models have made great progress in graph distribution learning by exploit- ing the capacity of deep neural networks. GraphRNN (You et al., 2018) and Graph VAE (Simonovsky & Komodakis, 2018) adopt sequential strategies generate nodes and edges. MolGAN (De Cao & Kipf, 2018) integrates generative adversarial networks (GANs) with reinforcement learning objec- tives synthesize molecules with desired chemical properties. Shi al. (2020) generates molecu- lar graphs using flow-based approach, while GraphDF (Luo et al., 2021) adopts an autoregressive flow-based model with discrete latent variables. Additionally, GraphEBM (Liu al., 2021) em- ploys -hared model for molecular graph generation. However, the end-to-end structure of these methods often makes them more challenging to train compared to diffusion- generative models.
Diffusion- based Generative Models. A leap in graph generative models has been marked by the re- cent progress diffusion-based generative models (Song et al., 2021). EDP-GNN (Niu al., 2020) generates the adjacency matrix by learning the score function of the denoising diffusion process, while GDSS (Jo et al., 2022) extends this framework by simultaneously generating node features and an adjacency matrix with joint score function capturing the node-edge dependency. DiGress (Vi- gnac et al., 2023) addresses the discretization challenge due to Gaussian noise, while CDGS (Huang et al., 2023) designs a conditional diffusion model based on discrete graph structures. GSDM (Luo et al., 2023a) introduces an efficient graph diffusion model driven by low-rank diffusion SDEs on the spectrum of adjacency matrices. HypDiff (Fu et al., 2024) introduces geometrically latent diffusion on hyperbolic space to preserve the anisotropy of the graph. Despite these advancements, current methods are ineffective modeling the topological properties of higher-order systems since
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learning to denoise the noisy samples does not explicitly lead to preserving the intricate structural dependencies required for generating realistic graphs.
Diffusion Bridge. Diffusion bridge processes, i.e., processes conditioned the endpoints, have been widely adopted in image-related domains, including image generation (De Bortoli et al., 2021) 1021), image translation (Zhou et al., 2024), and image restoration (Luo al., 2023b; Yue et al., 2024). Recently, several studies have improved the graph generative framework of diffusion models by leveraging the diffusion bridge processes. Wu et al. (2022) inject physical information into process by incorporating informative prior to the drift. GruM (Jo et al., 2024) utilizes the OU bridge to condition the diffusion endpoint as the weighted mean of all possible final graphs. GLAD (Boget al., 2025) employs the Brownian bridge data samples. However, existing methods often overlook or inadvertently disrupt the higher-order topological structures in the graph generation process.
discrete latent space with endpoints conditioned
Hierarchical and Fragment- based Generation. Several recent studies have also explored hierar- chical and fragment-based generative frameworks. HiGen (Karami, 2024) decomposes graph gener- ation into multiple layers of abstraction, using separate neural networks to model intra-community structures and comer-community connections at each level. GPrinFlowNet (Mo et al., 2024) proposes semantic-preed preseeringg framework based low-to-high frequency generation curriculum, where the k-th intermediate generation state corresponds to the smallest principal components of the adjacency matrices. Dymond (Zeno al., 2021) focuses temporal motifs in dynamic graph gen- eration. HierDiff (Qiang et al., 2023) progressively generates fragment-level 3D geometries, refines them into fine-gre ine-grained fragments, and then assembles these fragments into complete molecules. Mi- CaM (Geng et al., 2023) synthesizes molecules by iteratively merging motifs.
Higher-order Generative Models. Since higher-order structures are intrinsic to many real-world systems, incorporating them into generative models could yield more faithful representations of complex phenomena. Existing efforts have primarily explored higher-order information through hypergraphs. HypeBoy (Kim al., 2024) introduces self-supervised hypergraph representation framework based on hyperedge filling task, which enhances embeddings rather than perform- ing direct generation. Hygene (Gailhard al., 2025) reduces hypergraph generation to standard graph generation via a bipartite representation. However, no prior approach has explicitly integrated higher-order topology due the stricter challenges of modeling multi-way rather than pairwise dependencies.
C DETAILS OF HOG-DIFF
This section elucidates spectral diffusion methodology, the parameterization of the score net- work, and the associated training and sampling procedures.
C.I OVERVIEW
As shown in Fig. HOG-Diff employs hierarchical, coarse-to- toare--t-ffine generation curriculum, where both forward diffusion and reverse denoising processes are decomposed into K easy-to-learn sub- processes. Each subprocess is realized using the GOU bridge process.
Higher-order Structures
dG, Og Mer, (G. G:)dt sw,
Forward diffusion
Go
G
G
G
Reverse denoising
dG, [(,, (G.) oZ.2 Pa,logppGG G. G. dw,dW;
Authentic
Random
Figure 6: Illustration of the coarse-to-fin generation process in HOG-Diff using the generalized OU bridge.
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C.2
SPECTRAL DIFFUSION
Classical graph diffusion approaches typically isotropii Gaussian noise directly into adjaceney matrices A, leading to various fundamental challenges. Firstly, the inherent non-uniqueness of graph representations implies that graph with n vertices can be equivalently modelled by up n! distinct adjacency matrices. This ambiguity requires a generative model that assigns probabilities uniformly across all equivalent adjacencies to accurately capture the graph's inherent symmetry. Additionally, unlike densely distributed image data, graphs typically follow a Pareto distribution and exhibit sparsity (Ghavasieh & De Domenico, 2024), that adjacency score functions lie on a low- liwddmmensional manifold. Consequently, noise injected into out-of-support regions of the full adjacency space severely degrades the signal-to-noise ratio, impairing the training of the score- matching process. Even for densely connected graphs, isotropic noise distorts global message- passing patterns by encouraging message-passing on sparsely connected regions. Moreover, the adjacency matrix scales quadratically with the number of nodes, making the direct generation of adjacency matrices computationally prohibitive for large-scale graphs.
To address these challenges, inspired by Martinkus et al. (2022) and Luo et al. (2023a), we introduce noise in the eigenvalue domain of the graph Laplacian matrix L D A, instead of the adjacency matrix A, where D denotes the diagonal degree matrix. As symmetric positive semi-definite matrix, the graph Laplacian can be diagonalized as L UAU Here, the orthogonal matrix U = [u,.... _ comprises the eigenvectors, and the diagonal matrix A diag(A);"" A')) holds the corresponding eigenvalues. The relationship between the Laplacian spectrum and the graph's 's topology has been extensively explored (Chung, 1997). For instance, the low-frequency components of the spectrum capture the global structural properties such as connectivity and clustering, whereas the figh-frey components are crucial for reconstructing local connectivity patterns. Therefore, the target graph distribution p(Gg) represents joint distribution of Xo and Au. exploiting the permutation invariance and structural robustness of the Laplacian spectrum.
Consequently, we split the reverse-time SDE into two parts that share drift and diffusion coefficients
as
2 dxx- = [Fa, - Ih.t Vx x log (Gr
log ...)) dī + g.,
(68)
dA, [F Gā. V log GG )] dī + dwW
Here, the superscript of x(()) and A (k) are dropped for simplicity, and fg, is determined according to Eq. (7). Note that, for molecular datasets, the intermediate endpoint is obtained by cell complex filtering: it preserves coarse higher-order connectivity, but the resulting node features can carry only an approximate atom-type signal. To stabilize training, we therefore add a small masked Gaussian perturbation to its node features during training
In addition, we conduct a comparative evaluation of HOG-Diff under two generative settings: one operating directly in the adjacency matrix domain, and the other in the Laplacian spectral domain. Using a consistent hyperparameter search space, the results summarized Tab. 12 show that gen- eration in the spectral domain generally achieves comparable performance across most evaluation metrics. We adopt the Laplacian spectral domain as the default diffusion space in HOG-Dif as the spectral approach is more efficient and better aligned with theoretical principles such as permutation invariance and signal concentration on low-dimensional manifolds.
C.3 SCORE NETWORK PARAMETRIZATION
The score network in HOG-Diff is a critical component responsible for estimating the score func- tions required to reverse the diffusion process effectively. The architecture of the proposed score network is depicted in Fig. 7. The input A, is computed from Un and (A) using the relation A, (())) (R) where the Laplacian matrix is given by { U
(R) UT and the diagonal
degree matrix is given by D." (k) = diag L To enhance the input to the Attention module,
we derive enriched node and edge features using the 1-step random walk matrix obtained from the binarized A, Specifically. the arrival probability vector is incorporated additional node features, while the truncated shortest path distance derived from the same matrix is employed as edge fea- tures. Temporal information is integrated into the outputs of the Attention and GCN modules using
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Attention
FILM
Forward
Feed
Feed Forward
Time Encoding
Concat
Vy loget (Ge1G())
Forward
Feed
GCN
FILM
GCN
FILM
Vxlogp: (Gr]6r)
Figure 7: Score Network Architecture of HOG-Diff. The score network integrates GCN and At- tention blocks to capture both local and global features, and further incorporates time information through FiLM layers. These enriched outputs are subsequently concatenated and processed by sepa- rate fecd-forward networks to produce predictions for V Vx, log Ge Gr.) and VA, log p (G, G..): respectively.
Feature-wise Linear Modulation (FiLM) (Perez et al., 2018) layers, following sinusoidal position embeddings (Waswani al., 2017).
A graph processing module is considered permutation invariant if its output remains unchanged under any permutation of its input, formally expressed as f(G) = x + ((x(G) = x, where (G') represents permutation of the input graph G. It is permutation equivariant when the output undergoes the same permutation as the input, formally defined as f(x(G)) = (f(()). It is worth noting that our network model is permutation equivalent, each model component avoids any node deeeenetd operations.
C.4 TRAINING AND SAMPLING PROCEDURE
The diffusion process in HOG-Diff is divided into K hierarchical intervals, denoted by 7 where 0 TO < < Th-1 < < .... < Tk T. Within each interval [Tk-1, we employ the GOU bridge process to ensure smooth transitions between intermediate states GT and G.,- We apply the cell complex filtering (CCF) operation at each interval to obtain structured, topologically meaningful intermediate states G. == CCF (G, S.p). Specifically, CCF prunes nodes and edges that are not contained in the closure of any p-cell within given cell com- plex S. At the initial state, the filtering operation is defined as CCF((, S, 0) = G, i.e., the filtering operation leaves the input unchanged. A special case arises at the final step, where the intermediate state is initialized from Gaussian noise, i.e., CCF (G,S S, K) (0, I). Since the GOU bridge process naturally reduces to a standard diffusion process when the terminal distribution is Gaussian noise, we omit the GOU bridge in the final segment [T [TK-1: TK.. and instead use the Variance Pre- serving (VP) SDE (Ho al., 2020; Song et al., 2021). In our experiments, we adopt a two-stage generation process, i.e., K = 2. The intermediate state G., is obtained via 2-cell complex filtering for molecule generation tasks, or via the 3-simplicial complex filtering for generic graph generation tasks. The rationale for this choice of filtering strategy is discussed in App. F.4.
To approximate the score functions Vx, G., and VA, log Peggee(,,l (G.|Gr,), we employ (k) (G., G.,,t), as introduced in App. C.3. This model consists of a node (s x (Gr, t)) and a spectrum (B (G,, GTt, t)) output. The network is trained by mini- mizing the following mcore-matching loss:
a neural network $9
(k)
glk) (0) B.,G,.G G. (w(t)cc]lss (k)
8.x
Vx log
(69)
+ca
A VA log (G, G. I11))
(b)
where (1) is a positive weighting function, and e1, 02 control the relative importance of vertices and spectrum.
The overall generation procedure is as follows. We sample (Xxx) Årx) from the prior distribu- tion and select Un as eigenbasis drawn from the training set. Reverse diffusion is then ap-
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plied across multiple stages, sequentially generating (X+x-1- ATx-1) ...... (X1, A1). (Xo, Au), where each stage implemented via the diffusion bridge and initialized from the output of the previous step. Finally, plausible samples with higher-order structures can be reconstructed as Go (x,, Lo ).
We provide the pseudo-code of the training and sampling process in Alg. I and Alg. 2, respectively.
Algorithm Training Algorithm of HOG-Diff
Input: Score network 8( training graph datasct G, training epochs M..
For the k-th step:
I: for m
M₁ do
2: Sample Gu = (Xu; Au) ~
3: S + (
4:
G <- CCF (Co. k), and GT <- CCF(Go, S, Vectorss Do Ao)
D Cell complex filtering
5: Uo
Ara < EigenDecoompesitionn DrH Arr)
7:
A + EigenDeconne -iigenDecompostioon D A=g-1)
8:
Sample ~ Uni TO Tx-1))
9:
x ~ ~((x, X XI,1I, X.,)
>> Eq. (6)
10:
A (K) p(A.l A11-1,
Eq. (6)
( < U
(1)
UT
A D L,
13: (() + 1 Vx log.. )11 calls [.
14:
2+ optimizer (R( (0))
15: end for
16: Return: (R)
Algorithm 2 Sampling Algorithm of HOG-Diff
Input: Trained score network #8 (k)
diffusion time split {ro."" ,xh, number of sampling steps
M
1: 1 <-
X Nii0, I) and AN ~ N(0,
3: Un ~ Unif (Ug Eigen (ectorsec)ssL
4: G.N -(Xrx rk: ArR: Drn GoA U0
5: for K to do
m M
to
7:
Sx, SA + 80
(k)
(Gi, G. t)
X. + X.
+ 20:roh
(X Xx)) Sx ôt gh Val.o wx ~ NO
9:
R A.-
0.
1
(A12 Â.) SAA 8t + oo, wA N(0
10:
L. - 0,A.00
11:
A. + D. Lx
12:
t
13:
end for
14:
A quantize(A) A.)
Quantize if necessary
15:
G-I-I (X,, A., A.)
end
17:
Return: Xo. Ao
D Tb =
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D EFFICIENCY AND COMPLEXITY ANALYSIS
D.1 ONE-TIME PREPROCESSING COMPLEXITY
When the targeted graph is not in the desired higher-order forms, one should also consider the one-time preprocessing procedure for cell filtering. Unlike cell lifting, which enumerates all cell structures and can incur substantial computational overhead, cell filtering can be performed much more efficiently. This is because filtering does not require the explicit enumeration of all cells; instead, it only checks whether individual nodes and edges participate in cell. For instance, the 2-cell filter requires only checking whether each edge belongs to some cycle.
One method to achieve the 2-cell filter is to use depth-first search (DFS) strategy. Starting from the adjacency matrix, we temporarily remove the edge (i,j) and initiate a DFS from node i, keeping track of the path length. If the target node j is visited within path length of I, the edge (i,j) is marked as belonging to a 2-cell of length at most In sparse graphs with n nodes and m edges, the time complexity of single DFS is O(m n.). With the path length limited to I, the DFS may traverse up to layers of recursion in the worst case. Therefore, the complexity of a single DFS is (min(mm + n, 1 kmng)), where kmaz is the maximum degree of the graph. For all m edges, the total complexity is O (m min(m + n, kmaa)).
Alternatively, matrix operations can be utilized to accelerate this process. By removing the edge (i,j) from the adjacency matrix A to obtain A, the presence of a path of length / between i and This indicates that the edge (i,j) belongs j can be determined by checking whether A j Assuming the graph has n nodes and m edges, the to a 2-cell with maximum length of I + 1. complexity of sparse matrix multiplication is (mm)). Since ! matrix multiplications are required, the total complexity is: Oll m² n). While this complexity is theoretically higher than the DFS approach, matrix methods can benefit from significant parallel acceleration on modern hardware, such GPUs and TPUs. In practice, this makes the matrix-based method competitive, especially for large-scale graphs or cases where is large.
For simplicial complexes, the number of p-simplices in a graph with n nodes and m edges is upper- bounded by O(nP-1), and they can be enumerated in O(a (g)=-3 m) time (Chiba & Nishizeki, 1985), where a (g) is the arboricity of the graph G, a measure of graph sparsity. Since arboricity is demonstrated to be at most ((m!/2) /2) and m I n², all p-simplices can thus be listed in O (n²-3m) 3m). Besides, the complexity of finding 2-simplex is estimated to be ((k)) m) with the Bron- Kerbosch algorithm (Bron & Kerbosch, 1973), where (k) denotes the average node degree, typically small value for empirical networks.
D.2 EMPIRICAL EFFICIENCY: PREPROCESSING TRAINING, AND SAMPLING
To complement the theoretical analysis, we empirically evaluate the scalability of our approach on standard benchmarks of varying scales. As shown in Fig. 8, we record the wall-clock time for the filtering procedure on datasets ranging from small community graphs to large-scale molecular datasets such as ZINC250K and MOSES. The results indicate that the filtering time scales reason- ably with data size. Given that this is one-time preprocessing cost, it confirms that HOG-Diff is computationally feasible for large-scale graph generation tasks.
After preprocessing, the training loop is identical to that of standard diffusion- generative models. The proposed coarse-to-fine strategy splits a long diffusion trajectory into shorter segments, each trained with smaller smoothness constant, which by Theorem 3 provably leads to faster convergence than classical models in idealized settings. Furthermore, since the sub-processes are independent, they can be trained in parallel for additional efficiency gains. In practice we observe no measurable slowdown relative to GDSS or DiGress.
Table 5: Training time comparison between GDSS and HOG-Diff.
Method Community-smm small Ego-small
Enzymes
ZINC250k QM9
GDSS HOG-Diff
13 min
19 min
1.9 h
1.2 h
14.0 h
20 min
12 min
0.5 h
3.1 h
6.5 h
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For inference, Tab. 6 shows that the extra guidance logic does not slow sampling. HOG-Diff matches DiGress on smaller graphs and achieves up to 12xspeedup over both baselines on the protein-seale Enzymes dataset.
GuacaMol (56 min 20 s)
10º
MOSES (15 min 12
Filtering Time (s)
10
ZINC250K (2 min 44 s)
10"
QM9 (22.084 s)
SBM (1.971 s)
10
Enzymes (0.298 s)
10
Community-small (0.06) 067 s) Ego-small (0. 035 s)
10
10
10"
10
10
Data Size
Figure 8: Empirical scalability of the filtering procedure. We report the wall-clock time required for the filtering (preprocessing) operation across datasets of varying sizes. The results show that the one-time preprocessing cost remains tractable even for large-scale datasets.
Table 6: Sampling time (s).
Method
Community-small
Enzymes
Ego-small
GDSS
41
110
28
DiGress
8
301
13
HOG-Diff
11
26
15
Scalability Large Benchmarks. As shown in Fig. 8, the one-time filtering cost remains tractable across large-scale benchmarks, including datasets with substantially more training sam- ples (MOSES) and larger graphs (SBM and GuacaMol). Importantly, this preprocessing is incurred only once; after that, the training loop is identical to standard diffusion models and the sampling overhead remains negligible. Detailed generation results on MOSES, GuacaMol, and SBM are ported in App. F.2.
In short, HOG-Diff scales gracefully: the extra cost is confined one-time preprocessing step. while the end-to-end runtime is dominated by standard GPU diffusion training and sampling.
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E EXPERIMENTAL SETUP
E. COMPUTING RESOURCES
In this work, all experiments conducted using PyTorch (Paszke, 2019) single NVIDIA L40S GPU with 46 GB memory and AMD EPYC 9374F 32-Core Processor.
E.2 OVERVIEW OF COMPARED METHODS
For benchmarking, we include various representative molecular generation models. Autoregressive models include GraphAF (Shi et al., 2020), GraphDF (Luo et al., 2021), and GraphArm (Kong al., 2023); the fragment-ber based MiCaM (Geng et al., 2023) creatively leverages motif information. In contrast, the remaining methods adopt a one-shot generation paradigm: GraphEBM serves as an energy-based model, SPECTRE (Martinkus et al., 2022) and GSDM (Luo et al., 2023a) incorporate spectral conditioning within GAN and diffusion frameworks, respectively, while EDP-GNN (Niu et al., 2020), GDSS (Jo et al., 2022), DiGress (Vignac al., 2023), and Cometh (Siraudin et al., 2025) represent diffusion-basse models, including MoFlow (Zang & Wang, 2020), CatFlow (Bijkelboom al., 2024), and De- FoG (Qin et al., 2025).
generation models. We also compare against advanced flow-based
E.3 MOLECULE GENERATION
Early efforts in molecule generation introduce sequence-based generative models and represent molecules as SMILES strings (Kusner al., 2017). Nevertheless, this representation frequently encounters challenges related to long dependency modelling and low validity issues, as the SMILES string fails ensure absolute validity. Therefore, in recent studies, graph representations are more commonly employed for molecule structures where atoms are represented as nodes and chemical bonds as connecting edges (Jo al., 2022). Consequently, this shift has driven the development of graph-based methodologies for molecule generation, which aim to produce valid, meaningful, and diverse molecules.
QM9 and ZINC250k. We evaluate the quality of generated molecules on two well-known molec- ular datasets: QM9 (Ramakrishnan et al., 2014) and ZINC250K (Irwin et al., 2012), and obtain the intermediate higher-order skeletons using the 2-cell complex filtering. In experiments, each molecule is preprocessed into graph comprising adjacency matrix A E {0, 2, and node feature matrix X {0, 1]mxd, where n denotes the maximum number of atoms in molecule of the dataset, and d is the number of possible atom types. The entries of A indicate the bond types: for no bond, 1 for the single bond, 2 for the double bond, and for the triple bond. Further, we scale A with constant scale of 3 in order to bound the input of the model in the interval [0, 1], and rescale the final sample of the generation process to recover the bond types. Following the stan- dard procedure (Shi et al., 2020; Luo et al., 2021), all molecules are kekulized by the RDKit library (Landrum et al., 2016) with hydrogen atoms removed. In addition, we make use of the valency correction proposed by Zang & Wang (2020). After generating samples by simulating the reverse diffusion process, the adjacency matrix entries are quantized to discrete values 0, 1, 2, 3 by applying value clipping. Specifically, values in (-o0, 0.5) mapped O, [0.5, 1.5) 1, [1.5, 2.5) 2, and [2.5, + to 3, ensuring the bond types align with their respective categories.
We report the baseline results taken from Jo et al. (2022) and Kong et (2023) or reproduced using publicly available code.
To comprehensively assess the quality of the generated molecules across datasets, we evaluate 10,000 generated samples using several key metrics: validity, validity with correction (Val. w/ corr.), uniqueness, novelty (Jo et al., 2022), Frechet ChemNet Distance (FCD) (Preuer al., 2018), and Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) MMD (Costa & Grave, 2010).
Validity computes the fraction of valid molecules before any corrections Jo et al. (2022), providing insight into the intrinsic quality of the generative process. While validity with correction (Val. w/ corr.) is measured as the fraction of valid molecules to all generated molecules after applying procesiooss corrections such valency adjustments or edge
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resampling. Whether molecules valid is generally determined by compliance with the
valence rules in RDKit (Landrum et al., 2016).
Novelty assesses the model's ability to generalize by calculating the percentage of gener- ated graphs that are not subgraphs of the training set, with two graphs considered identical if they are isomorphic.
Uniqueness quantifies the diversity of generated molecules as the ratio of unique samples to valid samples, removing duplicates that are subgraph -itgrapp-isomoppicc ensuring variety in the output.
FCD quantifies the similarity between generated and test molecules by leveraging the acti- vations ChemNet's penultimate layer, assessing the generation quality within the chem- ical space.
NSPDK-MMD evaluates the generation quality from the graph topology perspective by computing the MMD between the generated and test sets while considering both node and edge features.
MOSES. We conducted experiments the MOSES dataset (Polykovskiy et al., 2020), refine- ment of the ZINC database for molecular generation containing approximately 1.9 million lead-like molecules. This dataset serves rigorous testbed for assessing whether the proposed higher-order guidance remains computationally feasible and effective at scale. Following Vignac et al. (2023), the reported scores for FCD, SNN, and Scaffold similarity are computed the dataset made of separate scaffolds, which measures the ability of the networks to predict new ring structures.
GuacaMol. GuacaMol (Brown et al., 2019) is derived from the ChEMBL database and contains 1.4M molecules, from which I.IM are used for training. We apply preprocessing step similar to Vignac et al. (2023), filtering out molecules that cannot be mapped from SMILES to a graph and back to SMILES. In Fig. 3(efft), we report the scores of the FCD and KL metries following Brown et al. (2019), Note that, unlike other settings, the FCD score is normalized using Dsoore = exp(-0.2-FCI) to obtain final value between 0 and I, meaning that higher values indicate better performance.
For the KI. score, compute the following descriptors for both the generated and reference molecules: BertzCT, MolLogP, MolWt, TPSA, NumHAAceptorss, NumHDonors, NumRotatable- Bonds, AummAliphaiicRings NumAromaticRings, and the ECFP4 fingerprint-based similarity the nearest neighbor. The KL divergence DKL,i is computed for each descriptor to measure the dif- ference between the two molecular sets. These divergences are then combined produce a final normalized score = 1 E exp(-DKL,)).
E.4 GENERIC GRAPH GENERATION
To display the topology distribution learning ability, assess HOG-Diff over four common generic
graph datasets: (1) Community-smon containing 100 randomly generated community graphs; (2) Ego-small, comprising 200 small ego graphs derived from the Citeseer network dataset; (3) En- zymes, featuring 587 protein graphs representing tertiary structures of enzymes from the BRENDA database; and (4) Stochastic Block Model (SBM), larger-scale dataset.
Community-sin small, Ego-small, and Enzymes. We follow the standard experimental and evaluation settings from Jo et al. (2022), including the same train/test splits and the use of the Gaussian Earth Mover's Distance (EMD) kernel for MMD computation, to ensure fair comparisons with baseline models- - except for the Stochastic Block Model (SBM) dataset. We use node degree and spectral features of the graph Laplacian decomposition hand-crafted input features. Baseline results sourced from Jo et al. (2022); Kong et al. (2023) or reproduced using the corresponding publicly available code.
SBMs. To further evaluate the scalability and robustness of HOG-Diff, we conduct experiments on the Stochastic Block Model (SBM) dataset, which comprises graphs of larger scale and evaluated under distinct experimental protocol from that used in Sec. 4.2. The dataset consists of 200 syn- thetic graphs generated using the stochastic block model. The number of communities uniformly sampled between 2 and 5, and the number of nodes within each community is uniformly sampled
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between 20 and 40. Edges created with probabilities of 0.3 for intra-community connections and 0.05 for inter-community connections.
We adopt the evaluation setting introduced by Martinkus et al. (2022), including the same data splits, feature initialization, and the use of Total Variation (TV) distance to compute the MMDs. The TV kernel is adopted since it offers higher computational efficiency compared to the Earth Mover's Distance (EMD) kernel, especially for large graphs. We also report the percentage of valid, unique, and novel (V.U.N.) samples among the generated graphs further assess the ability of our model to capture the properties of the targeted distributions correctly. Baseline results are sourced from Vignac al. (2023); Jo al. (2024) or reproduced using the corresponding publicly available code.
E.5 DATA STATISTICS
Tab. 7 summarizes the key characteristics of the datasets utilized in this study. The table outlines the type dataset, the total number of graphs, and the range of graph sizes (VV). Additionally, it also provides the number of distinct node types and edge types for each dataset. Notably, the Community-smon and Ego-small datasets contain relatively small graphs, whereas Enzymes, SBM and the molecular datasets exhibit greater diversity in terms of graph size and complexity.
Table 7:
Dataset summary.
Node types
Edge types
Dataset
Graph type
#Graphs
#Nodes
Community-
Synthetic
100
12 <
[V
<
20
Ego-small
Citation
200
4<
18
1
Enzymes
Protein
587
10 5
V
S
125
1
1
SBM
Synthetic
200
44 IV]
S 187
I
QM9
Molecule
133,885
1< V
S
4
3
ZINC250K
Molecule
249,455
6 2 2 8
V
< 38
9
3
GuacaMol
Molecule
1,398,223
V
S 88
12
3
MOSES
Molecule
1,936,962
S 27
7
3
F ADDITIONAL EXPERIMENTAL RESULTS
F.1 EXTENDED RESULTS STANDARD MOLECULAR BENCHMARKS: Q.99 AND ZINC250K
To visually assess the capability the proposed method in molecular generation, Fig. compares distributional results the molecular datasets. Specifically, first calculate the Morgan finger- prints (Rogers & Hahn, 2010) of molecules, which have been widely utilized in drug discovery for capturing structural information. Subsequently, apply Uniform Manifold Approximation and Projection (UMAP), a nonlinear dimensionality-reaee reducttion method that preserves local similarities, embed the fingerprints into two dimensions and plot the resulting distributions.
On the QM9 dataset, the distributions generated by our model closely align with those of the refer- set. On the more complex ZINC250k dataset, the generated distributions show slight deviations but remain well aligned. These results provide an intuitive demonstration of strong generative per- formance and complement the findings Tab.
We provide the standard deviation results and the additional metric validity with correction (Val. w/ corr.) in Tabs, and 9. Baseline results are sourced from Jo al. (2022); Kong al. (2023) or reproduced using the corresponding publicly available code.
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(a)
(b)
(c)
10
10
8-
6-
UMAR22
4-
QM9 ref ZINC250K ref
QM9 ref
ZINC250K ref ZINC250K gen
2-
QM9 gen
10
0.0
2.5
5.0
7'5
12.5
12 UMAP 1
14
UMAP
UMAP
Figure 9: Visualization of chemical space distributions. Molecular representations are obtained us- ing Morgan fingerprints and subsequently visualized through dimensionality reduction with Uniform Manifold Approximation and Projection (UMAP). Contour lines denote the probability distributions of reference (ref) and generated (gen) molecules.
(a) illustrates the distributional shift across QM9 (green) and ZINC250K (blue), while (b) and (c) show that our generative model faithfully captures the distinct distributions of each dataset.
Table 8: Comparison of different methods QM9. We report the means and standard deviations of 3 runs. The best results are highlighted in bold. Asterisks (*) indicate that the source did not report standard deviations.
Method
Val.f
com.1
Uni.f
Nov:
FCD]
NSPDK
GraphAF (Sli et 2020)
74.43
100.00:0.000
88.64 38.64
86.59 16.59911.95
5.625::0.259
0.001
GrahDFF (Lan et al., 2021)
93.88-+4.76
100.00+00000
98.58-+0.25
98.54=0.48
10.928-0.038
0.064++0.000
GraphAmm (Kong
2023)
90.25
100.00
95.62
70.39
1,220
0.002
MiCaM (Geng ct al., 2023)
99.93
100.00
93.89
83.25
1.045
0,001
GraphEBM (Liu al., 2021)
8.22-#2.24
100.00=0.000
97.90=10.05
97.01:0.11
6.143:20.411
0.030:10.004
SPECTRE" (Martinkus et 2022)
87,30
100.00
35.70
97.28
47,960
0.163
GSDM (Luo et al., 2023a)
99.90
100.00
2.650
0.003
EDP-GNN (Niu et al., 2020)
47.52::360
100.00::0.00
99.25:10.55
86.581 1,85
2.680:10221
0.005
GDSS (Jo et al., 2022)
95.72:
100.00::0.00
98.46:40.61
86.27
2.900::0.282
0.003
DiGress (Vignac al., 2023)
99:00
100.00
96.66
33.40
0,360
0.00n5
MoFlow (Zang & Wang, 2020)
91. 01.36:21.25
100.00 100.00==0.00
98 03.655
94.72:0.77
4, 4,77
0.017
CatFlow (Eijkelboom
2024)
99.81
100.00
99.95
0.441
Cometh (Siraudin et al., 2025)
99.59110.09
100.0.1.000
96.75:10.21
72.06:1..19
0,248:10.012
0.000
DeFoG (Qin al., 2025)
99.26++0.10
100.00+=0.00
72.57+11899
HOG-Diff (Ours)
98.74=20.22
100.00=00.00
17.14=
75.12=20.99
0.172:00.002
0.0003=000000
Table 9: Comparison of different methods on ZINC250k. We report the means and standard devi- ations of 3 runs. The best results are highlighted in bold. Asterisks (*) indicate that the source did not report standard deviations.
Method
Val.t
Val. w/ corr:
Uni.1
Nov
FCDJ
NSPDKL
GraphAF (Shi et al., 2020)
68.47+00.99
100.00++0.00
98.64+0.60
99.99-0001
16.023+0.451
0.044++0005
GrapDDF (Luo al., 2021)
90.61+:4.00
100.00=00.00
99.63==001
100.00=0:00
13.546+0.150
0.177=0000
GraphArm (Kong al. 2023)
88.23
100.00
99.46
100.00
16,260
0.055
MiCaM (Geng al., 2023)
100.00
100.00
88.48
99.98
31.495
0.166
GraphEBM (Liu et al., 2021)
5,29:13.83
99, 90.96.10.02
18.79.1..15
100.00:1..00
45.43333
0.212:0.075
SPECTRE* (Martinkus
2022)
18,440
0.109
GSDM (1 et al., 2023a)
2023aa)
92.70
100.00
12.956
0.017
EDP-GNN (Niu et al. 2020)
82.97+22.73
100.00=00.00
79:00000
100.00:0.000
16.737+
0.049:00.006
GDSS (Jo et al., 2022)
97.01100777
100.00::0.00
19.64.1..13
100.0010.000
14 65661006680
0.01910.001
DiGress (Vignac al., 2023)
91.02
100.00
81.23
100.00
0.082
MoFlow (Zang& & Wang, 2020)
63.1111..17
100.00::0.00
99.9910.01
100:0010.00
20.931
0.046:10.02
CatFlow (Eijkelboom el al., 2024)
99.95
99,99
100.00
13.211
DeFoG (Qin al., 2025)
94.97-44.026
99 19.99-+0.01
99.98-+0.02
100.00+0.00
2.030-00083
0.002+0001
HOG-Diff (Ours)
98. 08.56:20.12
100.00=00000
99,9 99.96=20.02
99.53=00.07
1.633-00012
0.001
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F.2 SCALABILITY EVALUATION ON LARGE BENCHMARKS: MOSES AND SBM
To address concerns regarding scalability and performance on larger benchmarks, we extended our evaluation encompass both large-scale molecular datasets and generic graph benchmarks.
MOSES. We conducted experiments the MOSES dataset (Polykovskiy et al., 2020). The results are presented Tab, 10. HOG-Diff demonstrates superior performance in terms of distribution
learning, achieving the lowest FCD score. This indicates that method more accurately captures the underlying chemical and topological distribution of the large-scale training set.
Table 10: Generation performance on the large-scale MOSES dataset. Top results are highlighted in bold.
Method
Val. f
Uni. 1
Nov. f
Filters f
FCD
SNN
Scaf
GraphINVENT (Mercado al., 2021)
96.4
99.8
95.0
1.22
0.54
DiGress (Vignac et al., 2023)
85.7
100.0
95.0
97.1
1.19
0.52
14.8
DisCo (Xu ct al., 2024)
88.3
100.0
97.7
95.6
1.44
0.50
15.1
Cometh (Siraudin et al., 2025)
90.5
99.9
92.6
99.1
1.27
0.54
16.0
DeFoG (Qin et al., 2025)
92.8
99.9
92.1
98.9
1.95
0.55
14.4
HOG-Diff (Ours)
99.7=0022
100.0
89.1
96.7
0.94
0.52
8.6
SBMs. To further evaluate the scalability and robustness of HOG-Diff, we report results on the
Stochastic Block Model (SBM) dataset in Tab. 11, which comprises graphs of larger scale and is evaluated under distinct experimental protocol from that used in Sec. 4.2. As shown in Tab. 11, HOG-Diff achieves competitive performance compared to the state-of-the-art.
Table Generation results on the SBM dataset. The best and second- best results are highlighted in bold and underlined, respectively.
Method
Deg.l
Clus
Orb.
Spec.
Avg.J
V.U.N.1
GraphRNN (You et al., 2018)
0.0055
0,0584
0.0785
0.0065
0.0372
0.05
GRAN (Liao et al., 2019)
0.0113
0.0553
0.0540
0.0054
0.0315
0.25
EDP-GNN (Niu et al., 2020)
0.0011
0.0552
0.0520
0.0070
0.0288
SPECTRE (Martinkus et al., 2022)
0.0015
0.0521
0.0412
0.0056
0.0251
0.53
GDSS (Jo et al., 2022)
0.0212
0.0646
0.0894
0.0128
0.0470
0.05
RefineGen (Bergmeister et al., 2024)
0.0141
0.0528
0.0809
0.0071
0.0387
0.75
GruM (Jo et al., 2024)
0.0015
0.0589
0.0450
0.0077
0.0283
0.85
DeFoG (Qin et al., 2025)
0.0006
0.0517
0.0556
0.0054
0.0283
0.80
HOG-DifT (Ours)
0.0028
0,0500
0.0428
0.0043
0.0249
0.83
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F.3 ANALYSIS OF DIFFUSION DOMAIN CHOICE
To investigate the impact of diffusion domain choice, we perform ablation experiments comparing two variants of HOG-Diff: one that operates directly in the adjacency matrix domain, and another in the Laplacian spectral domain. Both variants are trained under the same hyperparameter search space for fair comparison.
As shown in Tab. 12, the spectral variant achieves comparable performance to the odjacency-based approach across most evaluation metries on QM9 and ZINC250K. Despite similar results, we adopt the Laplacian spectral domain as the default diffusion space in HOG-Diff due to its theoretical and practical advantages. Specifically, the spectral domain offers greater efficiency and aligns natu- rally with core graph principles such permutation invariance and signal concentration on low- dimensional manifolds.
Table 12: Comparison of diffusion domains in HOG-Diff.
Dataset
Domain
/SPDK
FCD|
Val. f
Val. w/ corr.
Uni.1
Nov.
Adjacency matrix
0.0004
0.264
99.08
100.00
95.90
67.78
QM9
Laplacian spectrum
0.0003
0.172
98.74
100.00
97.10
75.12
Adjacency matrix
0.006
4.259
96.75
100.00
99.78
99.98
ZINC250K
Laplacian spectrum
0.001
1.633
98.56
100.00
99.96
99.53
F.4 RATIONALE FOR SELECTING TOPOLOGICAL FILTERS
The filtering strategy in HOG-Diff is primarily driven by data statistics and complemented by do- main knowledge to ensure meaningful choices.
In molecular generation experiments, we employ 2-cells as guides because they are ubiquitous in molecular graphs and capture critical chemical information, e.g., functional groups. In particular, 2-cells play a crucial role in determining the three-dimensional conformation, electron distribution, and target binding mode compounds, making them highly informative. Consequently. Tab. shows that conditioning 2-cells alone provides strong guidance and achieves competitive results across all metrics.
In contrast, cells with dimension > 2 are extremely sparse in the standard benchmarks (see Tab. 13). Identifying them requires increasingly complex preprocessing, such as detecting candidate higher- order topological structures and verifying their validity, which introduces non-trivial computational overhead. Moreover, given that these benchmarks contain only a few hundred nodes, guides with higher-dimensional cells would cover less than 0.05% of possible structures, introducing both un- necessary computation and a risk of overfitting without evident benefit. In practice, 2-cells already capture virtually all of the higher-order structure present in these datasets.
We further tested guiding molecular generation with 2-simplices but observed worse performance. This expected, summarized in Tab. 13, higher-dimensional simplices are extremely rare in molecular datasets (i.e., QM9 and ZINC250K), while 2-cells are comparatively more abundant. Moreover, most functional structures in molecules do not satisfy the requirements of simplices. These findings confirm that aligning the guide with the natural higher-order structures of the data improves both training efficiency and sample quality, whereas mismatched guides provide limited benefit.
In contrast, generic graphs (particularly those in social or biological domains) often exhibit diverse simplicial structures, making simplicial filtering more suitable. This demonstrates the adaptability of HOG-Diff across domains.
Overall, the proposed framework is most effective when the intermediate skeletons reflect structures naturally present the data. This aligns with prior work in graph representation learning, which shows that the benefits of higher-order representations are most pronounced in datasets that rich in such structures (Huang et al., 2024). Indeed, many real-world networks, including social, biologi- cal, and citation networks, exhibit abundant higher-order cells and simplices. Thus, many real- world
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tasks naturally fall into the "topologically rich" regime in which method is particularly advanta- geous.
Table 13: Average counts of simplices and cells across datasets. "_" indicates computation exceeded 10h.
Dataset
O-simplices 1-simplices 2- simplicess 3-simplices simmplies 2-cells 3-cells
Community
15.28
35.15
27.45
8.24
0.76
40.92
0.00
Ego-small
6.41
8.70
4.61
1.64
0.35
5.66
0.00
Enzymes
33.03
63.27
25.93
3.05
0.01
SBM
104.01
500.41
441.25
90.88
4.42
QM9
8.80
9.40
0.47
0.00
0.00
1.84
0.10
ZINC250K
23.15
24.90
0.06
0.00
0.00
2.77
0.00
F.5
ADDITIONAL VISUALIZATIONS
Fig. 10 provides qualitative view of the generative trajectory of HOG-Diff for the model trained on ZINC250K. shows several intermediate graph states from the initial noisy prior the final molecule, illustrating the coarse-to-fing to-fie refinement process.
X dlag(A)
A
G
T2 T
(,,T2)//2
122
T3/2
to =
Figure 10: Visualization of molecular graphs at different stages of the reverse generative process. Model trained on the ZINC250k dataset.
In this section, we additionally provide the visualizations of the generated graphs for both molecule generation tasks and generic graph generation tasks. Figs. 11-16 illustrate non-curated generated samples. HOG-Diff demonstrates the capability to generate high-quality samples that closely re- semble the topological properties empirical data while preserving essential structural details.
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14
QH
CH
(a) Training set
(b) HOG-Diff (Ours)
Figure 11: Visualization of random samples taken from the HOG-Diff trained on the QM9 dataset.
Eour 2zo
fuo
Pruodd
EVOR Eoo
xHXX"
(a) Training set
(b) HOG-Diff (Ours)
Figure 12: Visualization of random samples taken from the HOG-Diff trained on the ZINC250K dataset.
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(a) Training set
(b) HOG-Diff (Ours)
Figure 13: Visual comparison between training graph samples and generated graph samples produced by HOG-Diff on the Community-sma dataset.
(a) Training set
(b) HOG-Diff (Ours)
Figure 14: Visual comparison between training set graph samples and generated graph samples produced by HOG-Diff on the Ego-small dataset.
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CLUU
(a) Training set
(b) HOG-Diff (Ours)
Figure 15: Visual comparison between training graph samples and generated graph samples produced by HOG-Diff on the Enzymes dataset.
(a) Training set
(b) HOG-Diff (Ours)
Figure 16: Visual comparison between training set graph samples and generated graph samples produced by HOG-Diff on the SBM dataset.
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G LIMITATIONS AND FURTHER WORK
We propose a principled graph generation framework that explicitly exploits higher-order topologi- cal cues guide the generative process. This design enables HOG-Diff to achieve strong empirical performance across various tasks, including molecule and generic graph generation. While HOG- Diff shows superior performance, future work would benefit from improving our framework.
As discussed in Sec. 4.2, the performance of the proposed framework depends on the presence of explicit higher-order structures. Although previous studies have shown that such structures are prevalent in many empirical systems, certain types of graphs, such as the ego-small dataset, lack this topological richness. In these cases, the benefits of higher-order diffusion guidance diminish, and the performance advantage becomes less pronounced.
In addition, our framework built around the use of higher-order structures as diffusion guides, enabled by the Cell Complex Filtering (CCF) mechanism. As detailed in App. D, we introduce simplified computational fornulation of the CCF that facilitates efficient implementation of both cell complex and simplicial complex filters. However, extending this framework to capture more intricate topologial elements, such as motifs, higher-order cells (beyond second order). and topo- logical cavities, poses significant computational challenges. We leave this scalability bottleneck as future work.
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Reviews:
[{"review_id": "uAqIBmWLM6", "reviewer": "Reviewer_J3EW", "summary": "The paper proposes HOG-Diff, a diffusion-based graph generation framework that explicitly incorporates higher-order topological structures (e.g., cell complexes) through a coarse-to-fine generation curriculum. The method leverages generalized Ornstein-Uhlenbeck (GOU) bridges in the spectral domain and demonstrates superior performance on molecular and generic graph generation tasks, supported by theoretical guarantees.", "questions": "Can the proposed filtering and bridge approach scale to larger graphs?\n\nHow sensitive are results to the number of hierarchical stages？", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Introduces a Higher-Order Guided Diffusion model, which for the first time explicitly incorporates higher-order structures (e.g., cell complexes) into graph generation tasks, addressing the limitations of existing diffusion models in capturing graph topological properties.\n\nDesigns a Cell Complex Filtering (CCF) operation to enable a coarse-to-fine generation process, simplifying the modeling of complex graph distributions.\n\nAchieves  superior results on molecular datasets (QM9, ZINC250k) and generic graph datasets.", "weaknesses": "The proposed coarse-to-fine process and cell complex filtering could be computationally expensive, especially for large graphs or dense higher-order structures. No runtime is provided.\n\nAlthough the authors emphasize interpretability as a motivation, there are few visual or quantitative analyses explaining what “higher-order topology” the model actually captures or how it improves the generation quality.", "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": "TZNVTnUWbl", "reviewer": "Reviewer_zqT6", "summary": "This paper introduces HOG-Diff, a novel diffusion framework for graph generation. The core methodology involves \"lifting\" graphs into a higher-order cell complex representation and make the diffusion process respect higher-order topological structures. The authors posit that this higher-order diffusion better captures the intrinsic topological structure of graphs. The paper provides a theoretical analysis of the proposed methodology and presents experimental results to demonstrate its performance.", "questions": "Most of my questions are raised in weakness. This includes the problem of writing, the main motivation of the paper, the significance of the empirical studies and the scalability. Please refer to that section.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The idea of graph generation via lifted is refreshing.\n- The paper is built upon a solid mathematical formulation.\n- The theoretical foundations, including theorems and propositions, are well-presented, and the accompanying proofs provided in the appendix appear to be correct.", "weaknesses": "Despite its theoretical grounding, the paper suffers from significant weaknesses in its justification, clarity, and empirical evaluation.\n\n- **Clarity of Writing and Notation.** A primary concern is the clarity of the writing and the logical flow. The notation, in particular, seems inconsistent and potentially overloaded. For instance, Proposition 2 introduces the $p$-cell complex filtered graph $G_p$, where the subscript $p$ denotes the cell dimension. However, in subsequent sections, graphs are denoted as $G_t$, where $t$ represents time. It is unclear if $G_t$ retains a connection to a specific $p$-cell filtration or if this notation is dropped. This ambiguity makes the methodology difficult to follow, thus hinders a precise understanding of the manuscript’s main contribution and the relevant algorithm design. \n- **Justification of Core Motivation and Theoretical Bound.** The central motivation of the paper is that lifting graphs to a higher-order space improves generative fidelity, which I believe requires stronger justification. While higher-order representations are beneficial for systems with inherent group-level interactions (e.g., co-authorship networks), this does not mean converting graphs to their high-order counterpart would be beneficial for downstream tasks. As shown in [1], a hypergraph and its corresponding graph clique expansion can be information-theoretically equivalent in certain contexts. The paper does not sufficiently argue why this lifting operation provides a superior inductive bias for learning the data distribution. This concern extends to the primary theoretical result (Theorem 4). The analysis demonstrates that the final step of the HOG-Diff curriculum (from $\\tau_1$ to $0$) achieves a sharper error bound than a classical model's entire process (from $T$ to $0$). This comparison does not seem equitable. The proof relies on $\\tau_1 < T$ and a superior (coarsened) starting point which induces $\\mathcal{E}(\\tau_1) \\le \\mathcal{E}^{\\prime}(T)$. A more convincing analysis would compare the entire $K$-step generative process of HOG-Diff (from $T \\to 0$) against the classical $T \\to 0$ process, rather than isolating the final, and arguably simpler, refinement step. I might have misunderstanding on the theoretical bound and would appreciate if the authors can correct me. \n- **Critique of Diffusion Model Baselines.** The paper's motivation appears to be based on a critique of graph diffusion models. The claim that diffusion models are \"ineffective at modeling the topological properties\" (Lines 70-72) overlooks recent successes in discrete-state diffusion. Similarly, the criticism that \"the graph adjacency matrix quickly degrades into a dense matrix\" (Line 72) and that injecting \"isotropic Gaussian noise...is detrimental\" (Line 76) primarily applies to early continuous diffusion models. This critique does not hold for more recent discrete frameworks such as DiGress [3], Cometh [2], and DeFoG [4], which deviate from the Gaussian noise paradigm. The paper should position its contribution against these more relevant state-of-the-art discrete models, as the problems it claims to solve have, to a large extent, already been addressed.\n- **Significance and Fairness of Empirical Evaluation.** The empirical evaluation lacks significance due to the omission of standard benchmarks for graph generation. Datasets such as SBM, Planar, and Tree graphs are critical as they possess ground-truth topological properties (e.g., planarity, acyclicity). Evaluating validity on these datasets is fundamental to assessing a model's ability to capture intrinsic graph patterns. The paper's reliance on statistics ratio (degree, clustering) and molecular validity (which is basically valency validity) does not sufficiently demonstrate that the model can learn and respect these fine-grained topological constraints, especially in plain graph generation tasks. Furthermore, the comparison to existing work is incomplete. Key SOTA diffusion models, such as Cometh [2], are not included. In the molecular benchmarks (QM9, ZINC250k), the reported improvement over DeFoG [4] appears marginal, but a proper comparison is impossible as standard deviations are not provided.\n- **Scalability and Computational Complexity.** Scalability remains an unanswered question. All experiments are conducted on relatively small-scale graphs. Standard, larger benchmarks (e.g., MOSES, GUACAMOL for molecules; SBMs, the larger version of COMM-20) are absent. This is critical as the core premise of HOG-Diff relies on computing higher-order structures. It is unclear (1) whether computing these structures is computationally feasible for large graphs, and (2) how the performance and complexity of HOG-Diff scale relative to standard graph generation models on such data. Although Appendix D discussed computational complexity, it would be good to see empirical validation as well. \n- **Clarity of Algorithmic Details.** The paper would benefit from a more precise definition of its algorithmic settings in the main text. Key details, such as the exact dimensions and representations of $G$ and $G_p$, are not sufficiently clear. This lack of detail hinders a rigorous analysis of the algorithm's space and time complexity.\n\nReferences\n\n[1] B. Tang et al. Training-free message passing for learning on hypergraphs. ICLR 2025\n\n[2] A. Siraudin et al. Cometh: A continuous-time discrete-state graph diffusion model. TMLR 2025\n\n[3] C. Vignac et al. DiGress: Discrete Denoising diffusion for graph generation. ICLR 2023\n\n[4] Y. Qin et al. DeFoG: Discrete Flow Matching for Graph Generation. ICML 2025.", "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": "JxHPmiTHLw", "reviewer": "Reviewer_NCCR", "summary": "This paper introduces a new diffusion-based graph generative model that integrates higher-order topological information to guide the generation process. The proposed mechanism explicitly leverages cell complex filtering to learn higher-order structures and employs diffusion bridges to enable a coarse-to-fine generation scheme. This design aims to capture richer topological properties and improve the structural coherence of the generated graphs. Experimental results on both molecular and generic graph benchmarks demonstrate performance comparable to recent state-of-the-art graph generative methods.", "questions": "- **Hierarchical time windows:** What is the specific role of the number of hierarchical time windows $K$ in the model? If it is treated as a hyperparameter, how is its value determined and how it is related to the number of noising steps? How do changes in $K$ affect the model‚ as generative dynamics?\n\n- **Computational cost:** The model introduces several components that could significantly impact computational efficiency, such as the Cell Complex Filtering operation, the injection of noise into the Laplacian instead of the adjacency matrix (which requires recomputing the Laplacian at each noising step), and the different modules in the score network (GNC and Transformer blocks). Although the authors provide a theoretical argument that the framework converges to the target distribution at an equal or faster rate than classical models, there is no empirical evidence supporting this claim. Having this analysis would be particularly relevant, as the reported performance gains over existing methods are not substantial. Therefore, demonstrating that the added complexity of using higher-order structures translates into practical efficiency or quality improvements (beyond the provided ablation) would strengthen the paper.\n\n### Minor questions\n\n- Why do the authors use different sets of baselines across experiments? Is there a specific reason for this?\n\n- The authors claim that existing graph generative models are ill-suited to capture topological properties because they are largely adapted from image generation frameworks. They further argue that the adjacency matrix quickly degrades into a dense matrix with uniformly distributed entries, which is detrimental to graph structure preservation. However, the paper would benefit from clearer empirical or theoretical evidence supporting these claims. In particular, several recent graph diffusion models employ diffusion processes on discrete state-spaces that are adapted for graph generation and have shown strong performance in maintaining graph topology. As a justification for the need to learn higher-order topology, the paper mentions that the framework should ensure equivariance. However, many existing graph generative models already satisfy equivariance, so this motivation is a bit unclear. In particular, it would be relevant to have proof that the proposed model respects equivariance.\n\n- Could the higher-order information be further exploited to enhance scalability, for example, by incorporating higher-order structure generation within a latent diffusion framework?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The use of higher-order topological structures to guide the diffusion process represents a novel contribution. The alignment of the different diffusion stages with hierarchical graph structures seems an innovative way of capturing complex structural patterns.\n\n- The proposed framework is theoretically grounded and proves to be an effective extension of standard diffusion models to non-Euclidean domains.\n\n- Empirical results are competitive with state-of-the-art methods, showing the potential of the proposed approach.", "weaknesses": "- The reported performance improvements are relatively modest. Without an accompanying analysis of computational efficiency or scalability, the practical advantages of introducing higher-order topology remain unclear.\n\n- The methodological presentation of the framework could be more intuitive. Although the paper explains how higher-order structures are obtained and integrated into the diffusion process, the overall exposition is difficult to follow, as it often mixes high-level intuition with theoretical descriptions.\n\n- The paper would benefit from a clearer discussion of the potential limitations and possible extensions of the proposed approach, such as scalability, conditional generation for broader applicability across domains, and sensitivity to the choice of higher-order representations. It would be relevant to examine how the model scales with graph size (the SBM dataset, while described in the paper as large-scale, is of relatively modest size) and how conditional generation could be incorporated alongside the higher-order structures.", "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": "T7qB0R5PDX", "reviewer": "Reviewer_ftpE", "summary": "The authors propose the Higher-order Guided Diffusion (HOG-Diff) framework that guides the diffusion process to preserve higher-order topological structures in a coarse-to-fine manner. HOG-Diff utilizes cell complex filtering (CCF), which extracts hierarchical skeletons used as a generation guidance and enables efficient computation. The effectiveness of HOG-Diff is demonstrated through extensive experiments on molecular and benchmark graph datasets with autoregressive and one-shot graph generation methods.", "questions": "- As mentioned in the weakness section, can you provide quantitative and qualitative evaluation on the generated graphs that could verify the successful preservation of higher-order structures? Such information will support the claim that the performance improvement lies on the preservation of higher-order structure. (e.g., MSE of persistence image, Curvature Filtrations [1], etc.)\n- What advantages does guidance have over direct conditioning on higher-order information in the diffusion process? (other than interpretability)\n- The term “consistently outperforms” in L400 seems little excessive, as only the NSPDK and FCD scores in Table 1 shows improvement whereas the other three metrics differ.\n- What exactly do you mean by explicitly preserves higher-order structures in L399 about Fig.3? I don’t see the alignment of specific structures like the connected components or loops. (Also, the light-blue edges are extremely hard to see. I recommend changing the color to be more visible.)\n- Can you clarify the novelty of CCF? I am aware of the idea of simplifying simplicial complex while preserving topological invariants used in discrete morse theory or other TDL works. It would be nice to show how CCF differs from such methods.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Novel and interesting idea of integrating topology and diffusion as a guiding structure.\n- Solid theoretical grounding suggests how GOU process and Doob’s h-transform can be derived to guide topological information in diffusion process, as well as the error bounds.\n- Comprehensive evaluation on various graph datasets using a wide range of baseline graph generative methods.\n- Insightful dataset-level analysis, in terms of the discussion on dataset-dependent effects (e.g., weaker improvement on Ego-small due to limited higher-order structures).", "weaknesses": "- Lack of topological assessment: The authors do not provide evaluation(quantitative or qualitative) on higher-order topology preservation in the generated graphs.\n- Notation and presentation complexity: I understand it may be somewhat inevitable, but the excessive use of notations is hard to follow for readers.", "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)