DMol: A Highly Efficient and Chemical Motif-Preserving Molecule Generation Platform
Authors: Peizhi Niu
Yu-Hsiang Wang
Vishal Rana
Chetan Rupakheti
Abhishek Pandey
Olgica Milenkovic
Venue: NeurIPS 2025 poster ()
Keywords: Molecule Generation
Diffusion Model
Primary area: machine_learning_for_sciences

Abstract: We introduce a new graph diffusion model for small drug molecule generation which simultaneously offers a 10-fold reduction in the number of diffusion steps when compared to existing methods, preservation of small molecule graph motifs via motif compression, and an average 3\% improvement in SMILES validity over the DiGress model across all real-world molecule benchmarking datasets. Furthermore, our approach outperforms the state-of-the-art DeFoG method with respect to motif-conservation by roughly 4\%, as evidenced by high ChEMBL-likeness, QED and newly introduced shingles distance scores. The key ideas behind the approach are to use a combination of deterministic and random subgraph perturbations, so that the node and edge noise schedules are codependent; to modify the loss function of the training process in order to exploit the deterministic component of the schedule; and, to ''compress'' a collection of highly relevant carbon ring and other motif structures into supernodes in a way that allows for simple subsequent integration into the molecular scaffold.

DMol: A Highly Efficient and Chemical Motif-Preserving Molecule Generation Platform
Peizhi Niu', Yu-Hsiang Wang', Vishal Rana',
Chetan Rupakheti', Abhishek Pandey', Olgica Milenkovic 'University of Illinois Urbana-Champaign Abbbie
{peizhin2, yw121, vishalr, milenkov]@illlneee edu echetan.rupakiie abhishek. bandishe..anddy/)ee..
Abstract
We introduce a new graph diffusion model for small drug molecule generation which simultaneously offers 10-fold reduction in the number of diffusion steps when compared to existing methods, preservation of small molecule graph motifs via motif compression, and an average 3% improvement SMILES validity over the DiGress model across all real-world molecule benchmarking datasets. Furthermore, our approach outperforms the state-of-the- of-the-art DeFoG method with respect to motif-conservatii likeness, QED and newly introduced shingles distance scores. The key ideas behind the approach are to use combination of deterministic and random subgraph perturbations, so that the node and edge noise schedules are codependent; modify the loss function of the training process in order to exploit the deterministic component of the schedule; and, to "compress"" collection of highly relevant carbon ring and other motif structures into supernodes in way that allows for simple subsequent integration into the molecular scaffold"
by roughly 4%, as evidenced by high ChEMBL-
Introduction
Graphs are fundamental data structures used to model wide range of complex interactions, including molecules and drugs, biological pathways, and social and co-purchase networks Wu et al. (202T).
Zhang et al. [2024] Yang al. [2024), Waikhom and Patgiri [2021]. Hence, the ability to generate
graphs that accurately capture domain-specific distributions is of great significance T'sai et al. [0033)
You et al. 2018 Gamage al. [2020]. While generative models for images and texts are being used with great success Dhariwal and Nichol [221], Ho et al. [2022, Waikhom and Patgiri [021] ]. graph generation remains challenging frontier due to the discrete nature of the models, the need for permutation invariance and other distinctive properties Velikonivtsev et al. [2024). Wang et al. [2023]. This is particularly the case when trying to generate graph samples that innovate moleculan
compounds Jin et al. [2018), Mitton et al. [2021], Vignac and Frossard 2021), 3 Klómez-Bombare
ett al. [0018 K Kwon et al. 2020]. Wu et al. [2024], You al. [2018] Liao et al. [2019], /Seller
et al. 2018 Popova et al. [2019]. Madhawa et al. [2019], Zang and Wang [2020], in which case
the generated samples can fail obey necessary biochemical and physical constraints and preserve
important motif compounds.
One way to address these challenges is to use diffusion models Sohl-Dickstein et al. [2015] Ho
ct al. 22020], Fan et al. [2023], Chen et al. (2024). Trippe et al. 2023 Haefeli et al. [2033], Niu al. 2020b), Huang al. 2022 Liu et al. [202cc] which rely gradual noise injection process that perturbs the input samples and then reverses the process using a learned denoising network.
'The link to the code can be found at: http://ghhhaccoommidoo/orreeeeee
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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In the context of small molecule drug generation, existing methods focus on discrete diffusion in order to avoid the loss of key structural properties that arise when embedding graph structures into continuous spaces Niu et al. [2000a]. Jo et al. [022b]] Discrete diffusion models incorporate "transition mechanisms" that explicitly consider the categorical attributes of nodes and edges, ensuring fine-grained structural changes during the noise injection process [iu et al. (2024b), , Xu et al. [2024a] They also allow for continuous and discrete denoising steps that diversify the outputs Huang et al.) 2022 Vignac et al. [2023] Jo et al. [2022a]. However, they also typically require very large number of diffusion steps, resulting in large running times and extremely slow graph generation Kong et al.
2023 More importantly, they lead to structures that have low chemical validity even when measured through exceptionally coarse SMILES nomenclature constraints Hoogeboom et al. (2022), Liu et al.
[021]. Most learning methods consequently fail to produce molecules that can be chemically synthesized or that can properly dock on target proteins.
One well-known graph diffusion method, DiGress Vignac et al. [2023], relies on cosine noise injection mechanism that controls the amount of perturbations during the different diffusion steps. In the process, every node and edge in the graph is allowed to be modified at each step. DiGress evaluates the generated drug sample quality through SMILES validity, novelty, and uniqueness all of which provide highly limited measure of the utility of molecules. Also, the training and sampling time of the method is excessive. Furthermore, in its current form, and unlike some nondiffusion based methods Jin et al. [2018]. (Simonovsky and Komodakis [2018), Mitton et al. [202] Vignac and Frossard [021]. J. Gomez- Bombarelli et al. [2018], Kwon et al. [2020], Wu et al. [2024), You et al. et al. (2018). Popova al. [2019]. DiGress cannot preserve the structures and frequencies of important drug network motifs (including carbon rings, which are the most important building blocks of organic molecules).
[2018]. Liao et al.
We describe a new small molecule drug diffusion model, Diffusion Models for Molecular Motifs, DMol, which significantly outperforms DiGress across all benchmarking datasets in terms of running time, SMILES validity, and novelty scores and at the same time, allows for network motif preservation via compressed representations of subsets of motifs. At the same time, the DMol improves the more informative chemical validity scores when compared to the very recent state-of-ther the-art DeFoG Qin et al. 22225) method by roughly 4%, since the latter cannot guarantee preservation of motifs. Its main features are as follows:
1. DMol directly relates the diffusion time-steps with the number of nodes and edges that are allowed to undergo state changes in that step. This approach effectively increases the learning rate of the model as it learns increasingly larger submodels in a gradual manner. It also significantly reduces the time required for molecule generation, leading to an order magnitude reduction in the number of steps required by diffusion models.
2. The number of nodes and edges that DMol perturbs at each step deterministic, but the nodes themselves are selected at random. Furthermore, the perturbed edges are confined to the subgraph induced by the selected vertices, which couples the node and edge perturbations and leads to better preservation of subgraph structures.
3. DMol uses new penalty terms in the objective function that penalize mismatches in the determin- istic counts of nodes and edges perturbed during the forward and backward steps. Such penalties lead to poor results when the number of perturbed nodes is random, as in DiGress and other methods, and motivates the use of a mixed deternninstticrandoom noise schedule. This allows for further "forced motif" preservation and more flexible control of the noise schedule/distribution, as evidenced by straightforward theoretical analysis.
4. One of the most important features of DMol is motif compression which allows one to con- vert chemically important node motifs into supernodes that are diffused similarly to regular atom nodes. The chosen motifs (which are mostly carbon rings) have the property that they can always be reattached to the molecular scaffold, and the number of such molecules is bounded to retain compu- tational savings guaranteed by the first three innovations. Due their careful chemical selection, only one type of bond is possible during reconstruction, and "decoding" is extremely simple. Motif compression differs both from JT-VAEs Jin et al. (2019) and ring freezing proposed in DiGress (which failed to perform in a satisfactory manner), since in our case one is still allowed to directly replace one valid motif structure by single atom or another valid motif structure (in addition, the choices of substructures used differ widely).
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As a result, DMol improves the SMILES validity of DiGress by roughly 3% over all real-world molecular benchmarking datasets while keeping the number of diffusion steps at an order that is one magnitude smaller. At the same time, the significantly more meaningful chemical likeness scores (e.g., [aadrru et al. [006] ChEMBL likeness, QED and shingle distances, which indicate how likely the generated molecule is to have desired chemical motifs and similar biological activity properties) of DMol are consistently better than those of DiGress. These scores for DMol are roughly 4% higher, on average, than those of the recent state-of- the-ttrr flow matching method DeFoG Qin ct al. [222]] since the latter does not tend to preserve chemical motifs.
As final remark, it is important to point out that in the biochemistry literature. it has been long recognized that SMILES validity optimization may not be as practically relevant from the chemical synthesis/properie point of few: motif, and especially carbon ring structure preservation. is deemed much more important, but tangible evaluation metrics for true chemical validity are still lacking. See for example the recent work Skinnider [2024] that discusses pros and cons of SMILES validity.
2 Related Works
Viewing small molecules as graphs has inspired myriad of generative models for drug design
including JT-VAE Jin et al. (2018), MoFlow Zang and Wang [2020], CDGS Huang et al. 2023).
EDM Hoogeboom et al. [2022], etc Cornet et al. [.024], Shh et al. [002] Xu et al. [2024a] Chen
et al.
2023 Wu et al. 2022]. Bao et al. [2023], You et al. [2018 Liao et al. [2019]
Mercado et al.
202 Segler et al. [2018). K Kwon et al. [2020] Jensen [2019). Jo et al.
[224].
Furthermore, numerous surveys on the subject are readily available Tang et al. 2024]_ Du al.)
[2022). Yang et al.] [2024). We partition and review prior works on generative models for drug
discovery based on the methodology used.
Traditional Methods, Traditional approaches to graph-based molecular generation relied on the use of Variational Autoencoders (VAEs) as generative models Jin al.] [2018) (Simonovsky and
Komodakis 2018), Mitton et al. [2021]. Vignac and Frossard [202]]]Gómez GGmmzz Bombarelli et al. [2018].
Kwon al. 2020), Wu al. 2024]. VAES comprise an encoder that maps the input graph into
latent space and a decoder that reconstructs the graph from the latent embedding. Many methods
also involve autoregressive models, which generate molecular graphs by sequentially predicting the
next element in the sequence based on previous outputs You et al. [2018) Liao et al. [2019], Segler ct al. 20088. Popova al. [2019]. On the other hand, flow-based methods leverage the concept of normalizing flows for graph generation Madhawa et al. 2019), Zang and Wang (2020). [uo et al.
202lb) and Gavves (02T). Shi et al. 20220.. These techniques apply a series of invertible transformations to model complicated distributions based on simpler ones (such as a Gaussian). Additionally, Generative Adversarial Networks (GANs) approaches have also been used for molecule generation De Cao and Kipi (2018). Lukasz Maziarka et al.] [2019), Martinkus et al. [2022a]. GANs include a generator, which learns to create realistic molecular lar graphs, and a discriminator, which learns to differentiate between real and generated samples.
Diffusion-Based Methods. Diffusion models for graph generation can be broadly categorized as follows. Denoising Diffusion Probabilistic Models (DDPM) utilize Markov chain for the diffusion
process Vignac et al. 2023 Xu et al. 22022] Liu al. (0224). Xu et al. 2024a] Liu al. (024a))
Hoogeboom et al. [022] Jo et al. 2024]. Score-Based Generative Models (SGM) leverage score
matching techniques to model the data distribution, and generate graphs by reversing a diffusion
process guided by the score of the data distribution et al. [2021a], Chen et al. [2023). Wu et al.
2022]. Furthermore, by replacing discrete time steps with continuous time, one can use stochastic differential equations to model the diffusion and denoising processes Do et al.] [2022a] [ee et al.
20ua).Huang et al. [2023). J. Bao et al. 22023). J. Furthermore, studies have shown that different noise schedules may sigmificantly affect molecule generation quality (hh et al. [
Nichol and Dhariwal
2021 Comprchensive surveys on the subjects include Mengchun Zhang et al. [003 Fan et al.
2023
Chen et al. [2024].
DiGress Vignac et al. (2023) uses a denoising diffusion probabilistic model within a discrete state space and is considered one of the practically effective models for molecular generation tasks. It represents graphs in the discrete space of node and edge attributes by assigning categorical labels to each node and edge. DiGress models the noise addition process as Markov process, where each node and edge label evolves independently of the others. This assumption mirrors the approach in maage-based diffusion models. Denoising is performed using a graph transformer network, trained by
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minimizing the cross-entropy loss between the predicted probabilities for nodes and edges and the ground-truth graph. The transformer network reconstructs denoised graph from a noisy input. To generate new graphs, noisy graph is first sampled according to the model's limiting distribution, and the trained denoising transformer is then used to produce a graph with the desired properties. The DiGress model is discussed further in Section B
DisCo Xu et al. 2024b) uses discrete-state continuous-time diffusion, addressing limitations of discrete-time approaches like DiGress. The model formulates graph generation as a continuous-time Markov Chain (CTMC) that preserves the discrete nature of graph graph-structured data while enabling flexible sampling trade-offs. Unlike discrete-time models with fixed sampling steps, DisCo can adjust sampling steps after training without retraining the model. DisCo enables numerical approaches like r-leaping for efficient simulation of the reverse process, allowing practitioners to dynamically balance generation quality and computational efficiency during inference. Despite its innovations, DisCo suffers from quadratic computational complexity with respect to the number of nodes, limiting its scalability for generating large graphs. Moreover, DisCo reduces the number of sampling steps at the cost of degraded generation quality.
Flow Matching-basee Methods. Flow Matching (FM) has recently emerged as an alternative to diffusion models for generative tasks Lipman et al. 2023 Dao el al. [0033) FM frameworks define continuous probability paths between a simple prior distribution and the target data distribution, offering comparable performance and efficiency compared to diffusion based approaches. To address discrete state spaces, Discrete Flow Matching (DFM) formulations have been developed Gat et al.) [2024), providing a streamlined approach with more flexible sampling procedures. DeFoG Qin et al. (2025) extends the DFM framework specifically for graph generation tasks. It employs a linear interpolation noising process and CTMC-based denoising process while preserving the inherent symmetries of graphs. The key innovation in DeFoG is its disentanglement of training and sampling stages, which enables independent optimization of each component. Despite its innovation, DeFoG still requires dataset-specific hyperparameter tuning for optimal sampling strategies, making its adoption potentially challenging for new graph domains without extensive experimentation. Importantly, experiments to be described in this work reveal that DeFoG under-perform on key biochemical motif molecular property metries such QED.
3
DMol Diffusion
3.1 Review of Standard Graph Difftusion Models
The focal concept in diffusion models is the noise model q, designed to enable T forward diffusion steps. Graph diffusion relies on progressively introducing noisy perturbations to undirected graph g". encoded as z", resulting in a sequence of increasingly noisy data points (z) (z.,..... z'" The noise
addition process is Markovian, which translates to g(z'...., |=)) = II=
qzz" Iz" 1)
More precisely, the encoding of graph involves categorical information about the nodes V and vertices E of the graph G = (V, E). The assumptions are that each node belongs to one of a classes assumed to
and is represented as one-hot vector in Ra. The same is true for edges, which belong to one of b classes (with one class indicating the absence of the edge). We follow the notation from Vignac et al. 20233, where X; is used denote the one-hot encoding of the class of node The one-hot encodings are arranged in a node matrix X of dimensions n X a. Again, in similar manner, b. The Markovian state space for each node and edge is the set of classes of the nodes and edges, respectively. With a slight abuse of notation, x stands for the class (state) of node I, while ets stands for the class (state) of edge ls.
edges encoded into a tensor E of dimension
Diffusion is performed separately on nodes and edges, while the noisy perturbations in the forward process are governed by a state transition matrix. For node-level noise we use transition matrices (Q'...... .... )x), where each entry [Q'x ls defines the probability of an arbitrary node transitioning from state to state j at time step t. A similar model is used for edges, where the transition matrices (Q) ) describe the probabilities O'gls of an arbitrary edge transitioning from state i to state j. Formally, Q = q(x"
x²11 = i) and [Qrles = q(e' = j e = i). where x and e correspond to generic node and edge, while the superscript indicates the time stamp.
Consequently, the transition probability at time & is given by q(z' | d z' z'-1Q', where z stands for either and the subscript of Q is set accordingly. Adding noise to graph Gr-1 at

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time t 1 results in new graph G" = E"), and involves sampling each node and edge class from categorical distribution succinctly described as q(g² gg = (x* -x,
and ((" | = Qx" LE). where Q'x Ox O'x and Q Q O' The state transition matrices are invertible, which ensures the reversibility of the noise model: the graphs at time and time 0 can be transformed into each other using the state transition matrix. Reversibility guaranteess consistency between the forward and backward processes and simplifies backward denoising.
common issue with most known diffusion models is the large number of steps needed for conver- gence Hang et al. [2023] Wang et al.] [2024). For large training sets the computational complexity may be overwhe mming (see our Results section). Hence, the first important question is how to reduce the number of diffusion steps for graph generation, and at the same time, preserve the chemical utility of the model. These issues are addressed by DMol.
3.2 DMol Forward Noise Adding Strategy
The key idea behind our approach is as follows: at each forward diffusion step I, we randomly select (t) nodes and (t) edges from go contained in the complete subgraph induced by the selected nodes (note that the absence of edge is represented by special label so that we are effectively dealing with a complete graph), and change their classes according to fixed state transition matrices Qx and Qg. respectively, to obtain the graph G'. Here, (() and (t) are deterministic functions of the time step index í. Note that we effectively run the diffusion process on a determinist number of randomly selected subsets of nodes and edges, with the perturbed edges confined to the subgraph induced by the selected nodes. This effectively couples the node and edge diffusion processes.
Sclotin of My My odge
60
PP
Sondl Mus ars
NTT) MT))
(499
(a) At each time step t, we randomly select a predeter- mined deterministic number of nodes and add noise only to the class labels of these nodes and fixed num- her of edges in the subgraph induced by the nodes. Both selection numbers are controlled by the time in- dex L. The top row depicts how the number of selected nodes (L) and edges (1) of the selected subgraph evolves with (we set k 2,r 0.2).
(b) DMol diffusion relies on randomly selecting (t) nodes and (t) edges from the subgraph induced by the ((t)) nodes at each time step f, and diffusing their labels to generate noisy graph G². The denoising network do learns to predict the denoised graph from g'. During inference, the denoised graph is combined with " ( to predict G-1
Figure 1: (a) The forward process; (b) DMol illustration.
For (t) and (t), we adopt cosine schedules, N(t) [(1 a)nl and M() [(1 a) N()) (N 1) 0.5r], where number of nodes in the graph, r is hyperparameter and e is small positive constant. The total number of diffusion steps is fixed at T = n, where & is small hyperparameter (typically set to I or 2 so as to keep the number of diffusion steps as small as possible; details behind the selection of appropriate values of & can be found in Appendix G.3) The maximum number of diffusion steps is proportional to the number of nodes n (typically around 40 for small molecule drugs). Hence, DMol requires significantly fewer diffusion steps than other methods, such as Digress (which requires 500 or 1000 steps), and adapts itself to the size of the graph.
co.5(0.. + c)/(1 c)), and stands for the the
Additionally, our forward process constrains the class changes of edges to the subgraph induced by the nodes that underwent state change. This ensures that the sets of altered nodes and edges are co-dependent. Still, once the selections are made, the actual state changes of nodes and edges are governed by independent processes. We next turn our attention to the choice of the transition probability matrices. The results of DiGress Vignac et al. 20023 have shown that making the
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transition probability from class to class j proportional to the marginal probability of class j within the training dataset results in more effective learning of the true data distribution compared when using uniform transitions. We follow the same procedure and define the marginal probabilities of node classes the training dataset as my E RIxa and those of edge classes as m. Rx respectively. Hence,
meld M Ergs m.
if J. if j.
meld Erge m. [
if i J
Qxli,
QFli,
if
During the forward process, compute the distributions of all node and edge classes after state transitions. However, we only focus on the distributions corresponding to the selected [t)) nodes and (t) edges, sampling from these distributions update their states. We preserve the states of the remaining nodes from the previous step. Algorithm and Figure Tb illustrate our forward process.
Algorithm Forward Process
Algorithm Sampling Process
Input:
go (X", E.), state
Input: Denoising Network Φ9, node&edge class distribu-
transition matrices &x&Qg.
tion m = [p, 2 1 &&m"" = 1pp o state transition matrices
node&edge selection functions (.))&
xx&QF-
Process:
Process:
Sample GT from m m for == T to do
Sample & E =11 (1...7).
Sample g' (X'; F') from
+ f("",t) (Structural and spectral features).
X x EE (()) and v
p x 60 60 2, Ra, Re) (Prediction).
Calculate (().
Sample 8 - from 8x p
Sample g' (X E
from X Qx x E M(t-1),
Randomly select u nodes and edges from the subgraph induced by the nodes and generate "fil- ter" matrices K2 and Ke which serve as indicators of the selec- tions.
Calculate u = N(t 1). U.
Randomly select u nodes and edges from the subgraph induced by the u nodes and generate corresponding "fil- ter" matrices K, and K..
Set G'-1 = g" and replace K with K,X K, É f-1
Set g = go and replace K,X' with K, ' and K2 with K.E.
with K, K
end for
3.3 Denoising/Sampling Process
During the denoising process, first sample random graph based the marginal distributions of nodes and edges at f T. This random graph is then used as the input to the denoising network ée to predict denoised graph. Based on the prediction, graph at t = T 1. Subsequently, this noisy graph is fed into the denoising network, and the above process is repeated iteratively until new graph sampled.
use the noise model to obtain the noisy
To start the sampling process, we need compute the marginal distributions of nodes and edges at ( = T. At time ( - T, node class i at / T is result of transitions from some other node classes at = 0. As result, the marginal probability of the i-th node class at time equals
pj
Σ
P
1-
where pf and y are the marginal probabilities of node classes and j at time = 0. respectively. For edges, we similarly have that the marginal probabilities of the i-th edge class at f T equals
P3 T = =1= r Σ Σ1-
Pf
where pf and pf are the marginal probabilities of edge classes and at = 0.
After sampling the random graph, we use the denoising model to predict the clean graph and continue this process iteratively. At time step I, the input to the denoising model includes not only the noisy graph and the time t but also two additional functions: Rx (tt/nn and

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(e( = (0t) 1)), which describe the proportion of nodes and edges in the noisy graph time that undergo state transitions relative to the total number of nodes and edges, respectively. They indicate to the denoising network that the input noisy graph and the predicted graph should differ by (t) nodes and M(t) edges, thereby helping the network make more accurate predictions. The sampling steps are listed in Algorithm 2 Figure [ in the Appendix is an example of the sampling process from a random graph to a denoised graph. Note that the posterior distribution used by the
DMol sampling process, q "0"""""""0" .0)) factorizes as (()")119)) (0("")")) This decomposition improves both computational efficiency and inference accuracy.
The loss we use to train the denoising network differs from that of DiGress in so far that it contains two additional penalties,
1(p", .9) Σ CE(X, P. )
+ A1
Σ CE(eij, P.j
IsiSh
ISiJSh
+ 22 ASSE(D(argmann
argmaax (X),, + À3
argmax (
where p²= (p* p denotes the predicted probability distributions, while x; and eij represent the one-hot encodings of nodes and edges, respectively. CE stands for the cross-entropy loss, while MSE refers to the mean squared error loss. The term argmax(p"* equals the class index of highest probability for each node, while argmax ()) effectively converts the one-hot encoding of each node into its corresponding class index. D)+:-) is equal to the number of nodes (edges) in the two arguments that have different classes. The first two terms are the standard node classification and edge classification losses. The third and fourth terms ensure that the differences in the number of nodes and edges of the predicted and the ground truth graphs, respectively, are as close as possible to the number of nodes and edges modified during the noise addition process. This is possible since (t), (t) are deterministic the added losses degrade performance otherwise.
We also briefly remark that it is easy to see that the DMol diffusion model and the modified loss function are permutation invariant. To see why the second claim is true, observe that our loss function comprises two components, a cross-entropy loss and an MSE loss. For the cross-entropy loss terms,
IcE(p () = Σ CE P +
Σ 1S
CE(e Pij):
ISiSh
Since the overall cross-entropy loss is obtained by summing up the individual cross-entropy values for each node and edge, and the sum operation is permutation invariant, then the cross-entropy loss is also invariant. We arrive at a similar conclusion for the MSE terms in (MSE(p" .9),
X MSE ((argmax X) argmax (X)), N()) + A3
(argmax (p )) argmax(E)), M()),
where invariance follows based on the definition of D(::)) and the fact that the loss is computed separately for nodes/edges with different indices and then summed up.
3.4 Sampling Effficieccy of DMol
To explain the sampling efficiency of DMol, we analyze the minimum time step increment öt required to perturb exactly one node (i.e., the smallest discrete noise unit) during the diffusion process. We compare DMol with DiGress; however, is important to note that the conclusion holds for all discrete diffusion models, including DiGress, DisCo, etc. The ratio of the required time step increments for
DiGress and DMol satisfies
MtDsGress
(I)
80pMa
E.r
where Pfd denotes the probability of node type i (for derivations, see Appendix D.2). The inequality demonstrates that DMol requires fewer steps to perturb single node. Since the total number of diffusion steps in DMol is solely determined by the number of nodes, the efficient addition of noise to nodes directly contributes to the overall efficiency of DMol. For further discussion of likelihood computations, noise distribution evolution, and expressivity refer to the Appendices C
Motif Compression
To preserve motif compositions (e.g., carbon rings), we use a new motif compression feature, depicted in Figure 2 The process is designed to better preserve motifs in molecular graphs.
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Table 1: Performance comparison on the QM9 dataset (see Appendix G.4
for more details).
MODEL
VT
V.U.t
V.U.N.f
GRAPHNVP
83.5
18.4
GDSS
96.0
94.6
GRUM
99.4
85.1
22.2
DIGRESS
97.8
95.2
31.8
DISCO
98.2
95.6
57.5
DEFOG
97.9
95.9
62.8
DMOL(OURS)
98.3
96.0
75.1
Table 2: Performance comparison on the same version of MOSES. For fair comparisons, we ran the DisCo code without any modifications, and DeFoG with 50 diffusion steps to ensure similar computational complexity to our method (DeFoG also reports their results for 50 diffusion steps).
MODEL
Vt
U1
N1
FILTERST
FCDJ
SNNT
SCAFT
DIGRESS
84.8
100
94.5
97.2
1.18
0.55
14.6
DisCo
85.7
100
97.4
95.8
1.40
0.51
14.5
DEFOG
84.2
100
97.2
96.9
1.89
0.50
14.8
DMOL (OURS)
87.8
100
100
97.8
1.12
0.58
14.8
Specifically, we convert a small predefined number (3 - 15) of most frequently occurring sub- graphs/motifs (which may differ for different training sets) that only form single bonds with the scaffolds through their available carbons into supernodes, introducing in the process new node classes to represent the compressed motifs. The compressed graph representations fed into our DMol diffusion model. During the sampling process, the supernodes are converted back to their respective motifs and integrated into the graph scaffold in chemically valid manner. Although superficially similar to part of the JT- VAE Jin et al. [2019) approach, motif compres- sion is significantly different. The JT approach uses not just motifs but large number (roughly several hundreds) of submolecular structures, and requires finding minimum spanning tree during the encoding process; furthermore, VAEs do not directly generate graphs, and one has to perform complicated decoding. The main issue that each compressed molecular substructure form many different types of bonds with different atoms during the reconstruction pro-
directly
Codebook C C
Encoding
OH Preprcemsing
Nede trazaécium
Diffasion
Figure 2: Motifs (i.e., substructures occurring with high frequencies or frequencies higher than predicted by random models) that also allow for unique scaffold integration are compressed into su- pernodes with their own labels. During diffusion, supernodes are either converted into other classes of supermodes or into atomic nodes, and vice versa. During sampling, the supernode decoded back into its corresponding motif.
The options ranked according to special scoring function and the overall process computationally demanding. The motifs used in DMol for different datasets are available in the Appendix, Figures 4
Experiments
We present next the results of running DMol on several benchmarking datasets. For experiments per- taining other graph models (c.g.. SBMs and planar graphs), the reader is referred to Appendix G.I
Benchmarks. We compared DMol to several graph generation methods. The benchmarking models used in the experiments include GraphVAE Simonovsky and Komodakis 2011 GT- VAE Mitton
et al. 2021], Set2Graph Vignac and Frossard [2021]. SPECTRE Martinkus et al. [022b].
Graph NWP
al. 2019 GDSS Jo al. 2022b). GruM et al. 2024 DiGress Vignac
ct al. 2023 DisCo Xu et al. 2024b) and DeFoG Qin et al. 2025
Data. We tested the performance of different generative models on the QM9 Wu et al. 2018
MOSES/oolykooskee et al. [2020), and GUACAMOL, ( et al. (2019) datasets. QM9 a small dataset: we used 100K molecules for training, 20K for validation, and 13K for testing. Both MOSES

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and GUACAMOL contain millions of molecules, with the number of heavy atoms inside a single molecule bounded by 26 and 88, respectively. We used 85% of the molecules for training, 5% for validation, and 10% for testing.
Setup. For each molecule dataset, we choose different (relatively small) number of motifs to be converted into supernodes. The selected motif structures can be found in the Appendix [A] We set the hyperparameters of DMol in both setting to k 2. 0.2, A = 5, X2 = λ3 = 1. The number of training epochs is set to 500. The experiments were conducted using NVIDIA H100 GPUs. The QM9 runs were executed on single GPU, while the MOSES and GUACAMOL experiments were processed in parallel using 4 GPUs.
Evaluation Metries. The metrics used in the experiments include validity, uniqueness, and novelty. Validity (V) measures the proportion of generated molecules that pass basic valency checks (e.g., correspond to valid SMILES files). Uniqueness (U) quantifies the proportion of gener- ated molecules that are distinct. Novelty (N) represents the percentage of generated molecules that do not appear in the training dataset. Furthermore, V.U. stands for the product of validity and uniqueness scores, while V.U.N. stands for the product of validity, uniqueness, and novelty. Since MOSES and GUACAMOL are benchmarking datasets, they intro- duce their own set of metrics for re- porting results. In addition to V. U, and N, MOSES incorporates a filter score, the Frechet ChemNet Distance (FCD) Preuer et al. (018). SNN, and scaffold similarity scores, while GUA- CAMOL uses FCD and KI. diver- gence. The definition of these met- rics are in Appendix [E] The metrics mentioned above still have certain lim- itations. Since these metrics assess the overall quality of the generation pro- cess, but do not provide insights into the performance of individual molecules. To address this limitation, we also use the ChEMBL Likeness Score (CLscore) Bühlmann and Reymond (2020), Quantitative Estimation of Drug-likeness (QED) Bickerton et al. 2012] score, and our newly intro- duced Shingle distance (SD) (which quantifies the divergence between two molecular motif (shingle) distributions, see the Appendix D. The CLscore is calculated by considering how many substructures in a molecule also occur in the drug-like dataset ChEMBL (aaulton et al. [2016). QED evaluates ChEMBL(Gaulton drug- likeniiseess by considering eight widely recognized molecular properties, such as molecular mass and polar surface area. More details are available in Appendix F Another performance metrics used is the sample time (seconds) needed to generate batch of 64 molecules.
Table 3: Classical validity, uniqueness and novelty performance comparison on GUACAMOL, for comparable genera- tion complexity.
MoUEL
U
N1
DIGRESS
100
99.8
93.1
68.4
DisCo
86.0
100
99.8
92.8
59.7
DEFRO
86.7
100
99.5
92.5
58.1
DM (OURs)
86.7
100
100
94.2
69.3
Table 4: Biochemical performance comparison on MOSES and GUACAMOL. Here, CL=ChEMBL likeness score, SD=shingle distance, ANS=average number of diffusion steps, ST=sample time, and TT=training time. Note the significantly lower ANS, ST and TT times of DMol com- pared to DeFoG.
DAFASET
MODEL
CL1
QED1
SD1
ANS]
STISK
TT(ML
DiGienss
4,4965
0.8024
0.647
500
68. 24
136
MOSES
DeFoG
DisCo
4.0502
4,1373
0.7518
0.7475
0.585
6.693
500
50
65.36
5.81
152
115
DM (Orss)
4,5033
0.8085
0.583
38
1.51
60
DRURESS
4,3112
0,3630
0.565
300
10.32
162
GUACAMOL
DEFoG
DiaCo
3.8075
3.9093
0.4927
0.5366
0.687
0.879
500
50
68
7.62
155
132
DMOL (OURS)
4.2331
0.5786
0.678
48
1.84
88
The experimental results are presented in Tables I 2 3 and 4 On QM9, DMol achieves the best performance compared to other baseline models. In contrast, GruM, DiGress, DisCo, DeFoG have low novelty scores, indicating that most of the generated molecules may be duplicates or small perturbation of those already present in the training set. On MOSES and GUACAMOL, DMol demonstrates superior performance compared to other methods with respect to all metries.
Results.
Table 4 presents a comparison of diffusion models with respect to the CL/QED scores, average number of diffusion steps, sample time, shingle distance and training time. The results demonstrate that DMol consistently outperforms DiGress, DisCo, and DeFoG across multiple metrics. DMol produces samples with excellent chemical properties as evidenced by high CL scores and QED values. DMol also has larger SD than DiGress, indicating its ability to generate more novel molecules. Although DisCo and DeFoG achieve even higher SDs, as expected, due to lack of motif preservation, this comes at the cost of reduced chemical property preservation. Additionally, DMol exhibits excellent computational efficiency, reducing the average number of diffusion steps by an order of magnitude and significantly accelerating both sampling and training time. For a detailed ablation study, please see Appendix G.2

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Motif Distribution Analysis
To validate that motif compression does not introduce biases towards the selected compressed motifs, we conducted a comprehensive analysis comparing motif probabilities between training and generated molecular sets on the MOSES dataset. We examine the top 30 most frequent motifs, with the top 15 (IDs 0 14) structures chosen for compression into supernodes, and the remaining 15 (IDs 15 29) serving as a control group to assess potential distribution distortions.
Individual Motif Prohabilities. For each motif, compute the probability that molecule contains that specific motif. The results (detailed in Appendix Table T0 demonstrate a close alignment between training and generated set distributions. The average supernode motif (IDs 0 14) probability (IDs 0 - 14) equals 0.1189 in the training set and 0.1143 in the generated set. For nonsupernode motif's (IDs 15-29), these average probabilities equal 0.0189 and 0.0160, respectively. This indicates that our approach does not create systematic bias between compressed and uncompressed substructures.
Motif Co-occurrence Patterns. We examined joint probabilities of frequent motif pairs verify that our model maintains realistic structural relationships. For example, the co-occurrence probability of benzene (eleccecl) and pyridine (elcencel) 1) is 0.11 in the training versus 0.10 in the generated set, and similar results hold for other pairs. This demonstrates that DMol closely preserves not only individual motif frequencies but also their co-occurrence patterns.
Node and Supernode Marginal Distributions. We compared the marginal distributions of all node and supernode classes of training and generated molecules (see Appendix Table M... The distributions show strong alignment, confirming that DMol samples from the original data distribution without introducing biases. For instance, carbon atoms comprise 0.4897 of the training atom set, while for the generated set this value equals 0.4940; the most frequent supernode (elecceel, benzene) appears with probabilities 0.0736 and 0.0750, respectively.
We attribute this strong preservation of substructure statistics to two key design choices: (1) using marginal distributions from the training set as priors for both supernode and atom sampling, and (2) enforcing that (t) nodes change at each timestep, ensuring adequate creation of generating nonmotif substructures. These results demonstrate that motif compression effectively preserves molecular motif distributions without introducing systematic bias.
Conditional Generation Based on Scaffolds and Docking Results
To avoid issues associated with direct conditional generation, we instead proposed -nnffoeddd pipeline. The key ideas are to cluster the training set molecules according to either their RDKit
features, or embeddings generated by Mol2Vec or GroverJJeger et al. [2018), Rong et al. [2020] and then run DMol on the samples in sufficiently large cluster groups separately (see Appendix G.6 For relevant docking results, please see Appendix[]
Conclusion
We present DMol, highly computationally efficient motif motiff-presrvvng graph diffusion for small molecule drug generation. DMol couples node and edge noise through combination of deterministic and stochastic mechanisms, adapts the loss function and includes motif compression. On biochemi- cally relevant performance metrics, such as the ChEMBL likeness and QED scores, it outperforms all other reported methods. Limitations and Future Work: Please refer to Appendix K and Appendix L
Acknowledgments
We gratefully acknowledge the financial support and computational resources provided by Abbvie and the NSF grant CCF 24-02815, which were essential for conducting this research. We also extend our sincere thanks to Matthew Krafezyk from the National Center for Supercomputing Applications (NCSA) at the University of Illinois Urbana-Champall technical simulation issues encountered during our experiments. Furthermore, we thank Jiwoong Jung and Ziheng Qi for useful discussion.
for his valuable assistance in resolving
10

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A
Motif Compression
NH
Figure 3: The 3 selected motifs for QM9,
NH
Figure 4: The 15 selected motifs for MOSES.
NH
Figure 5: The 15 selected motifs for GUACAMOL.
Since each dataset has different size of graph and complexity, choose a different number of motifs, to converted to supernodes. We select motifs for QM9 15 motifs for MOSES 4 and 13 motifs for GUACAMOL
B DiGress vs DMol
We now further compare DiGress and DMol. First, we observe that DiGress also constructs state transition matrices based marginal distributions, with the key difference with respect DMol being that its transition matrices vary over time as
0 = a f m.y Q = a f11,m..
where a' and 3 are dime-dependent scaling parameters, while Q'x and Q' are the state transition matrices used to transition from g- to g However, since the forward noise adding process is Markovian, DiGress directly utilizes O'x and Q' to achieve the transition from go to g², where
Ox Q ... Q'x and O .
Since (1m)² 1m, we have
a = -'' 8'1 1'1.m, m. C - 1,,m
where कं - a and 3' -.a. Additionally, DiGress also adopts cosine schedule for "!: बे cos (0.5x(tt c) + c)), where > 0 is constant.

---

In DMol, for graph dataset with nodes, the number of nodes undergoing state changes at time step t equals (t) = (1 a) n. At the same time step in Digress, the expected number of nodes undergoing state changes equals Σ, upp (1 - a1 pt ). where ye represents the marginal probability of node class i. Therefore, at the same time step, the number of nodes modified by DMol is (1
times that expected from Digress (the derivations can be found in Appendix D.2). Moreover, among the nodes whose states are changed by DMol, the proportion of nodes with class is p. whereas in DiGress, this proportion equals p ). The ratio of these fractions equals 1 p.
For edges, a similar analysis can be applied, leading to the conclusion that at the last time step T, the ratio of the number of edges modified by noise model and that modified by Digress equals (Lp)()-))) where r is the hyperparameter mentioned in Section 3.2. At each time step, this ratio
needs be adjusted by a correction factor that accounts for the fact that our method only selects edges from the complete graph induced by the chosen nodes. This correction factor is the ratio of the number of edges in the induced subgraph and the total number of edges. Additionally, among the modified edges, the proportion of edges with class index differs from the same in DiGress by constant factor of 1 - pf. where p equals the marginal probability of edge class j. As result, DMol can be made to replicate the average "behavior" of DiGress by simply adjusting specific hyperparameters, whereas the opposite claim is not true. Comparisons of the noise evolution distribution, efficiency and expressivity of DMol and Digress are available in Appendix D
The following modifications can be applied DMol to obtain Digress:
1. Fix the maximum number of sampling steps to a constant length instead of varying based on the number of nodes.
Scale the number of nodes to be changed each step by a factor of P1 (1-11) p?))
3. Set the noising processes for edges and nodes to be independent, i.e., select edges from the entire graph instead of restricting to the subgraph formed by the selected nodes.
4. Scale the number of edges to be changed at each step by a factor of E1,((1=)))
5. When generating random scores for nodes and edges to determine the objects to be selected, multiply the random scores for nodes by a correction weight of 1 - p and multiply the random scores for edges by 1 1-p
These modifications are straightforward implement in practice and can be achieved by simply adjusting model hyperparameters. However, Digress cannot be easily converted into DMol.
C Likelihood Computation
DMol can be visualized through the model depicted in Figure 6 informed by the dataset distribution and where nodes are perturbed first, while edges are conditionally perturbed based on the node selec- tion. This framework extends traditional hierarchical variational inference structures by incorporating a crucial dependency relationship: the selection of edges to perturb conditioned on the selection of nodes at each timestep.
n
GT
G¹
G
K,
Figure 6: The conditional noise-addition model of DMol, where K2 (t) captures the random node selection while K. (t) captures the random edge selection process; the dashed arrow indicating the dependency of edge selection on node selection.
For a molecular graph G, the model likelihood must account for this conditional relationship. according to:
25

---

log po( log p(in) p(n) / p(G In)pold ( ....,,.""". .")
(2)
nEN
=logp(nc) + log / p(c" (G) II pe pC (p' (G) ,
Note the key difference in the conditional probability po (G- (2),), K. (()),, which explicitly accounts for the selection matrices (t) and (1) that designate which nodes and edges receive perturbations at time t.
Following established variational inference principles, we derive tractable evidence lower bound (ELBO) adapted for this conditional structure:
log pe(G) 2 (ogg(n)) + DKLla(G" ())lax(na) x qe (nG))
(4)
Prior
T
C
(5)
Diffusion component
+ E (log pe (
(6)
Reconstraction component
Where the diffusion component L, now explicitly models the conditional relationship between node and edge noise:
(G) = Ey(ona DEL |, (1)pi-1))lppppppll1 Ka 1), (t- -))]]
(
This can further decomposed to highlight the conditional structure:
C. = Σ
Egaajae [DL [9(2)
(8)
itK=(t-1)
+
Σ
Egtetylew) [DKL lel, ee)l (G" Kt 1))]]
(9)
(tt)E
Note the critical difference the edge term, where the predicted probability po po(c I Kg
1))
explicitly conditioned on the node selection (6 This formulation captures the key aspect of DMol's noise process: edges selected for perturbation only from within the subgraph induced by the selected nodes.
The conditional dependency structure provides two key advantages for DMol: 1. It preserves local structural coherence (e.g. motifs) by ensuring that edge modifications are correlated with node changes. 2. It significantly reduces the computational complexity by restricting edge perturbations to smaller subgraphs.
In practice, each ELBO component remains computationally tractable: we calculate log p(ng) from empirical node count distributions across the molecular corpus. The prior term involves categor- ical KI. divergence calculations, while the diffusion component now incorporates the conditional selection of edges based on the node selection. The reconstruction component derives from generated probabilities of the original structure given its slightly perturbed version G'.
26

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D Theoretical Analysis of DMol
D.1 Noise Distribution Evolution
In DiGress, the node type distribution remains fixed at all forward time steps because the initial node type distribution is the marginal distribution mx. This is easy to because po mx. so P1 = @oPo + 3 mx = mx, and by induction, Pt = mxx for all t, where ps is the node type distribution step In DMol, the node type distribution changes over time. For node with type selected for perturbation, the probability of transitioning to type + P(j This leads to a different recurrence relation for the probability distribution over time.
px()) (-)x(()
D.2 Efficiency
To compare the efficiency of DMol and DiGress, we compare the time step increment required to change exactly one nodemminimal noise) under both settings. In DMol, the number of nodes changing at time is (t) = (1 - a²) n. To find parat such that (t) = 1, we need
(tt + 6t) (t) =1
The number of nodes that change time + 8t equals
(t 6t) (1 att + 6t)) n
For DMol, we use = kn and the cosine schedule for a,
0.5 0..5 (1/T +T
((t) = cos
1+s
Taking the derivative with respect to / and using a small angle approximation for small 5t, we obtain:
da(l) sin s(t(T+s)
((t/T++)
sin (m(+TT+e))
cos
1-s
14.8
1+8
dt
T(1 +
2T(++
For small őt, we can approximate:
do((t)
a(l + 81) a(l)
dt
Therefore,
da(() dt
((t + 8t) N(t) (a(t) alt + ōt))
6t
Setting this equal to and solving for òt, we can have the value of öt Da
1
2T(1 + 8) mnsin [m(t/T++s] 1+s
do(t)
de
For DiGress, we use a similar approach to find öt ót changed by DiGress time equals:
First, the expected number of nodes
Niigress (t) = n. a(t))
Σ pt . Σe
pf)
Following a similar derivation as for DMol, we arrive at
21 +
8t Digress da(t)
de
Σ.P. (1 -
snsin
(m((/T++ 1+s
Σ.pt pT)
Then, the ratio of the minimum time steps of interest equals Digress pl
Σ.rf (1-
Because Σ.P. (1 Dt. < for any probability distribution, we have that
8t Sigress
1
pt
This proves that öl Digress > p or, words, that DMol requires a smaller time step increment to change exactly one node. As result, the number of steps required by DMol to change a node will be smaller than that of required by Digress, which implies that DMol performs graph perturbations more efficiently.
27

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D.3 Expressivity
Both DiGress and DMol start with the same underlying node and edge type distributions, but differ in how efficiently they explore the space of possible graphs. DMol tends to produce more diverse graphs within its reachable set because it makes larger localized changes. The Hamming distance between the initial graph Go and the graph at time t, Gr, approximately dи (Go. Ge) 3 n. This Hamming distance captures the difference between the number of node/edge types in the two graphs. For DiGress, the expected Hamming distance EE|d (Gu, Ge)l ~ 3 Eiecx (i) -1-Px (i).. Since Execx P.x (i) (1 px())) 1, we conclude that DMol explores the graph space more efficiently per diffusion step.
E
The Metrics used in MOSES and GUACAMOL
The metrics used in MOSES and GUACAMOL dataset are defined as follows: The filter score measures the proportion of molecules that pass the same filters used to construct the test set. The FCD score measures the similarity between molecules in the training and test sets using embeddings learned by a neural network. SNN quantifies the similarity of a molecule to its nearest neighbor based on the Tanimoto distance. Scaffold similarity compares the frequency distributions of so-called Bemis-Murcko scaffolds. KL divergence measures the differences in the distributions various physicochemical descriptors.
F ChEMB Likeness, QED Scores, and Shingle Distance
The ChEMBL-Likeness Score (CLscore) Bühlmann and Reymond (0200 is a metric used assess the drug- drug-likeness of molecule. It is determined by analyzing how many of the molecule's substructures are present in the ChEMBL datase(Gaulton et al.) [2016). a database of bioactive molecules with drug-like like properties. Essentially, the CLscore quantifies the likelihood of a molecule being drug-like based on its substructure similarity to known compounds in ChEMBL.
Quantitative Estimation of Drug-likeness (QED) Bickerton et al. 2012 is a computational metric used in drug discovery assess how "drug-like" compound is. It combines eight physicochemical properties, including molecular mass (M,,), octanol-water partition coefficient (ALOGP), number of hydrogen bond donors (HBDs), number of hydrogen bond acceptors (HBAs), molecular polar surface area (PSA), number of rotatable bonds (ROTBs), number of aromatic rings (AROMs), and number of structural alerts (ALERTS), into a single score ranging from 0 (least drug-like) to (most drug-like) The score is measured through defined desirability functions of the eight properties. QED provides a balanced evaluation by weighting these properties based on their distributions in known drugs.
The shingles distance (SD) is a metric designed to quantify the structural divergence between generated molecules and reference dataset. It is computed by comparing the frequencies of important, predetined molccular drug substructures (motifss, known in RdKit as shingles, according to the ChEMBL dataset Gaulton al. [2016). The same of shingles is used in the computation of the ChEMBL-Likenes: Score (CLscore) Bühlmann and Reymond (2020). SD measures the cosine distance between the vectors of shingle occurrence counts for two molecular distributions.
By emphasizing differences in the presence of key drug-red relgrreleaant motifs, SD offers more nuanced assessment of both validity and novelty.
G Supplementary Experiments
G.1 General Graph Generation
We conducted experiments using the benchmark proposed by Martinkus et al. [2022a]. which comprises two datasets: SBM and Planar. Each dataset consists of 200 graphs. DMol's 's ability to accurately model various properties of these graphs was assessed using metrics such as the degree distribution (Deg.), clustering coefficient distribution (Clus.), and orbit count distribution (Orb., the number of occurrences of substructures with nodes), measured by the relative squared Maximum Mean Discrepancy (MMD).
28

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Table MMD Performance Comparison on SBM and Planar Graphs.
DATASET
MODEL
DEG J
CLUS
ORB 4
VALID
GRAPHRNN
6.7
1.6
3.2
5.2
GRAN
14.3
1.7
2.0
25.5
SBM
SPECTRE
1.9
1.7
1.5
100.0
DIGRESS
1.7
1.6
1.7
66.7
(MOL()URS)
1.6
1.5
1.5
66.1
GRAPHRNN
24.6
8.8
2534.0
0.0
GRAN
3.6
1.2
1.8
98.2
PLANAR
SPECTRE
2.6
2.5
2.5
100.0
DIGRESS
1.5
1.6
84.5
(MUL()URS)
1.8
1.6
1.4
83.9
Table 6: Ablation Study
METHOD
V
U1
N
ONLY CE LOSS BOTH CE MSE Loss
79.6
100
100
85.4
100
100
SELECTING EDGES FROM THE WHOLE GRAPH
55.6
100
100
SELECTING EDGES FROM THE INDUCED SUBGRAPH
85.4
100
MMD measures the discrepancy between two sets of distributions. The relative squared MMD is defined as follows:
MMD2 2 (geen ([est)
MMD 2 (Graain (test)
where Syens Strain and Grest are the sets of generated graphs, training graphs, and test graphs.
In our experiments, split the datasets into training, validation, and test sets with proportions of 64%, 16%, and 20%, respectively. The chosen hyperparameters were k = 2 and r = 0.01. The experimental results are presented in Table[3] These results demonstrate that DMol performs comparably to DiGress in generating general graphs.
G.2 Ablation Study
The ablation study highlights the performance improvements of DMol compared to DiGress due to the use of a new loss function that penalizes discrepancies in the counts of nodes and edges of training molecular graphs and sampled graphs; and performance improvements due to the of special subgraph selection procedures. The results are summarized in Table 6 The model we used DMol, and the dataset is GUACAMOL. In Table 6 "ONLY CE LOSS" stands for using only the cross-entropy loss (the loss used by DiGress), while "BOTH CE & MSE LOSS" refers to using both cross-entropy loss and mean squared error loss (the loss used by DMol). "SELECTING EDGES FROM THE WHOLE GRAPH" means selecting edges from the entire graph, whereas "SELECTING EDGES FROM THE INDUCED SUBGRAPH" means selecting edges only from the induced subgraph formed by the selected nodes. The results show that incorporating the MSE loss can improve the validity of the generated molecules. This because the MSE loss penalizes discrepancies between the number of altered nodes and edges and the noise added during the forward process, resulting more accurate predictions by the denoising model. Additionally, selecting edges from the induced subgraph significantly enhances the validity of the generated molecular graphs. This is achieved by making the sets of altered nodes and edges codependent. However, if edges are selected from the whole graph, the reduced number of diffusion steps significantly increases the learning difficulty for the denoising model, leading to lower validity in the generated molecular graphs.
29

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(N10
10
Figure 7: Choosing larger value of results less step-like (flatter) (t) function. As shown in the two figures, using k = 10 leads to more skewed distribution of 8t toward N (t). This means that during sampling, the model is more likely select during the early diffusion stages, which makes harder for the neural network model to learn how to denoise.
Table 7: DMol with different values of ls MOSES dataset
k
VALIDITY f UNIQUENESS 1
NOVELTY f
SAMPLE TIME L
1 85.1
100
100
2.53
2 85.6
100
100
5.28
3
84.9
100
100
7.81
4
84.7
100
100
10.37
5
84.4
100
100
12.69
10
83.4
100
100
24.76
G.3 Selection of the parameter k
The hyperparameter determines the total number of diffusion steps since we set = kni, where is the number of nodes. In practice, l is typically set to 1 or 2 reduce the number of diffusion steps and enhance performance (as explained in Figure 7 The latter observation can be intuitively explained follows: the number of nodes modified at each time step increases over time, so that by the final time step, all nodes are changed. As k increases, the x-axis of the diffusion process is effectively stretched by a factor of while the y-axis remains unchanged (Figure 7 Consequently, the number of diffusion steps required to modify a single node increases. This leads to higher frequency of generated samples during the early stages of diffusion, when fewer nodes are altered; furthermore, fewer samples are generated during the "middle stages." However, these additional early-stage samples introduce redundancy during the process of training the denoising network, ultimately degrading performance. Furthermore, these samples are generated at the cost of samples with larger number of node changes, further degrading the performance.
In addition, to empirically evaluate the impact of k on the performance of the diffusion model, we conducted experiments for different values of As shown in Table 7 the sample time approximately proportional to k. Moreover, setting to = 2 yields the highest validity.
G.4 Complete Experimental Results for the QM9 Dataset
For complete performance comparison the QM9 dataset, please refer to Table
G.S Forward Process Schedule
To examine impact the cosine schedule on different molecular scaffolds and topologies, we conducted an ablation study comparing different values of the hyperparameter e in the cosine schedule as well as comparing cosine versus linear schedules on the MOSES dataset. The results are presented in Table
30

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Table 8: Performance comparison on the QM9 dataset.
MODEL
VT
V.U.f
V.U.N.1
GRAPHVAE
56.1
42.8
26.5
GA-VAE
74.8
16.5
15.6
SET2GRAPHVAE
60.0
56.8
SPECTRE
87.5
31.4
29.0
GRAPHNVP
83.5
18.4
GDSS
96.0
94.6
GRUM
99.4
85.1
22.2
DIGRESS
97.8
95.2
31.8
DISCO
98.2
95.6
57.5
DEFoG
97.9
95.9
62.8
DMOL(OURS)
98.3
96.0
75.1
Table 9: Ablation study for forward process schedules on the MOSES dataset. The cosine schedule with 0.008 achieves optimal performance across all metries.
Method
Validity
Uniquenessf
Noveltyt
Filterst
FCD
SNNf
Scaft
ChEMBL.1
QED1
cos (c 0.004)
86.9
100
100
97.4
1.13
0.57
14,2
4,5022
0.8022
cos (c = 0.006)
87.5
100
100
97.7
1.12
0.57
14.6
4.5020
0.8037
cos (c 0.008)
87.8
100
100
97.8
1.12
0.58
14.8
4.5033
0.8055
cos (e 0.01)
87.3
100
100
97.0
1.19
0.55
0,8042
Linear
85.2
99.8
96.1
0.56
4.5011
0.8016
The experimental results demonstrate that varying the value of c has relatively small impact on crucial performance metrics, with an optimal performance achieved at = 0.008, which is the setting used in our experiments. This finding aligns with the findings by DiGress, which also adopts 0.008 as the default value.
When comparing cosine and linear schedules, the linear schedule. This advantage stems from the fact that the cosine schedule adds less noise per timestep during both the initial and final phases of the diffusion process, compared to interme- diate stages. This distribution better aligns with the learning characteristics diffusion models. Specifically, maintaining relatively small noise levels during the early and final diffusion phases facilitates the model's learning of fine-grained data features and ensures "smoother" convergence. In contrast, the uniform noise distribution in linear schedules may prove either overly aggressive or overly conservative at certain stages, leading to compromised training efficiency and performance.
that the cosine schedule clearly outperforms
G.6 Scaffold-Constrainee and Conditional Generation
To avoid issues associated with direct conditional generation, instead proposed scaffold-informed pipeline. The key ideas are to cluster the training set molecules according to either their RDKit features, embeddings generated by Mol2Vec Grover Jaeger al. (2088) Rong al.) [2020). and then run DMol on the samples in sufficiently large cluster groups separately. More details are provided below.
RDKit-Based Scaffold Classification: Molecules are categorized into four scaffold classes following RDKit recommendations: aromatic monocyclic, aromatic monocyclic heterocycle, fused bicyclic, and aromatic heterocyclic.
GNN, Transformer and Other Embedding Classification Methods: One can use Mol2 or Grover perform molecular graph embeddings, and follow up with Kmeans++ clustering.
The detailed analysis of the pros and cons of this approach compared to classical conditional generation will be discussed in more detail in companion paper.
G.7 Motif Probability Distribution Analysis
Table TO presents the complete per-motif probabilities for the top 30 most frequent motifs in the MOSES dataset. Table T shows the marginal distributions of node and supernode categories.
31

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Table 10: Comparison of motif probabilities between training and generated sets on the MOSES
dataset. IDs 14 correspond to supernodes, while IDs 15 29 are nonsupernode motifs.
Motif (SMILES)
Prob on Training Set Prob on Generated Set
Supernode Motifs (IDs 0 14)
clececel
0.7753
0.7861
cleencel
0,1829
0.1630
2 clenncl
0.1019
0.0912
CICCNCCI
0.0852
0.0783
CICCNC1
0.0798
0.0662
clescel
0.0763
0.0740
cleesnl
0.0715
0.0686
CICOCCNI
0.0625
0.0651
CICNCCNI
0.0620
0.0681
clecoel
0.0573
0.0429
clenenel
0.0559
0.0512
11
clenenl
0.0464
0.0416
12
clnconl
0.0459
0.0458
clnennl
0.0453
0.0415
14
CICCCCCI
0.0351
0.0301
Nonsupernode Motifs (IDs 15-29)
15
CICC1
0.0298
0.0242
16
clenocl
0.0289
0.0238
CICCCCI
0.0246
0.0281
CICCOC1
0.0231
0.0241
19
cleenel
0,0214
0.0191
20
cleCNCCI
0.0202
0.0169
cleennel
0.0191
0.0123
22
clnnnal
0.0177
0.0161
23
clnnesl
0.0164
0.0143
clencen
0.0154
0.0127
25
clennnl
0.0151
0.0104
26
clcocnl
0.0146
0.0088
clnncol
0.0135
0.0090
28
elcCCCC1
0.0123
0.0098
29
clcNCC1
0.0114
0.0102
Average Probabilities
Top 15 (Supernodes) mean Last 15 (Nonsupernodes) mean
0.1189
0.1143
0.0189
0.0160
H Examples of Generated Molecules.
Figure illustrates the sampling process.
Figure shows examples of generated molecules.
T
37/4
L=
L=7/4
0
Figure 8: Sampling process.
32

---

Table 11: Marginal distributions of node and supernode categories in the MOSES dataset. Supernodes are treated as new node types alongside individual atoms.
Node Supernode Type Training Set
Generated Set
Individual Atoms
C
0,4897
0.4940
0,1511
0.1506
S
0,0170
0.0167
0,1654
0.1644
F
0.0246
0.0243
Cl
0.0093
0.0091
Br
0.0025
0.0026
Supernode Motifs
cleceeel
0.0736
0.0750
eleencel
0,0132
0.0130
clennel
0.0081
0.0077
C1CCNCC1
0.0068
0.0070
C1CCNC1
0.0049
0.0045
clesee1
0.0055
0.0052
cleesnl
0,0056
0.0050
CICOCCNI
0.0035
0.0036
CICNCCNI
0,0036
0.0031
clecoc1
0.0041
0.0039
elenenel
0.0044
0.0040
elenenl
0.0010
0.0008
clncon1
0.0027
0.0025
cinennl
0.0024
0.0020
C1CCCC11
0.0010
0.0008
NH2
Ho
OH
Figure 9: Molecular graphs sampled from DMol trained the QM9 dataset.
olior
cupord
Quo
gro
Figure Molecular graphs sampled from DMol trained on the MOSES dataset.
33

---

Figure 11: Molecular graphs sampled from DMol trained on the GUACAMOL. dataset.
I Sample Docking Results
We used AutoDock Vina version 1.2.0 Goodsell and Olson 1990) to visualize the docking of two of the lop-scoring generated molecules (with respect to the ChEMBL likeness) that also exhibit strong biding affinities the IPXH and 4MQS proteins when compared to reference drug ligants (the reason behind the selection of these proteins is confidential information). The 4MQS complex represents the active human M2 muscarinic acetylcholine receptor bound to the antagonist iperoxo, while IPXH represents the protein tyrosine phosphatase IB with bidentate inhibitor compound 2. The free energy results are shown in Figures 12 and and they indicate that the generated molecules can significantly outperform or underperiorm existing reference drugs.
PLR
(a) Reference: Free energy kcal/mol
(b) Generated drug 1: Free energy
(c) Generated drug 2: Free energy kcal/mol
6.4 kcal/mol
Figure 12: The reference molecule (a) and two of our generated molecules (b) and (c) docked at the protein receptor 1PXH, and the corresponding free energies. The generated molecules significantly underperform with respect to the reference due to their smaller sizes/masses. As result, molecular weight constraints have to be considered as part of the design process.
(a) Reference: kcal/mol
Free energy = 6.4 (b) Generated drug Free energy 6.6 kcal/mol
(c) Generated drug 2: Free energy
-9.1 kcal/mol
Figure 13: The reference molecule (a) and two of our generated molecules (b) and (c) docked the protein receptor 4MQS, and the corresponding free energies. The generated molecules significantly outperform with respect to the reference, and the sizes/masses of the drug molecules are comparable. This confirms the importance of focusing on the right molecular weight/size.
34

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J Broader Impact
This work presents DMol, an efficient molecular graph generation framework with a strong potential for practical applications in accelerating drug discovery, designing novel materials, and advancing computational chemistry. By substantially reducing computational requirements while maintaining or improving molecular quality metrics, our approach could democratize access to molecular design tools, enabling researchers with limited computational resources to engage in this field. The efficiency gains may also enable exploration of larger chemical spaces, potentially leading to discoveries that address pressing challenges in healthcare, energy storage, and environmental remediation. However, acknowledge that molecular generation technologies could potentially be misused for designing harmful substances if deployed without appropriate safeguards. Additionally, as with many other Al systems, there is risk of amplifying biases present in training data, which could limit the diversity of generated molecules or reproduce historical biases in pharmaceutical development. We encourage future work to address these concerns through development of responsible use protocols, integration of toxicity and safety prediction tools, and careful curation of training datasets to ensure equitable representation across chemical domains. The machine learning community should collaborate with domain experts in chemistry, biology, and ethics to ensure that advances in molecular generation technologies maximize societal benefit while minimizing potential harms.
K
Limitations
Despite the many documented advantages of DMol in terms of efficiency and molecular quality, several limitations remain. First, our approach still struggles with generating certain complex motif structures and stereochemistries, which are crucial for therapeutic applications. Second, while we achieve high validity scores, we do not explicitly enforce chemical constraints during generation, occasionally producing molecules that are formally valid but synthetically impossible to bring into existence. Third, our evaluation focuses primarily on small drug-like molecules; the performance on larger biomolecules, polymers, or metal-organic frameworks remains unexplored. Finally, the model's s adaptability to conditional generation tasks (e.g., optimizing for specific target properties) requires additional architectural modifications that may affect the established efficiency gains. Future work should address these limitations to further bridge the gap between computational generation and practical chemical synthesis. Most importantly, this and all other related works should put more effort in identifying more comprehensive evaluation metries for the generated molecules, since an overwhelming number of created samples cannot be synthesized do not dock any known protein.
L Future Work
Future work on DMol will focus on extending the model to conditional molecule generation, where specific chemical properties can be targeted through controlled diffusion processes. We also plan to address the remaining stereochemistry challenges by incorporating 3D structural information into the diffusion process. Additionally, we aim develop more sophisticated motif compression strategies that can handle a wider variety of molecular scaffolds while preserving chemical feasibility. Exploring the application of DMol to larger biomolecular structures and integrating it with downstream property prediction models presents promising research directions. Finally, we intend to investigate dynamic scheduling of diffusion steps based on molecular complexity to further optimize computational efficiency.
35


Reviews:
[{"review_id": "XVPIYVnmz3", "reviewer": "Reviewer_pDy8", "summary": "This work proposes a novel graph diffusion model (DMol) for small molecule generation task that enhances semantic controllability and structural motif preservation. Motivated by the limitations of prior discrete diffusion models such as DiGress [1], the work introduce three key innovations:\n\n1. Forward Noise Injection: At each diffusion step, DMol perturbs only a deterministic number of nodes and their associated edges within an induced subgraph, using learned transition matrices.\n\n2. Denoising and Sampling: The model incorporates auxiliary functions that encode the expected level of perturbation at each time-step, guiding the denoising network toward more accurate reconstructions. A modified and permutation-invariant loss penalizes deviations from the scheduled number of node and edge changes.\n\n3. Ring Compression: Typical carbon ring motifs are encoded as super-nodes, allowing for explicit preservation of chemically meaningful structural motifs and efficient decoding into valid molecular graphs.\n\nExperiments conduced on multiple benchmark datasets demonstrate that DMol achieves competitive or superior performance compared to multiple state-of-the-art baselines. Besides standard set of V.U.N. (Validity, Uniqueness, Novelty) metrics, this work also measured chemically informed metrics, e.g., ChEMBL-Likeness Score [2], Quantitative Estimation of Drug-likeness (QED) [3], and a newly proposed Shingle Distance (SD).\n\n[1] Vignac, Clement, et al. \"Digress: Discrete denoising diffusion for graph generation.\" arXiv preprint arXiv:2209.14734 (2022).\n\n[2] Gaulton, Anna, et al. \"The ChEMBL database in 2017.\" Nucleic acids research 45.D1 (2017): D945-D954.\n\n[3] Bickerton, G. Richard, et al. \"Quantifying the chemical beauty of drugs.\" Nature chemistry 4.2 (2012): 90-98.", "questions": "Please refer the review in weakness.", "limitations": "yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1. Methodologically novel and chemically informed diffusion framework that incorporates principled constraints on chemical motifs, enabling structure-aware and efficient molecule generation.\n2. Chemically-justified ring compression method for important motif / scaffold preservation (consistently overlooked by prior similar generation task, to the best of my knowledge). \n3. Comprehensive benchmarking with prior models. \n4. The authors made strong efforts to incorporate more chemically meaningful metrics that go beyond standard validity measures. \n\nWeaknesses:\n1. The only major concern to this paper roots from its Significance. While the method improves on small molecule (<40 atoms) generation metrics, it lacks validation on actual Computer-aided Drug Design-related downstream tasks, so the gap between generation quality and real-world utility remains unaddressed.\n2. No direct ablation study is observed for the forward process cosine schedules. It might remains unclear whether this scheduling imposes constraints across molecules with varying scaffolds or topologies.", "was_revised": true, "final_justification": "My final justification is still 5: Accept.", "rebuttal": "We thank the reviewer for constructive suggestions. The letter.number W.X in our response refers to the bullet listed as weakness, followed by its number.\n\nW1. We completely agree that the validity of generated molecules should ultimately be judged based on their utility in downstream pharmaceutical tasks. Unfortunately, the generative AI community as a whole currently does not have validation metrics that can be used to guarantee downstream performance. As a small step in this direction, we would like to draw your attention to the downstream validation results presented in our Supplement. For proprietary reasons, we cannot comment about our target selection rationale, but we indeed evaluated the AI-generated drugs as inhibitors of  two proteins, 1PXH (Protein-tyrosine phosphatase) and 4MQS (human M2 muscarinic acetylcholine receptor). We ran DMol and identified molecules with fairly large  CheMBL scores (>5) and certain desirable similarity with known drugs targeting the same proteins. We then ran docking simulations with AutoDock Vina (a simulation software that is not as accurate as an experiment, yet still one of the most sophisticated docking methods available). As shown in Figures 12 and 13 of the Supplement, our generated drugs illustrate both the strengths and limitations of AI approaches. For the 1PXH target (Figure 12), our selected generated molecules showed higher (less favorable) free energies (-6.4 and -6.0 kcal/mol) compared to the reference drug (-8.7 kcal/mol), which may be, among other reasons, attributed to the small molecular size of the drug candidate compared to the size of the reference inhibitor. We therefore focused on generated molecules that have similar sizes, motifs and compositions as the reference drugs and for the 4MQS target (Figure 13), we ended up with a molecule that  significantly outperformed the best reference, achieving a free energy of -9.1 kcal/mol compared to the reference’s -6.4 kcal/mol. As a result, we made sure that for each target protein we sub-selected AI-generated drugs with appropriate molecular mass and motif structures. We chose not to emphasize this test in the main text due to space limitations and in order to maintain the coherent structure of our exposition. However, we can easily comment on more sophisticated downstream validation approaches, including similarity metrics for the molecules based on their masses, motives, 3D shapes as well as their “distances” of embeddings (generated by graph neural network) from known references. We also had guidance from molecular dynamics simulations, but did not include the conclusions to once again maintain a coherent exposition. Bridging the gap between generation quality metrics and real-world utility is and always was our biggest priority.\n\nTable 1: The result of ablation study for the forward process schedules on MOSES dataset\n| Method | Validity | Uniqueness | Novelty | Filters | FCD | SNN | SCAF | ChEMBL | QED |\n|--------|----------|------------|---------|---------|-----|-----|------|--------|-----|\n| cos (c=0.004) | 86.9 | 100 | 100 | 97.4 | 1.13 | 0.57 | 14.2 | 4.5022 | 0.8022 |\n| cos (c=0.006) | 87.5 | 100 | 100 | 97.7 | 1.12 | 0.57 | 14.6 | 4.5020 | 0.8037 |\n| cos (c=0.008) | 87.8 | 100 | 100 | 97.8 | 1.12 | 0.58 | 14.8 | 4.5033 | 0.8055 |\n| cos (c=0.01) | 87.3 | 100 | 100 | 97.0 | 1.19 | 0.55 | 14.3 | 4.5018 | 0.8042 |\n| Linear | 85.2 | 99.8 | 99.9 | 96.1 | 1.19 | 0.56 | 14.2 | 4.5011 | 0.8016 |\n\nTable 2: The result of scaffold-constrained generation on MOSES dataset\n| Method | Validity | Uniqueness | Novelty | Filters | FCD | SNN | SCAF | ChEMBL | QED |\n|--------|----------|------------|---------|---------|-----|-----|------|--------|-----|\n| First collector | 82.5 | 100 | 100 | 93.1 | 2.05 | 0.47 | 13.9 | 4.212 | 0.7842 |\n| Second collector | 83.1 | 100 | 100 | 93.0 | 1.93 | 0.48 | 13.6 | 4.1873 | 0.7885 |\n| Third collector | 82.3 | 100 | 100 | 93.2 | 1.94 | 0.45 | 13.7 | 4.1796 | 0.7921 |\n| Fourth collector | 81.8 | 100 | 100 | 93.6 | 1.99 | 0.43 | 13.0 | 4.2017 | 0.7804 |\n| Fifth collector | 80.7 | 100 | 100 | 93.5 | 1.92 | 0.48 | 13.7 | 4.1121 | 0.7882 |\n| Aromatic monocyclic | 77.6 | 100 | 100 | 89.2 | 2.23 | 0.43 | 12.4 | 3.8313 | 0.7348 |\n| Aromatic monocyclic heterocycle | 76.1 | 100 | 100 | 88.7 | 2.37 | 0.43 | 12.8 | 3.754 | 0.7257 |\n| Fused bicyclic | 75.9 | 100 | 100 | 87.3 | 2.31 | 0.40 | 12.2 | 3.6008 | 0.7233 |\n|Aromatic heterocyclic | 75.8 | 100 | 100 | 87.4 | 2.19 | 0.45 | 12.7 | 3.9126 | 0.7109 |\n\nW2. To address the question regarding the cosine schedule, we reran DMol on the MOSES dataset using the cosine schedule with different values of  the parameter $c$ (mentioned in line 217 of our paper) as well as a linear schedule ($\\alpha = t/T$). The experimental results are shown in Table 1. As can be observed from the experiments, varying the value of $c$ has small impact on the performance metrics, with the optimal performance achieved for $c=0.008$ (which is the result presented in our paper). This finding aligns with Digress's selection of the hyperparameter $c$, as its code also adopts $c=0.008$ as the default setting. When comparing cosine and linear schedules, the cosine schedule is clearly superior to linear schedules. This advantage primarily stems from the fact that the cosine schedule adds less noise per timestep during both the initial and final phases of the diffusion process compared to intermediate stages. This distribution better aligns with the learning characteristics of diffusion models. Specifically, maintaining relatively small noise levels during the early diffusion phase/the last diffusion phase facilitates the model's learning of fine-grained data features, and ensure smoother convergence. In contrast, the uniform noise distribution in linear schedules may prove either overly aggressive or overly conservative at certain stages, leading to compromised training efficiency and ultimate  performance. \n\nRegarding scaffold-constrained generation, we actually did multiple experiments on this topic already. Rather than resort to complicated and time-consuming conditional generation, we opted instead to classify the molecules based on their scaffold (using four classes, aromatic monocyclic, aromatic monocyclic heterocycle, fused bicyclic, aromatic heterocyclic, following RDKit recommendations) or based on their graph embeddings (obtained by using GraphSage GNNs, followed by Kmeans++ clustering). In the latter case, we selected five clusters with the largest number of samples and best cluster separations. The results demonstrate that DMol perform significantly better on the GNN-governed clusters (validity: 80.7-83.1%, ChEMBL: 4.11-4.21) compared to scaffold based clusters (validity: 75.8-77.6%, ChEMBL: 3.60-3.91). This is a consequence of the fact that  RDKit scaffold classes exhibit significant overlaps, while the GNN-based classes are well-separated. The GNN-based results also show slight degradations in performance compared to the full datasets which may be attributed to a significant reduction in the number of training samples. Furthermore, the excellent performance on individual GNN-classes shows that the cosine schedule does not impose special constraints for different scaffolds.\n\nThank you again for the extensive feedback on our work and for raising such interesting questions."}, {"review_id": "kfRGwpDKdk", "reviewer": "Reviewer_6j7c", "summary": "This paper introduces a novel graph diffusion model for molecule generation with fast sampling speed and motif-preserving property. In detail, it modifies the sampling steps w.r.t. the number of atoms for accelerating sampling and retain the dependency between nodes and edges in the forward process. Moreover, it develops a ring compression technique to preserve chemical motifs along the diffusion process.", "questions": "On Line 245, DMol samples the nodes and edges from specific marginal distributions. Why not introduce the motivation of the choice of the prior distributions?\nSince the paper mentions the JT-VAE on line 307, why doesn’t it compare with this model in the experiments?\nOne of your strengths is fast sampling. Why not plot a figure to show your sampling speed compared to other methods?\nOne of your strengths is motif-preserving. Why not show some experimental results in the main text to show this property?\nWhy not introduce your model’s name in the abstract?", "limitations": "Yes", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\nProviding deep analysis of sampling efficiency and proving that DMol would be more efficient than Digress when changing a node. \nIt proves the fast sampling properties of the proposed model empirically.\nThe paper is clearly written. It first introduces the framework of graph diffusion models, then develops on this base, which is clear for the readers.\n\nWeaknesses\nTechnically impressive, but of limited practical value. This work brings little improvement in validity and ChEMBL-likeness and accelerates the sampling process by a large margin, compared to other methods. However, (i) the wet-lab experiment takes much more time than the in silico one, (ii) the sampling time of current methods is acceptable (e.g. DEFOG 5.81s). It means that the sampling time should not be the bottleneck in drug development.\n Poor formatting. The authors try to plug too much information into the main text, resulting in a tight page layout (the algorithms, the equations, etc). Besides, the font size in Figure 1 is too small to read.\nAmbiguous notations. On Line 230, the paper uses index j to represent both state and class, which may lead to confusion. It’s better to assign the indexes properly throughout the paper.\nOver-simplified metrics on experimental tables. It seems that there is enough space for the tables to write down some of the full names of the evaluation metrics. For instance, the V, U, N metrics are really indirect for the readers who only read the tables.", "was_revised": true, "final_justification": "I appreciate the authors' rebuttal, which has addressed most of my concerns. At this stage, I intend to maintain my original score and would be interested in seeing the feedback from the other reviewers.", "rebuttal": "We thank the reviewer for insightful comments and for taking the time to review our work.\n\n### Weakness\n> Technically impressive, but of limited practical value. This work brings little improvement in validity and ChEMBL-likeness and accelerates the sampling process by a large margin, compared to other methods. However, (i) the wet-lab experiment takes much more time than the in silico one, (ii) the sampling time of current methods is acceptable (e.g. DEFOG 5.81s). It means that the sampling time should not be the bottleneck in drug development.\n\nWe thank the reviewer for acknowledging the large improvements in validity, ChEMBL-likeness, and sampling speed of DMol. We agree that wet-lab experiments dominate the timeline during the later stages of drug development. However, sampling efficiency remains a critical factor that influences early-stage drug design pipelines, particularly in the de novo hit-to-lead phase where no prior reference compounds are available for a target protein. This is especially the case for high-throughput screening scenarios, for which the availability of a very large number of diverse and chemically valid candidates is essential.\n\nFor example, among molecules generated by all diffusion models including DiGress and DMol, those with a ChEMBL-likeness (CLscore) above 5.5 account for fewer than 0.5% of total outputs. While DMol significantly outperforms other models in this regard, each sampling run typically yields only a few dozen molecules that meet minimal thresholds for chemical likeness (note that scores closer to 8/9 are desirable, yet still not sufficient guarantees that the molecules can actually be synthesized). For example, Contract Research Organizations (CROs) such as Evotec and Nuvisan offer high-quality drug compound libraries containing over one million molecules, which are purchased and screened by pharmaceutical and biotech companies and it is easy to see that to get to such numbers purely through AI generation poses significant time constraints. This highlights the scale of high-throughput requirements and the need to generate large volumes of candidate molecules in de novo pipelines.\n\nHence, improvements in sampling speed directly enable more comprehensive chemical space exploration. Although methods like DeFoG already demonstrate efficient generation times (e.g., 5.81s/64 molecule), one has to note, as pointed above, that most of the molecules will not be chemically valid to the point that they can be synthesized. DMol offers further acceleration as well as a significantly better chance that the molecules will be valid due to their motif conservation. We believe this performance gain is meaningful in practical settings, especially when millions of candidates must be explored.\n\nIn conclusion, the general goal is not to merely generate as many molecules as possible, but to generate high-quality molecules, which is an inherently difficult task. This is not a critique of DeFoG or any other method specifically, but rather a shared limitation among current generative approaches in the field.\n\n> Poor formatting. The authors try to plug too much information into the main text, resulting in a tight page layout (the algorithms, the equations, etc). Besides, the font size in Figure 1 is too small to read. Ambiguous notations. On Line 230, the paper uses index j to represent both state and class, which may lead to confusion. It’s better to assign the indexes properly throughout the paper.\n\nWe thank the reviewer for this helpful feedback and apologize for the lack of readability. We acknowledge that we included a large amount of technical content in the main text. Given the complexity of drug discovery—requiring both rigorous mathematical formulation and deep domain knowledge—we hoped that including all core ideas would demonstrate the multifaceted ideas we tried to integrate to make the model more successful. We will nevertheless follow the advice and revise the formatting to improve clarity. Regarding line 230, the state j and class j are both referring to node type j. We apologize for this ambiguous use of the same symbol.\n\n> Over-simplified metrics on experimental tables. It seems that there is enough space for the tables to write down some of the full names of the evaluation metrics. For instance, the V, U, N metrics are really indirect for the readers who only read the tables.\n\nWe will revise all the tables to include the full names of evaluation metrics, improving readability for readers unfamiliar with our shorthand notation.\n\n### Questions\n> On Line 245, DMol samples the nodes and edges from specific marginal distributions. Why not introduce the motivation of the choice of the prior distributions?\n\nIn standard diffusion models, Gaussian (for continuous) or uniform (for discrete) priors are commonly used. However, in molecular generation tasks, such priors can be suboptimal due to the highly skewed distribution of atom and bond types in real-world molecules (e.g., carbon atoms are predominant in molecular graphs). Using a uniform prior significantly increases the learning burden on the model, as it must transform a flat distribution into a highly structured one. By contrast, using the empirical marginal distributions as priors gives the model a better inductive starting point, improving convergence and sample quality. This approach is supported by the recommendation in the landmark paper DiGress (see Section 4 and Table 6), where marginal priors resulted in better performance than uniform ones. Note that our priors are marginals pertaining to both atoms and the selected motifs.\n\n> Since the paper mentions the JT-VAE on line 307, why doesn’t it compare with this model in the experiments?\n\nJT-VAE is a landmark model that has been widely used for benchmarking in earlier molecular generation work. However, it is based on VAEs, which operates in a latent space and requires highly complex decoding procedures to ensure chemical validity, such as enforcing valency constraints through hard-coded rules. DiGress also highlights these “hard-coded rules” in Table 2, which we believe makes direct comparison with diffusion-based models unfair. In contrast, our method—and the baselines we compare against—are based on more recent paradigms such as diffusion models and flow matching. To ensure a fair and up-to-date comparison, we followed the common practice in the field by benchmarking against these more recent and directly comparable diffusion approaches.\n\n> One of your strengths is fast sampling. Why not plot a figure to show your sampling speed compared to other methods?\n\nWe appreciate the reviewer’s suggestion. In our submitted paper, we indeed presented the sampling times (rather than speeds) of each method in Table 4. We believe that this table provides a direct and quantitative comparison and clearly demonstrates the efficiency of DMol.\n\n> One of your strengths is motif-preserving. Why not show some experimental results in the main text to show this property?\n\nWe thank the reviewer for highlighting DMol’s motif-preserving capabilities. Since molecular motifs (also known as shingles according to RDKit) play a crucial role in determining the chemical properties of a molecule, metrics such as ChEMBL-likeness and QED are specifically designed to reflect how well a molecule captures and preserves important biomedical shingles/motif substructures. For a more detailed explanation of the ChEMBL-likeness and QED metric, please refer to Appendix F of our paper.\n\n> Why not introduce your model’s name in the abstract?\n\nWe thank the reviewer’s suggestion - we will add the model’s name in the abstract for better recognition of the method.\n\nThank you again for the extensive feedback on our work and for raising such interesting questions."}, {"review_id": "9CdoCM1ea4", "reviewer": "Reviewer_iPx9", "summary": "The manuscripts presents two modifications of the DiGress method, a graph generative model:\n1.) the replacement of common substructures, mainly carbon rings, by supernodes\n2.) instead of swapping each atom in a graph every diffusion step by some probability, the method presented her suggest to first sample a subgraph in each diffusion step and then only swap atoms and edges in this subgraph\n\nBoth modifications aim to speed up the graph diffusion and improve its validity rate.", "questions": "- Can you comment on the possibility and probability that some edges are never changed in the forward process in case they are never part of the a induced subgraph?\n- It would improve the manuscript if also a conditional generation benchmark would be included, although that is probably a lot of work", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Weakness:\n- the replacement of motifs by supernodes is an old trick\n- these supernodes have the side effect that graph diffusion might produce with different sizes\n- the choice of motifs might have a huge impact on the generation process, a few examples:\n  - if a certain ring structure is part of a motif (say, a chlorinated carbon ring) and a certain ring structure is not (say bromated ring), it is not possible during diffusion to swap from one ring to the other. This might also mess up the probabilities with which certain substructures are generated\n- using different set of motifs for different datasets/benchmarks is also not very convincing\n- there are many aspects in DMol that are not entirely clear to me. For example, when only egdges of induced subgraphs are changed, there might be some or a lot of edges that are never changed during the whole diffusion process (i.e. when two nodes never occur in the same sampled subset).\n- the benchmarks only cover unguided diffusion and evaluate the novelty, uniqueness and molecule-likeness of the outputs. It is not shown that DMol also outperforms DiGress on property optimization benchmarks. However, also the original DiGress paper did only perform a single benchmark on conditional generation\n\nStrength:\n- One issue with DiGress is that atom class marginal distributions are very skewed with most atoms are carbon and, thus, most transitions are carbon to carbon and have no effect. Therefore, DiGress is doing very few modifications per diffusion step and needs a lot of steps to add enough noise to a graph. The method here claims to need less steps due to their different sampling strategy. As I understand it, the nodes in the sampled subgraph all have to change their state, so there are no transitions from, say, Carbon to Carbon\n- the number of steps in DMol is two times the number of nodes in the graph which is usually a magnitude less steps than DiGress\n- sample time seems to be two magnitudes faster than DiGress\n- compressing common motifs to supernodes might preserve these motifs better in the generated output. However, it feels like a poor workaround, as a good diffusion model should be able to preserve such motifs without such tricks.", "was_revised": true, "final_justification": "The authors could convince me that the supernodes do not introduce a strong bias in the generative model. I do not share other concerns about running times are not important raised by other referees - in my opinion the running time improvements might be important in some future applications we might not have in mind yet. Overall, this seems to be a valuable improvement over the Digress method.", "rebuttal": "We thank the reviewer for constructive suggestions. The letter.number W.X and Q.X in our response refers to the bullet listed as weakness and question, followed by its number.\n\nW1: We respectfully disagree with the “old trick” characterization. While concepts similar to supernodes have been employed in methods like JT-VAE, our approach addresses fundamental limitations of VAEs. JT-VAE requires minimum spanning tree construction during encoding and faces  significant challenges in decoding due to the complexity and ambiguities that need to be resolved when reattaching molecular substructures with multiple bond types to scaffolds. JT-VAE also uses several hundred supernodes, which is unacceptable for diffusion models. Furthermore, the authors of DiGress clearly stated that their attempts at supernode/motif masking failed (see their GitHub issue #15, and they remark that “It (meaning masking) would probably work better if we preserved the motif during diffusion in training,” but they did not come up with a satisfactory resolution). In contrast to JT-VAE, our method only uses 15 carbon ring motifs that can always be reintegrated into the scaffold in a chemically valid manner (our selection was based on ring frequencies and was done in consultation with chemists). Note that these motifs change for each dataset as different sets of drugs do have different motifs (e.g., high frequency substructures).\n\nW2: For DMol, there is no concern pertaining to different molecular sizes since this issue is addressed through its design which preserves the *joint* distribution of individual atoms and motifs. Our approach does not arbitrarily replace motifs with single atoms, but motifs with other motifs or  single atoms, because we explicitly model and preserve the distribution of motifs in the training samples. The diffusion process reconstructs both individual atoms and motifs according to their empirical distributions, ensuring that molecules maintain appropriate structural complexity. Additionally, our model incorporates a unique feature where the number of diffusion steps depends on the size of the molecule ($T = 2n$), and our objective function accounts for this relationship through penalty terms that ensure consistent node and edge count modifications. This design prevents problematic size shrinkage.\n\nW3: It appears that there is a misunderstanding about our approach. While our predefined motifs (such as chlorinated carbon rings) cannot be  directly transformed into nonmotif rings (such as brominated rings) during diffusion, this limitation does not impair our model's generative capabilities. Rings that are not used as motifs, of which there are many,  can still be generated through atom-by-atom and edge-by-edge modifications during standard atom diffusion, which is how ours and other diffusion models manage to generate new ring structures in the first place. For example, a brominated ring can be generated by modifying an individual chlorine atom to a bromine atom through single-node state transitions. Our hybrid approach preserves the probability distributions of both motif and nonmotif structures. The key idea is that motif compression enhances generation of frequent rings while retaining full flexibility for generating arbitrary molecular patterns through an atom-by-atom level diffusion process.\n\nW4. We respectfully disagree with the claim. By definition, a motif is a substructure that appears with significantly higher frequency than would be expected by random chance in a given dataset. Different molecular datasets naturally exhibit different motif structures/frequencies because they represent distinct chemical compounds with varying biological activities and synthetic accessibility constraints. Hence, MOSES, Guacamole etc all have different motifs (highest frequency substructures). As stated in line 302 of our paper, we systematically select the most frequently occurring carbon rings that can always form single bonds with scaffolds through available carbons. This dataset-specific selection is not only appropriate but necessary to capture the underlying chemical diversity of datasets. \n\nW5 & Q1: There may be a misunderstanding how our forward diffusion process operates. In Appendix B (lines 669-679), we provide a comprehensive comparison of the number of edges modified at each time step of DMol and DiGress. In our approach, since the selection of a *deterministic number* of nodes at each time step is random, every edge in the graph has a nonzero probability of being selected and modified during the diffusion process. The probability that any edge remains unchanged throughout the entire diffusion process is the product of these probabilities across all time steps and our number of steps ensures that each edge has a nonzero probability of being selected. It is nevertheless important to understand that no diffusion model will or has to modify every node and edge. \n\nTable 1: The result of conditional generation on MOSES dataset\n| Method | Validity | Uniqueness | Novelty | Filters | FCD | SNN | SCAF | ChEMBL | QED |\n|--------|----------|------------|---------|---------|-----|-----|------|--------|-----|\n| First collector | 82.5 | 100 | 100 | 93.1 | 2.05 | 0.47 | 13.9 | 4.212 | 0.7842 |\n| Second collector | 83.1 | 100 | 100 | 93.0 | 1.93 | 0.48 | 13.6 | 4.1873 | 0.7885 |\n| Third collector | 82.3 | 100 | 100 | 93.2 | 1.94 | 0.45 | 13.7 | 4.1796 | 0.7921 |\n| Fourth collector | 81.8 | 100 | 100 | 93.6 | 1.99 | 0.43 | 13.0 | 4.2017 | 0.7804 |\n| Fifth collector | 80.7 | 100 | 100 | 93.5 | 1.92 | 0.48 | 13.7 | 4.1121 | 0.7882 |\n| Aromatic monocyclic | 77.6 | 100 | 100 | 89.2 | 2.23 | 0.43 | 12.4 | 3.8313 | 0.7348 |\n| Aromatic monocyclic heterocycle | 76.1 | 100 | 100 | 88.7 | 2.37 | 0.43 | 12.8 | 3.754 | 0.7257 |\n| Fused bicyclic | 75.9 | 100 | 100 | 87.3 | 2.31 | 0.40 | 12.2 | 3.6008 | 0.7233 |\n|Aromatic heterocyclic | 75.8 | 100 | 100 | 87.4 | 2.19 | 0.45 | 12.7 | 3.9126 | 0.7109 |\n\nW6& Q2: Regarding conditional generation, we actually did multiple experiments on this topic already. Rather than resort to complicated and time-consuming conditional generation, we opted instead to classify the molecules based on their scaffold (using four classes, aromatic monocyclic, aromatic monocyclic heterocycle, fused bicyclic, aromatic heterocyclic, following RDKit recommendations) or based on their graph embeddings (obtained by using GraphSage GNNs, followed by Kmeans++ clustering). In the latter case, we selected five clusters with the largest number of samples and best cluster separations. The results(Table 1) demonstrate that DMol perform significantly better on the GNN-governed clusters (validity: 80.7-83.1%, ChEMBL: 4.11-4.21) compared to scaffold based clusters (validity: 75.8-77.6%, ChEMBL: 3.60-3.91). This is a consequence of the fact that  RDKit scaffold classes exhibit significant overlaps, while the GNN-based classes are well-separated. The GNN-based results also show slight degradations in performance compared to the full datasets which may be attributed to a significant reduction in the number of training samples.\n\nThank you again for the extensive feedback on our work and for raising such interesting questions."}, {"review_id": "aUhLOqRKVS", "reviewer": "Reviewer_FMwP", "summary": "This paper targets the problem of small drug molecule generation. The paper proposes DMol (Diffusion Models for Molecular Motifs), which reduce the computational cost of diffusion models by relating the diffusion time-steps with the number of nodes and edges. DMol applies node compression that allows one to convert chemically important node motifs into supernodes. Experimental results show the effectiveness of the proposed method on small drug molecule generation tasks.", "questions": "* How is DMol compared to FragFM[1]?\n\n[1]Lee et al.,  FragFM: Hierarchical Framework for Efficient Molecule Generation via Fragment-Level Discrete Flow Matching", "limitations": "yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "pros:\n* DMol introduces a hybrid noise schedule that balances deterministic perturbations with random noise. This approach improves motif preservation and ensures a flexible control of the diffusion process, resulting in better molecule quality.\n* DMol introduces a novel \"ring compression\" approach, enabling the preservation and integration of chemically significant substructures like carbon rings. This method outperforms existing methods like DiGress and DeFoG in maintaining chemical motifs, which are crucial for practical drug design.\n* Experiments show the emprical gain of DMol on small drug molecule generation tasks. DMol achieves a 3% improvement in SMILES validity over DiGress and a 4% improvement in motif-conservation scores compared to DeFoG. It also highlights the limitations of SMILES validity as a sole evaluation metric, focusing on more meaningful measures like chemical motif preservation.\n\ncons:\n* None", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive suggestion. \n\nFragFM is indeed a very recent paper that has not been published yet (to the best of our knowledge). We were unable to locate publicly available code for FragFM, which limits our ability to conduct a direct empirical comparison at this time. We will definitely add a citation to FragFM in our revised manuscript and acknowledge it, and, if time permits, try to implement their software. Also, upon reviewing the FragFM paper content, we note that while both DMol and FragFM address molecular generation, they employ fundamentally different methodologies: FragFM utilizes fragment-level discrete flow matching with a hierarchical coarse-to-fine autoencoder and stochastic fragment bag strategies, whereas DMol employs diffusion models with ring compression, codependent node-edge perturbations with deterministic number of changes per step which give rise to valuable loss penalties, and specialized noise schedules for motif preservation. We observe that FragFM's use of discrete flow matching is related to DeFoG, which we already included as a baseline in our comparisons.\n\nThank you again for the extensive feedback on our work and for raising such interesting questions."}]

Author rebuttal:

Meta-review: This submission presents a new graph diffusion model, DMol, for small drug molecule generation, offering significant efficiency improvements over the DiGress method. While the reviewer pointed out some weaknesses, the overall assessment is positive. The model accelerates the diffusion process and improves the validity of generated graphs by sampling subgraphs at each step and modifying only the atoms and bonds within them, as well as by introducing an approximation that replaces ring structures with supernodes. Although there were concerns about the frequency of substructures related to the latter, the authors provided sufficient clarification in their rebuttal, which the reviewers acknowledged, leading to a meaningful discussion.

Overall, despite some remaining weaknesses, the strengths of the paper clearly outweigh the limitations. Therefore, I support accepting this paper.
Decision: Accept (poster)