Cometh: A continuous-time discrete-state graph diffusion model
Authors: Antoine Siraudin
Fragkiskos D. Malliaros
Christopher Morris
Venue: Accepted by TMLR ()

Abstract: Discrete-state denoising diffusion models led to state-of-the-art performance in graph generation, especially in the molecular domain. Recently, they have been transposed to continuous time, allowing more flexibility in the reverse process and a better trade-off between sampling efficiency and quality. Here, to leverage the benefits of both approaches, we propose Cometh, a continuous-time discrete-state graph diffusion model, tailored to the specificities of graph data. In addition, we also successfully replaced the set of structural encodings previously used in the discrete graph diffusion model with a single random-walk-based encoding, providing a simple and principled way to boost the model's expressive power. Empirically, we show that integrating continuous time leads to significant improvements across various metrics over state-of-the-art discrete-state diffusion models on a large set of molecular and non-molecular benchmark datasets. In terms of VUN samples, Cometh obtains a near-perfect performance of 99.5% on the planar graph dataset and outperforms DiGress by 12.6% on the large GuacaMol dataset.

Published in Transactions on Machine Learning Research (04/2025)
Cometh: model
A continuous-time discrete-state graph diffusion
Antoine Siraudin RWTH Aachen
intoine.siraudin wwg.. woclllo.rtthhachhhe
Fragkiskos Malliaros
Suptral,,Suétee, Inria
Université Poris-Saclo
Christopher Morris
RWTH Aachen
Reviewed on OpenReview: https:
// openreview. net/ forum? id= nulimmRrrjj
Abstract
Discrete-state denoising diffusion models led to state-of-the-art performance in graph generu- (on,, especially in the molecular domain. Recently, they have been transposed to continuous time, allowing more flexibility in the reverse process and better trade-off between sampling efficiency and quality. However, they have not yet been applied to graphs. Here, to leverage the benefits of both approaches for graphs, we propose COMETH, continuous-tine discrete- state graph diffusion model, tailored to the specificities of graph data. In addition, we also successfully replaced the set structural features previously used in discrete graph diffusion models with a single random-walk- walk-basod encoding, providing simple and principled way to boost the model's expressive power. Empirically, we show that integrating continuous time leads to significant improvements across various metrics over state-of-than theor------rtt discrete-state diffusion models on large set of molecular and non-molecular benchmark datascts. In terms of valid, unique, and novel (VUN) samples, COMETH obtains near-perfect performance of 99.5% on the planar graph dataset and outperforms DIGRESS by 12.6% on the large GuacaMol dataset.
1 Introduction
Denoising diffusion models (Ho et al. 2020 Song ct al. 2020 among the most prominent and successful classes of generative models. Intuitively, these models aim to denoise diffusion trajectories and produce new samples by sampling noise and recursively denoising it, often outperforming competing architectures in tasks such as image and video generation (Sohl-Dickstein et al. 2015 Yang et al. 2023). Recently, a large set of works, e.g., Chen et al. (2023); Jo et al. (2022) 2024): Vignac et al. 20222 aimed to leverage diffusion models for graph generation, e.g.; the generation of molecular structures. One class of such models embeds the graphs into a continuous space and adds Gaussian noise the node features and graph adjacency matrix {Jo et al. 2022). However, such noise destroys the graphs' sparsity, resulting in complete, noisy graphs without meaningful structural information, making it difficult for the denoising network to capture the structural properties of the data. Therefore, däscrete-state graph diffusion models such as DIGRESS (Vignac et al. 2022 have been proposed, exhibiting competitive performance against their continuous-state counterparts. These models utilize a categorical corruption process Austin et al. 2021 making them more suited to the discrete structure of graph data.
In parallel, the above Caussian-noisee based diffusion models have been extended continuous time ( et al. 20201, i.e., relying continuous-time stochastic process (Capasso & Bakstein, 2021 by formulating

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the forward process stochastic differential equation. In addition, discrete-state diffusion models have recently been transposed to continuous time (Campbell et al. 2022 Sun et al. 2022): relying continuous- time Markov chains (CTMC). Unlike their discrete-ting counterparts, which define fixed time scale during training, they allow training using a contimuous- time scale and leave the choice of the time discrctization strategy for the sampling slage. Hence, incorporating continuous time enables a more optimal balance between sampling efficiency and quality while providing greater flexibility in designing the reverse process, as various CTMC simulation tools can be utilized (Campbell et al.) 2022] Sun al. 2022) However, extending continuous- time discrete-state diffusion models to graphs is not straightforward. Unlike other discrete data, such as text, where all tokens share the same support, nodes and edges in graphs have distinct attributes and must be handled separately. Furthermore, the noise models used for other data particularly the uniform noise model, which yields very dense graphs may be suboptimal for spanse, real-world graphs (Vignac el al., 2022
Present work Hence, to leverage the benefits of both approaches for graph generation, i.., discrete-state and continuous-time, we propose COMETH, continuous-lime discrete-state graph diffusion model, integrating graph data into continuous diffusion model framework; see Figure I for an overview of COMETH. Concretely, we
1. propose new noise model adapted to graph specificities, featuring distinct noising processes for nodes and edges, and we extend the marginal transitions previously proposed for graph data to the contimuous-time setting.
2. In addition, we successfully replace the structural features used in most previous discrete graph diffusion models with random- walk--asll encoding. We prove that it generalizes most of the features used in DIGRESS, representing a simple and elegant way to boost the model's expressivity and reach state-of- the-r-r--t----tt--ee performance.
3. Empirically, we show that integrating contimous-time into a discrete-state graph diffusion model leads to state-of-the-art results on synthetic and established molecular benchmark datasets across varions metrics. For example, in terms of VUN (valid, unique, and novel) samples, COMETH obtains a near-perfect performance of 99.5% on the planar graph dataset and outperforms DIGRESS by 12.6% on the large GuacaMol dataset.
Overall, COMETH a state-of- graph diffusion model that combines the benefits of a discrete-state space with the fleribility of a continuous-ting scole in its sampling algorithm design.
1.1 Related work
Diffusion models are a prominent class of generative models successfully applied to many data modalities, such as images, videos, or point clouds (Yang et al. 2023).
Graph generation is well-studied task applied to various application domains, such molecule generation, floorplan generation, or abstract syntax tree generation (Shabani et al. 2023 Shi et al. 2019). We can roughly categorize graph generation approaches into auto-regressive models such as Kong et al. (0023 ; You et al. (2018) Zhao et al.] (2024): Jang et al. (2024b) and one-shot models such as diffusion models. The main ad vvaatage of one-shot models over auto-regressive ones is that they generate the whole graph a single step and do not require any complex procodure to select a node ordering. On the other hand, auto-rogressive models are more flexible regarding the size of the generated graph, which can remain unknown beforehand, and they usually enjoy more favorable complexity since the don't require as many sampling steps as diffusion models.
While the first. diffusion models for graph generation leveraged continuuss-stal spaces NNu et al. 2020). they are now largely replaced by discrete-state diffusion models (Haefeli et al. 2022 Vignac et al. 2022), using a discreto-time scale. However, using discrete time constrains the sampling scheme a particular form called ancestral sampling, which prevents the exploration of alternative sampling strategics that could optimize sampling time or enhance sampling quality. Recently, there has been surge of interest in continuous-time
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CE loss
Predict GO
Compute
A G)
Sample ((
Figure 1: Overview of COMETH. During training, the graph transformer learns to predict the clean graph (! from noisy graph G Unlike previous discrete-time diffusion models, COMETH performs transitions at any time / E [0, 1]. During sampling, the clean graph GOD) is first predicted and used to compute the reverse rate Rt.0 R G) as defined Equation (2). Next, a r-leaping step performed to sample G(t-7), with the step length fixed to т. Optionally, multiple corrector steps can be applied at - т, which experimentally improves sample quality.
discrete-state diffusson models [Lou al. 2023 Gat al., 2024: Shi al. 2024), which have demonstrated
promising performance on text generation tasks, rivaling GPT-2 models of comparable scale.
Another line of research considers lifting the graphs into contimuous-staee space and applying Gaussian noise to the node and edge features matrices (Niu et al. 2020 Jo et al.) 2022 2024). Such continuous noise allows the generation of continuous features to be handled smoothly, such as the generation of atomic coordinates in molecular graphs (Jo et al., 2024). The above Gaussian- seissiaannnis--basee diffusion models have many successful applications computational biology (Corso et al. (2023); Yim et al. (2023))- However, they almost exclusively consider point-cloud generation, focusing on modeling the goometry of the molocules and ignoring structural information. In addition, some hybrid approaches also exist that consider jointly modeling the 2D molecular graphs and their 3D geometry (Hua et al. 2024 Le et al. 2023 Vignac al. 2023). These models usually roly on continuous noise for the atomic coordinates and discrete noisc for the atom and edge types.
Recent works have tried to scale graph generative models in size (Bergmeister et al. 2023 Luo et al.) 2023 Qin et al. 2023). Such frameworks are often built on top of previously proposed approaches, c.g., SPARSEDIFF (Qin et al. 2023) based on DIGRESS. Therefore, these scaling methods are likely to apply approach.
Concurrently with our work, Xu et al. 20244 proposed a continuous-time discrete-state graph diffusson model. While they experiment using an MPNN as the backbone with limited success, we effectively utilize RRWP (Relative Random Walk Probabilities) as a positional encoding. Moreover, our experimental evaluation of COMETH more comprehensive, as we successfully implement the predictor-corrector mechanism (Diethelm et al. 2004) and include a conditional generation experiment.
2 Background
In the following, we overview the continuous-time discrcto-state diffusion framework on which COMETH builds. We provide complete description of this framework in Appendiz B.] providing intuitions and technical details, and refer to Yang et al. (2023) for general introduction to diffusion models.
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2.1
Continuous-tima discrete diffusion
In the discrete diffusion setting, aim at modeling a discrete data distribution Pdata(2(0) where z(0) 2 and Z is a finite set with cardinality S = 2 A continuous-ting diffusion model (Campbell et al. 2022 Sun ict al. 2022 Lou et al. 2023) is a stochastic process, running from time t= 0 to t = 1. In the following, we denote the marginal distributions of the state 20 E Z at time t by (=x))), and q1)a(2 | (x()) denotes the conditional distribution of the state 2(1) given the state 2(+) E Z at some time E [0, 1]. We also denote ò2,- the Kronecker delta, which equals 1 if = and otherwise. We define a forward process which gradually transforms the marginal distribution (0(2(0)) = (=)ta( into qı (1()))), that is "close" to an easy-to-sample prior distribution Prot(=(1), e.g., a uniform categorical distribution.
We define the forward process as a continuous-lime Markov chain (CTMC). The current state z(() alternates between resting in the current state and transitioning to another state, where transition rate matrir R E R controls the dynamics of the CTMC. Formally, the forward process is defined through the infinitesimal transition probability from 2(t) to another state E 2, for a infinitesimal time step dt between time l and + dl,
qt+daje (2) (((()) == 85 + R (2(0) )).
where (2(1,, denotes the entry of Rit) that gives the rate from 20t) to Intuitively, the process is more likely to transition to a state where (2(t), 2) is high, and R is designed so that qi (z(1) closely
approximates the prior distribution Pref-
The generative process is formulated as the time reversal of the forward process. therefore interpolating from qi (2) to )))). The rate of this reverse CTMC, R2(0) is intractable (Campbell et al. 2022 and has to be modeled by a parametrized approximation, i.e.,
20jo (3 =00))) 210) 20 2(0)) Poje (=)(((00)) (200) for = + E
(2, 2) -2 (2,2)
where p/ (z(0) =) is the denoising neural network with parameters 0.
For efficient training, the conditional distribution ())((()) | 20)) needs to be analytically obtained; see Appendix B.1 for more details. As demonstrated in Campbell et al. (2022). , this property is achieved by choosing R Rt with B(t) E R being the noise schedule and R. RSS is constant base rate matrix. We can now generale samples by simulating the reverse process from (= 1 to ( 0. Different algorithms can be employed for this purpose, such as Euler's method (Sun et al.) 2022), r-leaping and the predictor-corrector scheme (Campbell et al., 2022). In particular, the predictor-corrector method consists in, after each r-leaping step using R()) (=d 10 Ed), in applying several corrector steps using the expression defined in Campbell al. (2022), i.e., A(02 = M10.0 + RO0, This rate admits ge (2())) as its stationary distribution, which means that applying the corrector steps brings the distribution noisy graphs at time t closer to the marginal distribution of the forward process.
3 A Continuous-Ting
Saccrete-State Graph Diffusion Model
Here, we present our COMETH framework, a continuous-time discrete-state diffusson model for graph generation. Let [m, n|] == (m...., n. C N. We denote n-order attributed graph as pair G (G, X, E), where G = (V) F(G)) is a graph, X E 10, 1)nxa, ; for > 0, is a node feature matriz, and E E (0, (0,1)*xx0 1xxxxb, for b > 0, is an edge feature tensor. Note that node and edge features are considered to be discrete and consequently encoded in a one-hot encoding. For notational convenience, in the following, we denote the graph at time E [0, by G(O, the node feature of node i E (GG) at time t by z!" e [[L,]] and similarly the edge feature of edge (i.j) E EG) at time t by et e [l. b]. In addition, we treat the absence of an edge as a special edge with a unique edge feature. By 1,, we denote an all-one vector of suitable size, by I, the identity matrix of suitable size, while o denotes the all-zero matrix of suitable size. Moreover, by a', we denote the transpose of the vector
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3.1 Forward process factorization
Considering the graph state-space would result in a prohibitively large state, making it impossible to construct transition matrix. Therefore, we consider that the forward process factorizes and that the noise propagates independently on each node and edge, enabling us to consider node and edge state spaces separately. Formally, let G (G, X, E) be n-order attributed graph, then we have
12
(G(++dtt) G(1))
II ae (a!
(c) I
II SKJ
dt+ (Cg
(1) C13
(t4-dt/2
The above factorization reveals challenge not yet addressed for the contimuous-time diffusion model. In other types of discrete data, such as text, all tokens share the same support. In contrast, nodes and edges have different attributes, and their respective sets of labels may have different sizes. We, therefore, need to define their respective forward processes differently.
We then define pair of rate matrices (R&2 R with R = Rx) and R = (g) where 0 is the noise schedule and Rx E Rdxd Rø E _exe are base rate matrices for nodes and edges, respectively. The two matrices differ in size and allow for controling the dynamics of the forward process in distinct ways for nodes and edges. Note that we followed the design choice of Campbell et al. 2022 introduced in Section
3.2 Noise model: extending marginal transitions to continuous time
In the following, we detail the proposed noise model.
Noise model Several noise models have been proposed for discrete diffusion models, including uniform transitions, absorbing transitions (Austin et al. 2021), and marginal transitions (Vignac et al.) 2022). We propose to use a rate matrix that is analogous to the marginal transition matrix, as is well adapted for graph data. However, marginal transitions have not yet been utilized in the continuous time framework. To address this, extend this concept by constructing the following rate matrices, i.e.,
Rx 1m'x - and Rg 1mm
where mx and mg vectors reprosenting the marginal distributions mx and mg of node and edge labels, i.e, they contain the frequency of the different labels in the dataset. With such rate, the transition rate to a particular state depends on its marginal probability. Specifically, the more common a node or edge label is in the dataset, the higher the transition rate to that label. Consequently, this approach helps preserve sparsity in noisy graphs by favoring transitions to the "no edge" label. Embedding this inductive bias in the noise model simplifies the model's task, as it no longer needs to reconstruct this sparsity during the generative process.
Deriving explicit expression for the forward process We now aim to understand the relationship between the rate matrix and the endpoint of the diffusion process at t= 1, qi (GO)). In fact, need to choose Pref so that (G) x (G) for determining the appropriate prior distribution that aligns with the chosen rate matrix. Ideally, we seek to use the product of distributions II, mx II mg, which, as demonstrated in Vignac et al. (2022, Theorem 4.1), is optimal within the space of distributions factorized over nodes and edges. In the following, we explain how to design the noise schedule 3 to achieve this.
To better understand the relationship between the rate matrix and (z()),, we require an explicit expression that readily links (2)) 2(0)) to ((t), However, Campbell et al. (2022) providess the following closed-form for the former, which does not easily allow for the direct deduction of appropriate prior distribution,
ahhl (2 = =k 1 =1) = (Ppesp A L. ((s)ds p-
{
(1)
where R, PAP-1 and exp refer to the matrix exponential. Given Equation [ it is not straightforward to determine the form of (10(=(1) 2(0)), and, consequently, the form of qı (z(1).
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We, therefore, prove that this expression can be refined, offering more precise insights into the behavior of the forward process.
Proposition 1. For CTMC (z()) )e((o1)) with rate matrix R(() = B(t and Rs 1lm I, the forward
process can be written as
(*) e-- 30
- I+ (1- -1- 1m',
where (0) = = =
and B(0 = - B(s)ds.
Therefore, if we can design 3(0) so that lim e -(0) O, we get that lime-11 Q(1) Im', which means that 2(1)) is sampled from the categorical distribution m whatever the value of 20)), ie., (=(1)) m. In our case, since Proposition ! holds for every node and edge, and given that the forward process is factorized, this would yield that Qi (G =I II mx Ih<s mE as desired. We provide a more detailed explanation in Appendix B.2
Proposed noise schedule Even though, in theory, one should set lim;-1 300) +oo so that lim,-1 Q(9) 1m', it considerably restricts the space possible noise schedules. Relying on the exponentially decreasing behavior of the cumulative noise schedule e - one only needs to ensure that ,(1) is high enough so that Q( satisfyingly approximates Im'.. (Cmmpblll ct al. (2022) proposed an exponential noise schedule for categorical data. Instead, we followed an older heuristic, i.e., using cosine-shaped schedule as introduced for discrete-time models in Nichol & Dhariwal (2021). We therefore propose to use cosine noise schedule, where
and L B(s)ds = aa(1 cos =t
π B(t) = 2 sin
2
Here, a is a constant factor. Since e - e given that a is high enough, di (G will be close to Im',, and we can therefore use Il. mx Iiis me as our prior distribution. We provide more intuition on our noise schedule in Appendix C.2 Finally, noising an n-order attributed graph G (G,X, X, E) amounts to sampling
from the following distribution,
( |G) =X _Q
4:j0
Since we consider only undirected graphs, we apply noise the upper-triangular part of and symmetrize. Note that we apply noise to graph in the same manner in discrete-time models, with the only difference being that t is no longer discrete.
3.3 Parametrization and optimization
We can now formulate the approximate reverse rate for our graph generation model. We set the uni-dimensional rates according to the parameterized approximation derived by Campbell al. (2022).
(8) Rx x ...))
-R x(0)
Σ
Z:jo
i |
Pojt
(()) I
(0)
(()).
for 2(") + Z..
(2)
4:j0
x
x
We elaborate using those rates to simulate the reverse process in Section C.1 Formally, we set the overall reverse rate to
and similarly for the edge reverse rate Red 0 (0)
cof Esj
= Σ S (
A.. (4 .)) +
B onyy R (c E.y),
i<3
where 6 Grs is if and & differ only on node and otherwise, and similarly &ghn (hy for edges. The reverse rate exhibits the classic diffusion model parametrization, which relies on predicting clean data point G given a noisy input G We, therefore, train a denoising neural network p (G G(9) for this purpose. The outputs are normalized into probability distributions for node and edge labels.

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The model can be optimized using the continuous-tine ELBO proposed in Campbell et al. 2022 Additionally, they incorporate direct model supervision by optimizing an auxiliary objective, i.e., the cross-entropy loss between the predicted clean graph and the ground truth G However, our preliminary experiments with this ELBO yielded poor empirical results, as detailed in Appendix E.3 Given that the ELBO and cross-cntropy share the same optinum Campbell et al. (2022 Proposition 8), and the cross-entropy loss alone has been successfully applied for graphs the discrete-time case (Vignac et al. 2022), we opted to use the cross-entropy loss LCE as our optimization objective,
Paat (G
Σ log Poje (2
G
log Poj (
(0) Cig
G
(3)
where A E R+ is scaling factor that controls the relative importance of edges and nodes in the loss. Tn practice, we set À > 1 so that the model prioritizes predicting correct graph structure over predicting correct node labels.
3.4 Simple and powerful positional encodings with RRWP
In all our experiments, we use the graph transformer proposed by Vignac et al. 2022); 13 see Figure in the appendix. Relying on the fact that discrete diffusion models preserve the sparsity of noisy graphs, they propose a large set of features to compute at each denoising step to boost the expressivity of the model. This set includes Laplacian features, connected components features, and node- and graph-level cycle counts. Even though this set of features has been successfully used in follow-up works, e.g., Vignac et al.] (2023), Qin ct al. (2023), Igashov et al. (2023), no theorctical nor experimental study exists to investigate the relevance of those particular features. In addition, rich literature on encodings in graph learning has been developed, e.g., LapPE [Kreuzer et al. 2021), SignNet (Lim et al. 2023), RWSE (Dwivedi et al. 2021), SPE (Huang ct al. transferred to the generative setting.
2023 which led us to believe that powerful encodings developed for discriminative models should be
Specifically, in our experiments, we leverage the relative random- walk probabilites (RRWP) encoding, intro- duced Ma et al. (0023). Denoting A the adjacency matrix of graph G, D the diagonal degree matrix, and M D A the degroe-normalized adjacency matrix, for each pair of nodes (i.j) E V())?, the RRWP encoding computes
plf = [4ij.MMs Mij. My ........... M, 11
(4)
where K refers to the maximum length of the random walks. The entry PK corresponds to the RRWP encoding of node i; therefore, we leverage them as node encodings. This encoding provides an efficient and elegant solution for boosting model expressivity and performance through a unified encoding.
In the following, we show that RRWP encoding can (approxinately) determine two nodes lie in the same connected components, approximate the size of the largest connected component, and count small cycles.
Theorem 2 (Informal). For n E N, let 9n denote the set of n-order graphs and for a graph G 9n let V(G)= = [[, Then RRWP composed with universally approximating ferd-forward neural network can
determine two vertices are in the same connected component,
approximate the size of the largest connected component in G,
3. approximately count the number p-cycles, for < 5, in which a node is contained.
See Appendix B.3 for the detailed formal statements.
However, we can also show that RRWP encodings cannot detect if a node is contained in a large cycle of a given graph. We say the RRWP encoding counts the number of p-cycles for p 2 2 if there do not exist two graphs, one containing at least one p-cycle while the other does not, while the RRWP encodings of the two graphs are equivalent.
Proposition 3. For p 2 8, the RRWP encoding does not count the number of p-cycles.
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Hence, for p 2 8 and K 2 there exists graph and two vertex pairs (r,s), V(G)² such that (r,s) is contained in C while (v,w) is not and P P.
3.5 Equivariance properties
Since graphs are invariant to node reordering, it essential to design methods that capture this fundamental properly of the data. Relying on the similarities between COMETH and DIGRESS, we establish that COMETH is cermutation-equivaat
and that our loss is permutation-invaree sampling scheme and the predictor-corrector yield exchangeable distributions, i.c., the model assigns each graph permutation the same probability. Since those results mainly stem from proofs outlined in Vignac et al. 2022). we moved them to Appendix B.4
We also establish that the r-leaping
3.6 Conditional generation
If unconditional generation is essential to designing an efficient diffussin model, conditioning the generation on some high-level property is critical in numerous real-world applications (Corso et al. 2023 Lee Min 2022). In addition, Vignac guided by regressor the target property. Instead, we rely the classifier-free guidance procedure (Ho & Salimans 2021 which consists in jointly optimizing conditional model p pf((G" (G G vy and an
[2022] used classifier guidance, which relies on trained unconditional model
unconditional model Poje (G G( by randomly dropping the conditioner during training, We provide further details concerning conditional generation in appendix B.5
4 Experiments
In the following, we empirically evaluate COMETH on synthetic and real-world graph generation benchmarks. For all datasets, the results obtained with the model are denoted as COMETH, while the results using the predictor-corrector are referred to as COMETH-PC. We sample from Cometh using the tau-leaping algorithm
as described in Campbell et al. (2022), procedure that we detail in Appendix C.1 We also conduct several
ablation studies in Appendix E Our code is available at ggithub.com//siraate. /asiraudiiia Cometh
4.1 Synthetic graph generation: Planar and SBM
Here, we outline experiments regarding synthetic graph generation.
Datasets and metrics We first validate COMETH on two benchmarks proposed by Martinkus et al. 2022 PLANAR and SBM. We measure four standard metrics to assess the ability of our model to capture structural propertics of the graph distributions, i.c., degree (Degrec), clustcring cocfficient (Cluster), orbit count(Orbit) and the eigenvalues of the graph Laplacian (Spectrum). The reported values are the maximum mean discrepancies (MMD) between those metrics evaluated the generated graphs and the test set. We also report the percentage of valid, unique, and novel (VUN) samples among the generated graphs to further assess the ability of our model to capture the properties of the targeted distributions correctly. We provide a detailed description of the metries in Appendix D.1
Baselines We evaluate our model against several graph diffusion models, namely DIGRESS (Vignac el al. 2022), GRUM (Jo et al. 2024h, two autoregressive models, GRAN (Liao et al. 2019), and GRAPHRNN (You ct al. 2018), and a GAN, SPECTRE (Martinkus et al. 2022). We re-evaluatod previous statc-of-the-art models over live runs, namely DIGRESS and GRUM, provide error bars for the results.
Results See T On PLANAR, our COMETH yields very good results over all metrics, being outperformed by a couple of baselines on Degree and Orbit, but with a much lower VUN. The sampling quality benefits from the predictor-corroctor scheme, with near-perfect VUN score. On SBM, we also obtain sato-of-thheart results on all metrics. However, we found that, overall, the predictor-correcto did not improve performance this dataset.
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Table Synthetic graph generation results. We report the mean of five runs, as well as 95% confidence intervals. We reproduced the baselines results for DIGRESS and GRUM; the rest is taken from Jo al. (2024).
The best results are highlighted in bold.
Model
Degree
Cluster
Orbit
Spectrum
VUN [%] f
Planar graphs
GRAPHRNN
24.3
9.0
2508
8.8
0
GRAN
3.5
1.4
1.8
1.4
0
SPECTRE
2.5
2.5
2.4
2.1
25
DIGRESS
0.810.0
4.1-0.3
0,5
76.010.1
GRUM
2,621.7
1,3=0.3
10.027.7
1.440.2
91.0as.7
DISCO
1.220.5
1.3=0.5
1.720.7
83.6+2.
COMETH
2.1=18
1,5=0.4
3.1+30
1.3+n.3
92.5+a.7
COMETH- PC
2.0=0.9
1.1=0.1
7.T=38
1.3+0.2
99.5+0.8
Stockastic block model
GRAPHRNN
6.9
1.7
3.1
1.0
5
GRAN
14.1
1.7
2.1
0.9
25
SPECTRE
1.9
1.6
1.6
53
DIGRESS
1.7=0.1
5.0=0.1
3.6+0.4
74.0+44.0
CRUM
2.611.0
1,5.0.0
1.810.4
0.910.2
69.018.5
0.8
0,8=0.4
2.0.0.3
66.211.4
COMETH
2.421.1
1.5=0.0
1.720.2
0.8
77.025.3
COMETH PC
1.420.4
1,3.0.0
2.0
0.8
75.524.0
Ablation Study We also conducted an ablation study on the positional encoding to compare the benefits of the RRWP encoding against the feature set used in Digress (see table 2). While both approaches achieve comparable results across most distribution metrics, RRWP significantly outperforms DiGress' set of features in terms of VUN.
Table 2: Ablation study the positional encoding for synthetic graphs. We report the mean of runs, well as 95% confidence intervals.
Noise Model
Degree | Cluster
Orbit 4
Spectrum
VUN [%]] f
Planar graphs
RRWP 2.1+13
1.510.3
3.143.0
1.2.0.2
3..67
DiGress' features
2,2+1.
2.240.3
18.0=7.
1.300
67.5=3.7
Stockastic block model
Marginal
2.4+1.1
1.520.0
1.740.2
0.8
77.0=5.06
DiGress' featares
2.3
1.520.0
1.320.2
0.920.
64.526.41
4.2 Small-molecule generation: QM9
Here, outline experiments regarding small-molecule gencration.
Datasets and metrics To the ability of method to model attributed graph distributions, we evaluate its performance the standard dataset QM9 (Wu al. (2018)). Molecules are kekulized using the RDKit library, removing hydrogen atoms. We the same split Vignac et al. (2022), with 100k molecules for training, 10k for testing, and the remaining data for the validation set. We want to stress that this split differs from Jo et al. (2022), which uses ~ 120k molecules for training and the rest as a test set. We choose to use the former version of this dataset because it allows for selecting the best checkpoints based on the evaluation of the ELBO on the validation set. We report the Validity over 10k molccules, as evaluated by RDKit sanitization, well the Uniqueness, FCD, using the MOSES benchmark, and NSPDK. Appendix D.2 provides complete description of the metrics.

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Published in Transactions on Machine Learning Research (04/2025)
Table Molecule generation results on QM9, We report the mean of five runs, as well as 95% confidence
intervals. The best results are highlighted in bold. Baseline results are taken from Jang al. (2024a)
Model
Validity
Uniqueness
Valid & Unique
FCD 4
NSPDK
GDSS
95.72
98.46
94.25
0.003
DIGRESS
99.01
96.34
95.39
0.001
GRAPHARM
90.25
95.26
85.97
1.22
0.002
HGGT
99.22
95.65
94.90
0.40
0.000
COMETH
99.57=0.07
76
96.34=20.2
0.25+ on
0.000+0.00
COMETH PC
99.69
96. 96.66620.08
96.36=0.07
0.25 o
0.000
Table 4: Molecule generation on MOSES. We report the mcan of five runs, as well as 95% confidence intervals. The best results are highlighted in bold, and the second-bes best results are underlined.
Model
Class
Val. f
Val. & Uni
VUN
Filters
FCD
SNN
Scaf
VAE
Smiles
97.7
97.5
67.8
99.7
0.57
0.58
5.9
JT-VAE
Fragment
100
100
99.9
97.8
1.00
0.53
GRAPHINVENT
Autoreg.
96.4
96.2
95.0
1.22
0.54
12.7
DIGRESS
One-shot
85.7
85.7
81.4
97.1
1.19
0.52
14.8
DISCO
Oue-shot
88.3
88.3
86.3
95.6
1.44
0.50
15.1
COMETH
One-shot
87.0=0.2
86.910.2
83.8.0.2
97.210.1
1.4410.02
0.5110.0
15.910.
COMETH- PC
One-shot
90.5=0.3
90.420.3
83.720.2
99.1+0.1
1,27#0.02
0.54+0.0
16.0=0.7
Baselines We evaluate our model against several recent graph generation models, including diffusion
models, DIGRESS Vignac et al. (2022) and GDSS (Jo et al. 2022) and two autoregressive models, HGGT
Jang et al. (024aa) and GRAPHA RM Kong et al. (2023)).
Results We report results using 500 denoising steps for fair comparison, as in Vignac et al. 2002 COMETH performs very well on this simple molecular dataset, notably outperforming its discrete- discrete-time counterpart DIGRESS in terms of valid and unique samples with similar FCD and NSPDK. We experimentally find that the predictor-correcte offers only a slight improvement in the number of valid and unique samples on this dataset, with performance comparable for FCD and NSPDK.
4.3 Molecule generation on large datasets: Moses and GuacaMol
Here, we outline experiments regarding large-scale molecule generation.
Datasets and benchmarks We also evaluate model two much larger molecular datasets, MOSES PPolykosskiy et al. 2020)) and GuacaMol (Brown et al.) 2019). The former is a refinement of the ZINC database and includes 1.9M molecules, with 1.6M allocated to training. The latter is derived from the ChEMBL database and comprises 1.4M molecules, from which 1.1M are used for training. We use a preprocessing step similar to Vignac et al. (2022) for the GuacaMol dataset, which filters molecules that cannot be mapped from SMILES to graph and back to SMILES.
Both datasets come with their own metrics and baselines, which briefly describe here. As for QM9, we report Validity, as well as the percentage of Valid & Unique (Val. & Uni.) samples, and Valid, Unique and Novel (VUN) samples for both datasets. We also report Filters, FCD, SNN, and Scaf for MOSES, as well as KL div and FCD for GuacaMol. These metrics are designed to evaluate the model's capability to capture the chemical properties of the learned distributions. We provide a detailed description of those metrics in Appendix D.3
Results Similarly to previous graph generation models, COMETH does not match the performance of molecule generation methods that incorporate domain-spocific knowledge, especially SMIIEES-based models (see Table 4p. However, COMETH further bridges the gap between graph diffusion models and those methods, outperforming DIGRESS in terms of validity by large margin.
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Table Molecule generation on GuacaMol. We report the mean of five runs, as well as 95% confidence intervals. Conversely to MOSES, the GuacaMol benchmark reports scores, so higher is better. The best results are highlighted in bold, and the second-best results are underlined.
Model
Class
Val.1
Val. & Uni.t
VUNT
KL divt
FCD1
LSTM
Smiles
95.9
95.9
87.4
99.1
91.3
NAGVAE
One-shot
92.9
95.5
88.7
38.4
0.9
MCTS
One-shot
100.0
100.0
95.4
82.2
1.5
DIGRESS
One-shot
85.2
85.2
85.1
92.9
68.0
DISCO
One-shot
86.6
86.6
86.5
92.6
COMETH
One-shot
94.4+0.2
94.4+0.2
93.5=2.3
94.1=204
67.4+0.3
COMETH
One-shot
98.940.1
98.9+0.1
97.6+0.2
96.7.20.2
72.7+0.2
Table Conditional molecule generation results on QM9. We report the mean of five runs, as well as 95% confidence intervals.
HOMO
& HOMO
Model
MAE
Val
MAE
Val
MAE
Val
DIGRESS 0.81
0.56
0.87
COMETH
0.67
88.800.5
0.32:0001
94.120.
0.58
92.5207
COMETH obtains excellent performance in terms of VUN samples, with an impressive 12.6% improvement over DIGRESS. The LSTM model still surpasses our graph diffusion model on the FCD metric. This may be because we train on a subset of the original datasct, whereas the LSTM is trained directly on SMILES.
On GuacaMol (see Table 5
4.4 Conditional generation
We perform conditional generation on QM9 following the setting of Vignac et al. (002 We target two molecular properties, the dipole moment and the highest occupied molecular orbital energy (HOMO). We sample 100 properties from the test set for each experiment and use them as conditioners to generate ten molecules. We estimate the properties of the sampled molecules using the Psi4 library (Smith et al. (0200) and report the mean absolute error (MAE) between the estimated properties from the generated set and the targeted properties.
We report our results against DIGRESS in Table 6 Overall, COMETH outperforms DIGRESS by large margin, with 18%, 43%, and 33% improvements on HOMO and both targets respectively. Those performance improvements indicate the superiority of classifier-free guidance over classifier guidance for conditional graph generation.
5 Conclusion
Here, to leverage the benefits of contimouus-iime and discrete-state diffusion model, we proposed COMETH, a contimuous-time discrete-state graph diffusion model, integrating graph data into a continuous-timg diffusion model framework. We introduced a new noise model adapted to graph specificities using different node and edge rates and a tailored marginal distribution and noise schedule. In addition, we successfully replaced the structural features of DIGRESS with a single encoding with provable expressivity guarantecs, removing unnecessary features. Empirically, showed that inlegrating continuous time leads significant. improvements across various metrics over state-of-the-art discrete-state diffusion models large set of molecular and non-molecular benchmark datascts.
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A
Appendix
We provide proofs and additional theoretical background in Appendix We give details on our implementation in Appendix C Finally, provide visualization for generated samples in Appendix
experimental details in Appendix D and additional experimental results in Appendix
B Theoretical details
Here, we outline the theorctical details of our COMETH architecture.
B.1
Continuous-timg discrete diffusion
Here, we provide more detailed description of the continuous-time, discrele-state diffusion model introduced in Campbell et al. (2022).
We begin by recalling our notations. We aim to model discrete data distribution (2)) where 2(0) E Z and is a finite set with cardinality == (2). In the following, the state is denoted by =(0) E Z, where time is denoted by E [0, 1], and 2tt) E {0, 13 its one-hot encoding. The marginal distributions at time are denoted by (2(1)) and the conditional distribution of the state (()) given the state 2(0) at some time [0.1] 1] by (2(() 2))).. We also denote ò2,- the Kronecker delta, which equals if 2 and 0 otherwise.
A contimuuusttime, discrete-state diffusion model builds upon contimuuu--iime Markov chains (CTMCs). CTMCss are continuoustingg processes in which the state 20)) alternates between remaining in the current state and transitioning to another state. The dynamics the CTMC are governed by a rate matrix R( E RxS where S represents the number of possible states. We denote (2)() ž) the transition rate from the state (1) to another state
Precisely, CTMC satisfies three differential equations:
(forward)
(z 12) Σ 94 2)) (y, (J,
(backward)
0.qu/(z (1)= Σ (2, w)
(forward, marginals)
0.q((2) = Σ (.) E..( (y.2).
where 2, 2, E 2 and f, s E [o, 1]. The infinitesimal probability of transition from 2(1) to another state i E 2, for infinitesimal time step dt between time t and + is given by
(4 201) = { 1 + R(O) (200
R (2(1) d)
if 4 (1) i (())
83,200 + ((t) (1) E)dl.
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The rate matrix must satisfy the following conditions,
(2 2 O, and ((0)) =- Σ (2000=0) < 0.
The second condition ensures that do+-dtje(- 20) sums to 1. Intuitively, the rate matrix contains instantancous transition rates, i.e., the number of transitions per unit of time. Therefore, the higher (=)( g); the more likely the transition from 2(0) to I. Since the diagonal coefficients of the rate matrix can be derived from the off-diagonal ones, we will define only the latter in the following.
The CTMC is initialized with ((=(")) (2 The key challenge in designing the rate matrix is ensuring that the forward process converges to a well-known categorical distribution we can use later as a prior distribution during the generative process. In the discrete case, such a distribution can be, e.g., a uniform distribution, a Srecretized-Gaussiin distribution, or an absorbing distribution (Campbell et al. 2022
The reverse process can also be formulated as CTMC. Using similar notations for the forward process, the reverse process defined through
de (2 [ dg00, & +
where (t)t) is the rate matrix for the reverse process. Similar to discrete-time diffusion models, this reverse rate can be expressed as (Campbell et al.
2022
Proposition 1),
400(3 20)) (=)(((()) 200) 4oje (of((z (0))), for 2 # 2.
(2, = (2
(0) E2
Since the true reverse process 90 (2 2) is intractable, we approximate it using a neural network (2(0) z) parameterized by 0, yielding the approximate reverse rate,
[no(= =00)) 200 90jo(> 200)) Poj (z =)) for + z.
(2)
Diffusson models are typically optimized by minimizing the negative ELBO on the negative log-likelihood, i.., - log lo (2))), Campbell et al. (2022 Proposition 2) provides an expression for the ELBO. Although it is not used in this work, we include for completeness,
Σ Re (2,2)) -2()) 2 log (120 (3=) C,
(Cr(O) 7 Be-.u
where C' is a constant independent of 8, z - = R((1) (2, (2, and r(z =2)= R(t) (z, 8) z().
For efficient optimization, it is essential to express (x(2) 2 22(0) )000)))))()) in closed form. In this context, the transition matrix R()) must be designed so that o: =())) has closod-form expression. Campbell et al. (2022) established that when R(O) and R((') commute for any and l', the transition probability matrix can be written as:
Q(0 (0))((())))) 20()) exp (f"))))s)
where (Qt) -(²1 = 1 j). This condition is met when the rate is written R(O) B(t where 3 is time-dependent scalar and R, is constant base rate matrix. In that case, the forward process can be further refined
= (1) = k 20 = -PPe Pexp A B(s)ds p-1)
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where R, = PAP and exp refer to the element-wise exponential.
Given one-hot encoded data sample z we can sample a noisy state 2()) by sampling categorical distribution with probability vector 20)) (()..
Since most real-world data is multi-dimensional, the above framework needs to be extended dimensions. This is done by assuming that each dimension noised independently so that the forward process factorizes as
D
Ge+aije( | x) = II
qi au-atjl22 (Za l za);
where @++Ace [ zd) is the uni-dimeensional forward process on the dth dimension.
Campbell et al. (2022 Proposition 3) establishes how write the forward and reverse rates in the D- dimensional case,
(2,
Σ
dg 2i Rug (Zds
(Zd, za);
D
(g Ea
[29) Pôj
(2, 8)
Σ
w R( (2dr (2s, za)
(2) =),
for za # Zd. In brief, assuming that a transition cannot occur in two different dimensions simultaneously, the multi-dimenssonal rates are equal to the uni-dimensional rates in the dimension of transition. Importantly, if the dimensions are independent in the forward process, they are not in the reverse process since the whole state is given as input in Poje z).
Finally, need practical way to simulate the reverse process over finite time intervals for D-dimensional data. To that extent, we follow Campbell et al. (2022) and use the r-leaping algorithm. The first step is to sample 2(1) from the prior (")) The sampling procedure is as follows. At cach itcration, we keep (!!) and (z, z) constant and simulate the reverse process for a time interval of length 7. means we count all the transitions between and t - and apply them simultaneously.
The number of transitions in each dimension 2d of the current state 2(9) between 2d and Ed is Poisson distributed with mean =f(-0(22)) (1) Za). In a state space with no ordinal structure, multiple transitions in one corrector scheme. After each predictor step using R(t). (1)) (=()) Ea), we can also apply several corrector steps using the expression defined in Campbell et al. (2022), i.e., R(00. = (())) + R(O). The transitions using the corrector rate are counted the same way as for the predictor. This rate admits (2(1)) as its stationary distribution, which means that applying the corrector steps brings the distribution of noisy graphs at time & closer to the marginal distribution of the forward process.
dimension are meaningless, and we reject them. In addition, we experiment using the predictor-corrector
B.2 Noise schedule
Here, we provide proof for Proposition [ as well as some intuition on the prior distribution.
Proposition 4. (Proposition ] in the main body) For CTMC (z(0))efa,)) with rate matrix R R.)
and R. Im - I, the forward process can be written as
Q = = I + -1- 1m'
where (Q(1).; 9(2(0= = 2() = j) and 3 Jn B(s)ds.
Proof. Since m' is rank-one matrix with trace it is diagonalizable and has only non-zero cigenvalue, equal to tr(1m')= 1. Therefore,
R. 1m - I PDP-1 P 1)) p-1
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with D 0......... Denoting 3r So B(s)ds,
Q = Pexp (D p-1
= P D ---- - e D) p-
e 3¹ I+ 3 1m'.
We now wish elaborate on the link between Proposition and the choice of the prior distribution. Recall our noise schedule,
(()) a (2t) and L. 3(s)ds = =a 1 cos (²:))
When t = 1, it holds that - = = and therefore ( = e - I
When a is large enough, then e and Q R m'm Denoting (Q the j-th row of Q it holds that (Q")), = ((2(1) =0(0) j) x m.
In other terms, whatever the value 2 20) z(1) is sampled from the same categorical distribution with probability vector Therefore,
(1(2(1)
Σ (1)((((1) (0(0) = 3)90(z = =])
jEz
mqu m =
JEZ
~ m = Pret(2(1)
The first line is obtained by marginalizing 200) the second line by using the factg that (1,0((()) z0()) j) for all possible 200), and the last one using that Ejez qo(2(0) = = 1 and m is independent of j. In the D-dimensional case, since the forward process factorizes, we get
(1)((((1)
20) II m (1))(2(1)),
where d e [0, D] denotes d-th dimonsion of z E 2
B.3 RRWP properties
In the following, we formally study the encodings of the RRWP encoding.
Notations A graph G is a pair (V(G), E(G)) with finite sets of vertices or nodes V(G) and edges E(G) S ((u, v) C VVG) A v}. If not otherwise stated, we set n = [V(G),, and the graph of order n. For ease of notation, we denote the edge w) E F(G) by (u, v) or (w, u). An n-order attributed graph G (G, E), where G (V) E(G) is a graph and X {0, 1jxxa for 0, node feature matris and E E (0, 1axxxbb for b > 0, is edge feature tensor. Here, set V(G) = [1,n]. The neighborhood of (GG is denoted by N(v) = (u V(G) (v,u) E F(G)).
pair
In our experiments, we leverage the relative random-walk probabilites (RRWP) encoding, introduced in Ma et al. (2023). Denoting A the adjacency matrix of graph G, and D the diagonal degree matrix, and M D- the degree-normalized adjacency matrix, for each pair of nodes (i,j), the RRWP encoding
computes
pk [Li Mij. .,.,, MK-1
55)
where K refers to the maximum length of the random walks. The entry pK corresponds the RRWP encoding of node therefore, we leverage them node encodings. This encoding alone can train our graph diffusion model and attain state-of-the-art results.
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In the following, we show that RWP encoding can (approximately) determine if two nodes lie in the same connected components and approximate the size of the largest connected component.
Proposition 5. For n E N, let Gn denote the set of n-order graphs and for graph G Gn let V(G) = [[, Assune that the graph G has connected components and let C' E {0, 1j be matrix such the ith row C, is one-hot encoding indicating which of the e connected components the vertex i belongs to. Then, for any 6 > 0, there exists a feed-forward neural network FNN: R + [0, 1) 1jaxe such that
||FNN(M* -ClI E.
Proof. Let R M First, since the graphs have n vertices, the longest path in the graphs has length n - 1. Hence, two vertices u w E V(G), with + w are in the same connected component if, and only, if Rew # O. Hence, applying sign activation function to R pointwisely, we get a matrix over {0, 1) with the same property. Further, by adding D, E {0, 1) an n x n diagonal matrix with ones on the main diagonal, to this matrix, this property also holds for the case of v = w. In addition, there exists a permutation matrix PR such that applying it to the above matrix results in block-diagonal matrix B E {0, 13"xm such that Bg = B for v, w E VVG), if, and only, if the vertices v, are in the same connected component. Since n is finite, the number of such B matrices is finite and hence compact. Hence, we can find a continuous function mapping each possible row of B.., for v E V(),, to the corresponding one-hot encoding of the connected component. Since all functions after applying the sign function are continuous, we can approximate the above composition of functions via a two-layer foed-forward neural network leveraging the universal approximation
theorem Cybenko 1992: Leshno et al. 1993).
Similarly, we can also approximate the size of the largest component in a given graph.
Proposition 6. For N, let Gn denote the set of n-order graphs and for G E Gn let V(G) = [I, Assume that S is the number of vertices in the largest connected component of the graph G. Then, for any e > 0, there exists a feed-forward neural network FNN: R=Xn > n),
(FNN(M*))) -| S..
Proof. By the proof of Proposition we get block-diagonal matrix B E {0,1)"xx such that Bu = 1 if, and only, if u, v are in the same connected components. Hence, by column-wise summation, we get the nunber of vertices in each connected component. Hence, there is an n X 1 matrix over {0, extracting the largest entry. Since all of the above functions are continuous, we can approximate the above composition of functions via a two-layer ferd-forward neural network leveraging the universal approximation theorem (Cybenko 1992
Leshno et al. 1993
Moreover, we can show RRWP encodings can (approximately) count the number p-cycles, for p < 5, in which a node is contained. A p-cycle a cycle on p vertices.
Proposition 7. For n E N, let Gn denote the set of n-order graphs and for G E Gn let V(G) = [1,n]. Assume that e E No contains the number of p-cycles a node is contained in for all vertices in G, for p E (3,4).
Then, for any & > 0, there exists a feed-forward neural network FNN: TRan -> Ra
|FNN(P"-1) ccl2 S.
Proof. For E (3,4), Vignac et al. (2022 Appendix B.2) provide simple linear-algebraic equations for the number of p-cycles each vertex of a given graph is contained based on powers of the adjacency matrix, which can be expressed as compositions of linear mappings, i.e., continuous functions. We can extract these matrices from p=-1, Further, note that the domain of these mappings is compact. Hence, we can approximate this composition of functions via a two-layer feed-forward neural network leveraging the universal approximation
theorem |Cybenko
1992 Leshno et al. 1993).
However, we can also show that RRWP encodings cannot detect if a node contained in a large cycle of a given graph. We say that encoding, e.g., RRWP, counts the number of p-cycles for p ' 2 if there do not.
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exist two graphs, one containing at least p-cycle while the other does not, while the RRWP encodings of the two graphs are equivalent.
Proposition 8. For p 8, the RRWP encoding does not count the number of p-cycles.
Proof. First, by Rattan Seppelt (2023). , the RRWP encoding does not distinguish more pairs of non- isomorphic graphs than the so-called (1, ) J-dimensional Weisfeiler Leman algorithm. Secondly, the latter algorithm is strictly weaker than the 3-dimensional Weisfeller-Leman algorithm in distinguishing non- isomorphic graphs (Rattan & Seppelt 2023 Theorem 1.4). However, by Fürer (2017) Theorem 4), the 3-dimensional Weisfeiler-Lenan algorithm cannot count 8-cycles. Hence, the results follows.
Hence, the above proposition implies the following results.
Corollary 9. For p 2 8 and K 2 0, there exists a graph G containing a p-cycle C, and two vertex pairs (r, (v, w) 2 such that (r, 8) is contained in C while (v, w) not and pk PK
B.4 Equivariance properties
In this section, we prove that our model is equivariant (Proposition 10 and that our loss permutation- invariant (Proposition 11 relying on Vignac et al. (2022 Lemma 3.1 and 3.2). We also prove exchangeability
with Proposition
I2
Let start by defining the notation for graph permutation. attributed graph G = E. as,
Denote permutation, acts the
rG = (V(aG), E(TG)) where V(nG)= {I(1),.... [()) and E(TG) ((=(1),(()) | (vr, uy) e E(G)),
T the matrix obtained by permutating the rows of X according to i.c. (x Zx-1(1)
Similarly, # the tensor obtained by the permutation of the components ei of E according to m
ie. (n --==())=1)))>
Proposition 10 (Equivariance). DIGRESS' graph transformer using RWSE as node encodings and RRWP as edge encodings is permutation equivariant.
Proof. We recall the sufficient three conditions stated in Vignac et al. 2022 for ensuring permutation- equivariance of the DIGRESS architecture, namely,
their sot structural and spectral features is oquivariant.
All the blocks of their graph transformer architecture are permutation equivariant.
The layer normalization is equivariant.
Replacing the first condition with the permutation--equivariaal nature of the RRWP-based node and edge encodings completes the proof.
We now derive more thorough proof of the permutation invariance of the loss compared to Vignac al. (2022) Lemma 3.2), relying the peimutaaioon-equivrratt nature of both the forward process and the denoising neural network.
Proposition 11 (Permutation invariance of the loss). The cross-entropy loss defined in Equation is invariant to the permutation of the input graph ((0).
Proof. Given a graph G (G,X,E) we denote by -= G (G, the predicted clean graph by the neural network and #G (=G, xX, E) permutation of this graph, for arbitrary permutation Let
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establish that the loss function is permutation-invarin We recall the loss function for permutation of the clean data sample (( is
n
n
LCE [. (1000)//pha
Σ log (= m(1) TT(()) -
_
log Pojs (e ((())(()))) |TG00)
Because dimensions are noised independently, the true data distribution Pdata (T Pdata (G(0)) is permutation-invariint, and the forward process is permutation-cquivariint, Thus, we write,
LCE = (60))).).pnnn(0)))))c(000000
Σ log e (x)((( xz(1)) - Σ
log Poj Poj (()))))))))))))))))))))))(()
xG(O)
isj
Using Proposition
10 also have that (x (0) G(t) poj (=
(0)
| xG and ( (0)
G())
Pope (e (0) m(()())) ( which concludes the proof.
Proposition 11 shows that, whatever permutation of the original graph we consider, the loss function remains the same, and so do the gradients. Hence, we do not have to consider all the permutations of the same graph during the optimization process.
Proposition 12 (Exchangeability)) COMETH yields exchangeable distributions.
Proof. To establish the exchangeability, we require two conditions, a permutation prior distribution and an equivariant reverse process.
Since nodes and edges are sampled iidd from the same distribution, our prior distribution is permutation-invarinn, i.c., each permutation of the same random graph has the same probability of being sampled. Hence Pre((TG(2)) Pret (G(T))
is straightforward to see that our reverse rate is permutation-equuiarre regarding the joint permutations of (()) and G(O), We illustrate this using the node reverse rate,
R' (Z,, (1) Σ gqo(i; x1
(x (0) (x: ((01)
Poje
(!
The forward rate, as well as the forward process, is permutation-cquivariani regarding the joint any permutation on (L)) and G(O), and the neural network permutation--qquiariinn Similarly, we can reason regarding the edge reverse rate. Therefore, the overall reverse rate permulation- equivariant. Since we sample independently across dimensions, the 7-leaping procedure is also permutation--qqiiarian..
B.5 Classifier-free guidance
In the conditional generation setting, one wants to generate samples satisfying a specific property y, which refer the conditioner. For example, in text-lo-inage diffusion models, the conditioner consists of textual description specifying the image the model is intended to generate. The most straightforward way to perform conditional generation for diffusion models is to inject the conditioner into the network-therefrr modeling p" (2(t-1) (()), y) hoping that the model will take it into account. However, the network might ignore y, and several efficient approaches to conditional generation for diffuusion models were consequently developed.
The approach leveraged by Vignac (002)) to perform conditional generation dassifier-guidance.. It relies trained unconditional diffusion model and regressor, classifier, depending the conditioner,
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trained to predict the conditioner given noisy imputs. As mentioned in Ho & Salimans (2021) it has the disadvantage of complicating the training pipeline, pre-trained classifier cannot be used during inference.
To avoid training a classifier to guide the sampling process, classifier-free guidance has been proposed in Ho & Salimans (2021) and then adapted for discrete data in Tang et al. (2022). A classifier-fie free conditional diffusion model jointly trains a conditional and unconditional model through conditional dropout. That is, the conditioner is randomly dropped with probability Puncond during training, in which the conditioner is set to mull vector. However, Tang et al. (2022) showed that learning the null conditioner jointly with the model's parameters is more efficient.
At the sampling stage, the next state is sampled through
log (=(0((((-11)) 100),)) = logg (ogg((x((11) (x(1-1) 2²0,0) .0) + (s + p) (2((1) |0(0) y) - log (0gp((2(1-1) 2=(1,)))
where 8 is the guidance strength. We refer to Tang et al. (2022) for deriving the above expression for the sampling process.
Let explain how apply classifier-free guidance in our setting. Denoting R G y), the conditional reverse rate can be written as
GG, G
E
f ot 511) + (x
Σ
8 Grn P/ (e]f,ex egg 1y),
icj
and
gep(ds 2,
R5d (1 (1) Rx; x
(oj (x(²)) (G for x, (6) gejo (x 10) Pope
+ Tr
and similarly for edges. At the sampling stage, we first compute the unconditional probability distribution (a (0) ((), 0), where 0 denotes the learned null vector, then the conditional distribution 16j (0)
G(
These two distributions are combined in the log- pog-robbaiility space in the following way,
logp'( (2 10( y) = loggr' (2 (x(1-1) 22(0.0) 0) + (s + 1)(log (o (x(11) 12 y) -oog p (2 (1 | ()). (6)
Finally, the distribution in Equation 6 is exponentiated and plugged into the reverse rate.
C Implementation details
Here, we provide some implementation details.
C.1 Reverse process: Tau-leaping and predictor-correcttor
At the sampling stage, we use the r-leaping algorithm generate new samples, as proposed by Campbell ct al. (2022). We first sample the graph size and then sample the noisy graph G(t) from the prior distribution. At each iteration, we keep (()) and (G, G) constant and simulate the reverse process for a time interval of length 7. In practice, we count all the transitions between and t - T and apply them simultaneously. o and ēxÿ) is Poisson distributed with mean T (a(") .i) (respectively T (e()) ε.))). Since our state space has no ordinal structure, multiple experiment using the
The mumber of transitions between 2!") and &, (respectively (e)
transitions in one dimension are meaningless, and we reject them. In addition, predictor-correctto scheme. After each predictor step using R(t). (G.G), we can also apply several corrector sleps using the expression defined in Campbell et al. (2022), i.e,, R(t). = f((1)0 + R(O) This rale admits qo( (G()) as its stationary distribution, which means that applying the corrector steps brings the distribution of noisy graphs at time closer the marginal distribution of the forward process. As fixed only during the sampling stage, its value can be adjusted to balance sample quality and efficiency, i.c., the number of model evaluations. We perform an ablation study in Appendix
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0.4
0.0
0.0
32
0.5
0.0
14
Time
Figure 2: Comparison between our cosine noise schedule and the constant noise schedule proposed by Campbell et al. (2022). Both schedules are plotted using rate constant 5.
C.2 Noise schedule
We plot our noise schedule against the constant noise schedule used for categorical data in Campbell et al. (2022) in Figure 2 Following Proposition we plot ã = e on the y-axis, quantifying the information level the original data sample retained at time t. Similarly to Nichol & Dhariwal (2021). we can see that the constant noise schedule converges towards zero faster than the cosine schedule, hence degrading the data faster. In our experiments, we perform a hyperparameter search to select the best rate constant for each dataset, with E {4,5, 6}. Following Campbell et al. 20022), set minimum time to fmin 0.017 because the reverse rates are ill-conditioned close to / 0.
C.3 Algorithms
We provide the pseudo-code for the training and sampling from COMETH in Figure 3 Similar to Campbell et al. (2022), we apply last neural network pass at = tmin and set the node and edge types to the types with the highest predicted probability. We omit the corrector steps in the sampling algorithm for conciscness. They are exactly the same as the predictor r-leaping steps, using the corrector rate (()). f((). + R(O), and applied after time update, ie. at - T. Since those steps are sampled from different CTMC with rate R(0,,, we have control over т when applying corrector steps. We provide additional details on the choice of this hyperparameter, denoted as Te: in Appendix D
C.4 Graph transformer
See Figure an overview of the used graph transformer, building on Vignac el al. (2022)
The transformer takes node features X as input, and updates them using the self attention mechanism, i.e., ATTENTION(X, X) softmax WoWo (Wxx))) Wv X. Edge features are incoporated into the attention scores i.c. Va W X(WxXWWX using a FiLM layer (Perez al. 2018), were FILM(M,, M2) M, WT + W2) @ M2 + M2. The attention scores with incorporated edge features are used to update edge features. Global features y are incorporated into the updated node and edge features using FiLM layers as well. The final outputs are obtained by passing the representations through fully connected networks.
We use the RRWP encoding, defined Equation 5 for synthetic graph generation. For molecule generation datasets, we additionally compute several molecular features used in Vignac et al. (2022), namely the valency and charge for each node and the molecular weight.
D Experimental details
Here, we outline the details of your experimental study.
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Sample random graph from prior distribution Published in Transactions on Machine Learning Research (04/2025)
Algorithm Training
Input: A graph G= (X, E)
Sample ~ 0,
Sumple G ~ XQxx E
Sample sparse noisy graph
Predict PGj[C G(9)
> Predict clean graph using neural network
CeB + Σ, log (1) G A Eie, log (0) ((
Update 0 using LoB
Algorithm 2: 7-leaping sampling of Cometh
Sample n from the training data distribution
Sample ( -I. mx II me
while 0.01 do for i= to do
for in X' do
R (x) (x) (i. x; 1 Exo mn)l)
mpuld 14 Por (x:" ((t), for o!!
Sample -, P (x)) (2(), .))
Count transitions node
end end
for i.j = to n. i < do
for in 8 do
it (Gg B(s) (ei,, o3j En
mute, (n Poj ( for ci * 65
Gnoteyle )
Sample Jal ~ P(r (3501/101 (6g 0g))
Count transitions on edge ij
end
end
for
do do
for
1.0 (- a
= and Eej = then
Apply unique transition or discard
end
end
end
for
ijj= = to n. i < do
for in 8 do
if (0) = 1 and Exi.g.
then
=ē
Apply unique transition or discard
end
end
1 t- f-
G II. arguax (x, ( arguax (e C(0)
Last pass
return G"
Figure 3: Training and Sampling algorithms of COMETH
D.1 Synthetic graph generation
We evaluate our method on two datasets from the SPECTRE benchmark (Martinkus al. 2022 with 200 graphs cach. PLANAR contains planar graphs of 64 nodes, and SBM contains graphs drawn from a stochastic block model with up to 187 nodes. We use the same split the original paper, which uses 128 graphs for training, 40 for training, and the rest validation set. Similar to Jo et al. (2024). we apply random permutations to the graphs at each training epoch.
We report five metrics from the SPECTRE benchmark, which include four MMD metries between the test set and the generated set and the VUN metric on the generated graphs. The MMD metrics measure the Maximum Mean Discrepancy between statistics from the test and the generated set, namely the Degree (Degree) distribution, the Clustering coefficient (Cluster) distribution, the count of orbit information regarding subgraphs of size four Orbit (Orb.) and the eigenvalues (Spectrum) of the graph
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E'
X
Lin
Lin
E
FILM
FILM
Node/ Edge wise MLP
GT Layer
Flatten heads
Flatten heads
Ease
GT Layer
Node/
Edge wise MLP
Softmax
Lin
FiLM
X
E
E
X'
E'
Outer product scaling
Encoding/ Features
Lin
Lin
x
X
X
E
y
Figure Overview of DIGRESS graph transformer.
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Laplacian. The Valid, Unique, and Novel (VUN) metric measures the percentage of valid, unique, and non-isomorphic graphs to any graph in the training set.
For the MMD metries, we report ratios between the MDD computed between the generated and the test set, over the MDD between the train and the test set. In fact, due to the small size of the datasets, the statistics that we compute vary slightly between the train and the test set. Formally,
MMD test)²
M (traiin, test)²
Note that we actually compute the MMD squared, following a common practice (Martinkus et al. 2022 Vignac et al. 2022).
On PLANAR, wo report results using 5 = 0.002, i.c. using 500 r-lcaping. We also evaluate our model using 10 corrector steps after each predictor step when 1 0.17, with 0.002, for 1000 r-leaping steps. We found our best results using Fe = 0.7.
On SBM, we report results using r = 0.001, i.c., using 1 000 r-lcaping steps.
D.2 Small molecule generation : QM9
We evaluate model QM9 (Wu et al. (2018) to assess the ability of our model to model attributed graph distributions. The molccules are kekulized using the RDKit library, and hydrogen atoms are removed, following the standard preprocessing pipeline for this dataset. Edges can have three lypes, namely simple bonds, double bonds, and triple bonds, as well as one additional type for the absence of edges. The atom types are listed in Table 7
We use the same split as Vignac et al. (022). i.., 100k molecules for training, 13k for testing, and the rest (20885 885 molecules) it leaves validation set to evaluate the ELBO and select the best checkpoints to minimize this quantity. In consequence, our training dataset contains roughly 20k molecules, which is less than what most graph generation works use.
validation set. We choose this split over the one proposed in Jo et al. (2022) because
At the sampling stage, we generate 10k molecules. We evaluate four metrics. The Validity is evaluated by sanitizing the molecules and converting them to SMILES string using the RDKit library. The largest, molecular fragment is selected as sample if it is disconnected. We then evaluate the Uniqueness among valid molecules. As stated in Vignac al. (2022), evaluating novelty QM9 bears little sense since this datasel consists of an enumeration of all stable molecules containing the atom above lypes with size nine or smaller. We also evaluate the Fréchet ChemNet Distance (FCD), which embeds the generated set and the test set using the ChemNet neural network and compares the resulting distributions using the Wasserstein-2 distance (Preuer et al. 2018)). ). Finally, we evaluate the Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) between the test set and the generated, which measures the structural similarities between those two distributions.
D.3 Molecule generation on large datasets
We further evaluate COMETH on two large molecule generation benchmarks, MOSES (Polykovskiy et al, 2020) and GuacaMol (Brown et al., 2019). The molecules are processed the same way as for QM9 and present the same edge types. The atom types for both datasets are listed in Table The filtration procedure for the GuacaMol consists of converting the SMILES into graphs and retrieving the original SMILES. The molecules
for which this conversion is not possible are discarded. We use the code of Vignac et al. (0222) to perform this procedure. We use the standard split provided for each dataset.
Both datasets are accompanied by their own benchmarking libraries. For GuacaMol, we use the distribution learning benchmark. During the sampling stage, we generate 25k molecules for MOSES and 18k for GuacaMol, which is sufficient for both datasets, as they evaluate metries based on 10k molecules sampled from the generated SMILES provided.
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Table 7: Details of the molecular datasets. The number of molecules for GuacaMol is computed after filtration.
Dataset
Number of molecules
Size
Atom types
QM9
133 885
to 9
N, C, O, CI, Br B, Br, CI, P, Sc, Si
MOSES
936 962
8 to 27
GuacaMol
398 223
to 88
We then claborate the metrics for cach datasct. For both datasets, we report Validity, defined in the same manner as for QM9, the percenlage of Valid and Unique (Val. & Uni.) samples, and the percentage of Valid, Unique, and Novel (VUN) samples. We prefer the latter two metrics over Uniqueness and Novelty alone, as they better assess model's performance compared to separately reporting all three metrics (Validity, Uniquenes,, and Novelty). The MOSES benchmark also computes metrics by comparing the generated set scaffold test set, from which we report the Fréchet. ChemNet Distance (FCD), the Similarity the nearest neighbor (SNN), which computes the average Tanimoto similarity between the molecular fingeprints of generated set and the fingerprints of the molecules of reference set, and Scaffold similarity (Scaf), which compares the frequencies of the Remis-Murcks scaffolds in the generated set and reference set. Finally, we report the metries of the Filters indicating the percentage of generated molecules successfully passing the filters applied when constructing the dataset. The GuacaMol benchmark computes two scores, the Fréchet ChemNet Distance (FCD) score and the KL divergences (KL) between the distributions of a set of physicochemical descriptors in the training set and the generated set.
We report results using 0.002, i.e., 500 demoising steps on both datasets. The predictor-corredd experiments used 0.004 and 10 corrector steps for 500 denoising steps. For both datasets, used
To 1.5.
D.4 Conditional generation
We perform conditional generation experiments on QM9, targeting two properties, the dipole moment, j and the highest occupied molecular orbital energy (HOMO). They are well suited for conditional generation evaluation because they can be estimated using the Psi4 library (Smith et al. 2020). We trained models sweeping over Puncond E {0.1, and explore different values for s in 1, 6] during sampling. We and s
obtained our best results using Puncond
We evaluated our method in the same setting Vignac et al. (2022) during inference. We sampled 100 molecules from the test set, extracted their dipole moment and HOMO values, and generated 10 molecules targeting those properties. We estimated the HOMO energy and the dipole moment of the generated molecules, and we report the Mean Absolute Error (MAE) between the estimated properties and the corresponding targets.
To efficiently incorporate the conditioner y, we implemented a couple of ideas proposed in Ninniri et al. (2023). Instead of using y solely as a global feature, we incorporated as an additional feature for each node and edge. Additionally, we trained two-layer neural network to predict the size of the molecule given the target properties rather than sampling it from the empirical dataset distribution. Our empirical observations indicate that this approach enhances performance. As of the time of writing, no official implementation has been released for Ninniri et al. (2023), rendering it impossible to reproduce their results. Additionally, since they do not report validity in their experiments on QM9, we choose not to include their results as a bascline to avoid unfair comparisons.
D.5 Compute Ressources
Experiments on QM9, PLANAR, and SBM were carried out using a single V100 or A10 GPU at the training and sampling stage. The training time on QM9 is 6 hours, while the training time on SBM and Planar is approximately 2 days and half.
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We trained models MOSES Guacamol using two A100 GPUs. To sample from these models, used single A100 GPU. The training time on MOSES is approximately two days while training GuacaMol requires 4 days.
E
Additional Experiments
E.1 Ablation on the number of steps
To demonstrate why the continuous-lime approach allows trade sampling quality and efficiency, we perform an ablation study on т. Results are presented in Tables S to We report the number of model evaluations equal to 1/r instead of r for readability.
Overall, the model achieves decent performance across all datasets with just. 50 steps. Increasing the number of model evaluations to 500-700 enhances performance to a state-of-the-art level. Beyond this point, performance saturates and models using 1000 steps do not necessarily outperform those with fewer evaluations, seen with SBM and PLANAR.
Table Ablation study on the number of steps for synthetic graphs. We report the mean of runs, well 95% confidence intervals. The best results highlighted in bold.
Number of steps
Degree + Cluster
Orbit +
Spectrum
VUN [%]
Planar graphs
103.0s6.
11.0a.1
305.4.10.6
5.3
0.0+0.0
50
3.611.3
3.3102
12.4145
1.310.2
41.5.14.51
100
3.1a112
2.31033
5.2-25
1.4102
76.012.23
300
3.4
1.7+0.5
6.2=39
1.3+0.1
86.5+3.82
500
2.1213
1.520.3
3.1=30
1.2:2.2
92.523.67
700
2.4+1.2
1.5=0.2
2.2+1.
1.1:0.2
94.0+2.23
1000
1.9210
1.910.2
2.7
1.5.0.2
89.5.3.35
Stockastic block model
10
166.6216.1
1.840.1
3.3=0.2
2.220.2
14.0:24.72
50
2.6:1.66
1.5.0.0
1.9=0.2
0.910.1
62.0.15 44
100
1.4=0.7
1.5=0.0
1.7=n.2
0.9
70.5
300
2.4.0.6
1.51000
1.8.0.3
0.8
65.5.15
500
2.411.
1.5.00
1.7.0.2
0.8.0.1
77.015.26
700
2.6=0.9
1.5+00
1.620.1
0.9
69.02:4.00
1000
1.840.7
1.5200
1.7-04
0.8:2..1
67.313 10
Table 9: Ablation study on the number of steps for QM9, We report the mean of 5 runs, as well as 95% confidence intervals.
Number of steps
Validity
Uniqueness f
Valid Unique t
FCD
NSPDK J
10
88.69+0.30
98 18
87, 47.42+0.46
0.84=a.02
0.001
50
99.07
96.78=0.16
95.88+0.21
0.25=0.01
0.000
100
99.4210.00
96 10.06
96. 24.1.05
0.26
0.00010.0
300
99.534c.0z
96.57=0.12
96. 12..11.
0.25=0001
0.00044.0
500
99.57.0.00
96.76.20. 17
96.34go.2
0.25=0.01
0.000
700
99.53+6.00
96. 10.655
96.2+0.15
0.25=
0.000
1000
99.57+6.07
96.79=a0.8
96.37+0.14
0.25=0..r
0.000
E.2 Ablation on the noise model
To emphasize the impact of using marginal transitions instead of uniform transitions, we trained models on PLANAR and SBM with the uniform noise model (see table [Tp. While the uniform model performs competitively with marginal transitions on SBM, the VUN score remains significantly higher with marginal transitions. On PLANAR, marginal transitions demonstrate substantially superior performance compared uniform transitions.
28

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Published in Transactions on Machine Learning Research (04/2025)
Table 10: Ablation study on the number of steps for MOSES We report the mean of five runs, as well as 95% confidence intervals. The best results are highlighted in bold.
Number of steps
Val.
Val. & Uni.
VUN
Filters
FCD 4
SNN
Scaf
26.1:0.2
26.1
26.010.2
59.920.8
7.8820.13
0.36220.0
8.9
50
$2.920.3
82.9+0.3
80.5.a.3
94.6=0.1
1.54=0.01
0.49=o.0
18.4+10
100
85.820.2
85.7+0.1
82.9=02
96.5+0.1
1.43+0.01
0.5+0.0
17.2+0.6
300
86.910.2
86,910.2
83.810.2
97.110.1
1.44.10.22
0.51..0
17.811.
500
S7.0+0.2
86.9+0.2
83.8+0.2
97.2=0.1
1.44=0.02
0.51
15.9+0.8
700
87 87.2
87.1
83.9.0.2
97.2=0.1
1.4320.02
0.51
15.920.4
1000
87 87.2
87.2+0.2
84.0+0.2
97.2
1.44=0.01
0.51
17.3#0.9
Table Ablation study on the noise model for synthetic graphs. We report the mean of runs, as well 95% confidence intervals.
Noise Model
Degree
Cluster 4
Orbit
Spectrum +
VUN [%]
Planar graphs
Marginal Uniform
2.1
1.5203
3.1230
1.2g02
92.5=3.67
143+42
3.8s0.6
14.2a6.1
1.750.2
32.5=4.4
Stockastic block model
Marginal
2.+111
1.5.0.0
1.740.
0.840.1
77.0
Uniform
1.620.5
1.520.0
1.820.
0.9g0.1
63.5-0.6
E.3 Ablation on the loss function
To justify choice to cross-entropy our loss function instead of ELBO, we provide the results of our experiments on QM9 using ELBO as our loss function. Figure 5 presents the validation performance of both approaches regarding validity over 512 samples. While the cross-entropy loss allows us to reach a near-perfect validity quickly, the model trained using the ELBO saturates below 80%. The significant performance gap on a simple dataset like QM9 underscores the inefficiency of using the ELBO as the loss function for Cometh.
Validation validity on QM9
100
95
Validity
85
80
75
EL90
CE
100
200
300
400
500
Epochs
Figure 5: Validation results on QM9 using the cross-entropy and ELBO losses.
E.4 Ablation the noise schedule
We also performed simple ablation study on QM9 to compare the performance of our cosine noise schedule against the exponential noise schedule ((t) = an log using 0.8 and 7 2. Overall, our cosine schedule performs better every metric except the Uniqueness.
29

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Published in Transactions on Machine Learning Research (04/2025)
Table 12: Ablation on the noise schedule on QM9. We report the mean of five runs, as well as 95% confidence intervals. The best results are highlighted in bold.
Model
Validity
Uniqueness
Valid & Unique
FCD +
NSPDK +
COSINE
99.57.0.07
96.7620.7
96.3420.2
0.25
0.000 00
ExP 98.28
97.0=0.18
95.34
0.31
0.001#0.00
F
Limitations
Although model advances the state-of-the-art across all considered benchmarks, it still faces quadratic complexity, common issue in graph diffusion modcls. This problem could be alloviated by adapting mothods like EDGE (Chen et al. 2023) used to scale DIGRESS for large graph generation. Additionally, our approach does not support the generation of continuous features and is restricted to categorical attributes. It should be combined with a contimuous-state diffusion model to generate contimuous featuros, resulting in an approach similar to Vignac
2023
G Samples
Figure 6: Samples from COMETH on PLANAR (top) and SBM (bottom)
HD
6H
Figure Samples from COMETH on QM9,
30

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Published in Transactions on Machine Learning Research (04/2025)
810
Figure 8: Samples from COMETH on MOSES.
axio
Figure 9: Samples from COMETH GuacaMol. The samples on this dataset exhibit some failure cases, such disconnected molecules 3-cycles of carbon atoms.
31


Reviews:
[{"review_id": "SBJ1CTDdGV", "reviewer": "Reviewer_GTmN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The graph generation task is an important task that is interesting to the TMLR audience.\n- The proposed method use sensible choices of diffusion processes and parameterization to achieve the goal of graph generation.\n- The evaluation results demonstrate competitive performance.\n- The presentation is clear and easy to follow.\n- Ablation studies are conducted to demonstrate the importance of number of diffusion steps and the positional embedding.\n\nWeaknesses:\n\n- The new components proposed are relatively incremental extension to existing approaches.\n- Evaluation results are competitive but does not seem to be a step-change compared to existing models. \n\nAlbeit relatively incremental, the combination of several sensible modeling choice makes feel this paper is still a nice contribution to the graph generation task.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a diffusion model for graph generation using a continuous-time diffusion process and discrete state for graph representation. Building upon existing works in continuous time diffusion models for discrete data and discrete diffusion models for graph data, this work extends these models with new noising/denoising processes, training framework, and positional embedding. the proposed method is evaluated over synthetic graph generation, QM9 small molecule generation, and Moses/GuacaMol molecular generation tasks. The proposed method achieves competitive performance across these tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It might be useful to have the ablation studies on positional embedding included in the main paper, as it seems to be one of the main contribution of this paper.", "broader_impact_concerns": "There is no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "U89VSL5o8t", "reviewer": "Reviewer_fvEe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The paper topic is highly relevant.\n- The paper appears to be technically sound.\n- Empirical performance of the proposed method is good in comparison with baselines, and the comparison seems fair.\n\nWeaknesses:\n\n- Novelty is limited. The paper is very similar to previous work, e.g. Vignac et al. (2022)\n- The authors do not provide a software implementation, which makes it hard to reproduce results.\n- Clarity could be improved. Many details are left to the appendix, which sometimes makes the exposition difficult to follow. Some implementation details are not clear.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a diffusion model for (discrete) graphs with node and discrete edge features. The main novelty lies in its formulation as a continuous-time diffusion (although this has also been proposed independently by Xu et al. at NeurIPS 2024). The paper includes a reasonably comprehensive and fair comparison with baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "\"Furthermore, the noise models used for other data modalities may be sub-optimal for graphs (Vignac et al., 2022).\" Could you elaborate this a bit - it is not clear to me what is meant with this comment without referring to Vignac et al. 2022.\n\n\"...into a continuous diffusion model framework...\" Should that be a \"...into a continuous-time diffusion model framework...\"\n\nPlease define VUN the first time it is used in the main text.\n\n\"Overall, Cometh is the first graph diffusion model allowing the benefits of using a discrete-state space and\nthe flexibility of a continuous-time scale in the design of the sampling algorithm.\" Is this true, considering Xu et al. [1]?\n\n\"...do not suffer from the quadratic complexity of one-shot models.\" I am unsure what this means exactly? Do you mean quadratic complexity in the number of nodes in the graph? How do the auto-regressive models scale in comparison? I suppose they could scale linearly if we have constant complexity per step, but I would assume it would be typical with linear complexity per step leading to overall quadratic complexity?\n\nConsider discussing masked diffusion (see e.g. Shi et al. [2] and references therein) in the related works section.\n\nPlease define RRWP when it is first used.\n\n\"...the predictor-corrector mechanism...\" Consider including a reference here.\n\n\"The rate of this reverse CTMC (...) has to be modeled by a parameterized approximation...\" Is this the only way this rate can be parameterized? It would be nice with a bit of explanation for this expression - is it derived from a detailed balance consideration? I know there are some more details in Campbell et al. but it would be nice with a short explanation here.\n\n\"As demonstrated in Campbell et al. (2022), this property is achieved by choosing R(t) = β(t)R...\" It would be nice with a bit of discussion about which other parameterizations are possible here.\n\n\"...where Rb = P ΛP −1 and exp refer to the element-wise exponential.\" Should this not technically be the matrix exponential, which can be computed by the element-wise exponential of the diagonal elements? (i.e. not compute the exponential of the off-diagonal zero elements.)\n\n\"Given that the ELBO and cross-entropy share the same optimum...\" Is this easy to see? Is there a reference, where this is shown?\n\nIt is not clear to me what theorem 2 and proposition 3 adds to the paper. Is it simply meant to provide some intuition about what is captured by the RRWP? Perhaps a comprehensive empirical study of different features would be more illuminating.\n\nThe technical description of the graph transformer architecture is very brief and not self-contained. For example FiLM is not defined and there is no reference to Perez (2018).\n\nIn table 1 under Degree, Disco 1.2 is in bold, but DiGress is lower at 0.8.\n\nWhy is Cometh-PC not included in the SBM experiment?\n\n\"...only outperformed by DiGress on Degree and Orbit, but with a much lower VUN.\" A bit misleading, since Cometh is outperformed by several competitors on Orbit.\n\nThe MMD numbers in table 1 for Planar and SBM seem to be on another scale than what is reported in Jo et al. and Martinkus et al. Do you explain somewhere, how these are computed?\n\nIs there full code available? I did not see a link to an implementation, which would be very much appreciated - also for the review process.\n\nPlease provide a public code repository with the implementation if the paper is accepted.\n\nPlease do include results for the PC sampling on QM9 - it is interesting to see if it is on par or worse.\n\nAt least some technical details regarding conditional generation should be included in the main text. As it is now, there is only a reference to appendix C.2 which then refers to appendix B.5 for the details.\n\nPlease check if all citations are correct. I noticed the paper by Xu et al. (2024) is listed as \"arXiv preprint\" but it was published at NeurIPS last year, I believe.\n\nPlease restructure the paper, so that sufficient information is available in the main body, at least regarding model architecture, conditional sampling, and the predictor-corrector mechanism.\n\n[1] Xu et al. \"Discrete-state Continuous-time Diffusion for Graph Generation\"\n[2] Shi et al \"Simplified and Generalized Masked Diffusion for Discrete Data\"", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "xaqSzFIv0C", "reviewer": "Reviewer_9gLW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: Addresses real world problem, likely to be interesting to subset of TMLR readers and appears to be correct. Therefore, acceptance is recommended after relatively minor revisions detailed below.\n\nWeakness: Theorem B.3 seems kind of trivial and not very relevant to the proposed work. It appears to be correct, but seems like it was added in just for the sake of having \"more theory\" and the detailed statemetns statements seem semi-tautological\n\nQuestions: Why is there a delta in the equation at the bottom of page 3? It seems like this term should be omitted. In the appendix (page 14) it says that this ensures summation to 1, but it seems like the equation should be something like q_{t+dt} (tilde(z), z^t) = q_{t} (tilde(z), z^t) + R( stuff ) dt\nWhy do you constain the feature matrices X and E to have value in {0,1}? This seems particularly strange for E", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work introduces continuous time diffusion models for graph structured data building off of previous work that was either discrete in time or not for graph generation\n\nIt utilizes a generation method based on a reversible SDE while enforcing sparsity in the generation process associated to the backwards equation", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address the weaknesses noted above.\n\nPlease add more details to the last three steps of the proof of proposition 4.\n\nClarify that prop 4 is a restatement of prop 1 (check throughout).", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper considers discrete graph generation with continuous-time diffusion models. All three reviewers gave insightful comments, and authors responded appropriately, and updated the manuscript. All reviewers agree on audience and claim evidence.
Decision: Accept (poster)