Title: DMol: A Highly Efficient and Chemical Motif-Preserving Molecule Generation Platform
Venue: NeurIPS 2025
Accepted: yes
Keywords: Molecule Generation

Diffusion Model
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

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 ChEMBLlikeness, 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 1 .

1 Introduction

Graphs are fundamental data structures used to model a wide range of complex interactions, including molecules and drugs, biological pathways, and social and co-purchase networks Wu et al. [2021], Zhang et al. [2024], Yang et al. [2024], Waikhom and Patgiri [2021]. Hence, the ability to generate graphs that accurately capture domain-specific distributions is of great significance Tsai et al. [2023], You et al. [2018], Gamage et al. [2020]. While generative models for images and texts are being used with great success Dhariwal and Nichol [2021], Ho et al. [2022], Waikhom and Patgiri [2021], graph generation remains a 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 molecular compounds Jin et al. [2018], Mitton et al. [2021], Vignac and Frossard [2021], Gómez-Bombarelli et al. [2018], Kwon et al. [2020], Wu et al. [2024], You et al. [2018], Liao et al. [2019], Segler et al. [2018], Popova et al. [2019], Madhawa et al. [2019], Zang and Wang [2020], in which case the generated samples can fail to 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 et al. [2020], Fan et al. [2023], Chen et al. [2024], Trippe et al. [2023], Haefeli et al. [2023], Niu et al. [2020b], Huang et al. [2022], Liu et al. [2024c] which rely on a gradual noise injection process that perturbs the input samples and then reverses the process using a learned denoising network.

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. [2020a], Jo et al. [2022b]. 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 Liu 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 a 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. [2021]. 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 a 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 a 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. [2021], Vignac and Frossard [2021], Gómez-Bombarelli et al. [2018], Kwon et al. [2020], Wu et al. [2024], You et al. [2018], Liao et al. [2019], Segler et al. [2018], Popova et 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).

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-the-art DeFoG Qin et al. [2025] 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 of magnitude reduction in the number of steps required by diffusion models.

2. The number of nodes and edges that DMol perturbs at each step is 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 deterministic 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 deterministic/random noise schedule. This allows for further 'forced motif' preservation and a more flexible control of the noise schedule/distribution, as evidenced by a straightforward theoretical analysis.

4. One of the most important features of DMol is motif compression which allows one to convert 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 computational savings guaranteed by the first three innovations. Due to 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 a single atom or another valid motif structure (in addition, the choices of substructures used differ widely).

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., RDKit's Landrum et al. [2006] 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-art flow matching method DeFoG Qin et al. [2025], since the latter does not tend to preserve chemical motifs.

As a 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/properties 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 a 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. [2024], Shi et al. [2021], 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. [2021], Segler et al. [2018], Kwon et al. [2020], Jensen [2019], Jo et al. [2024].

Furthermore, numerous surveys on the subject are readily available Tang et al. [2024], Du et 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 et al. [2018], Simonovsky and Komodakis [2018], Mitton et al. [2021], Vignac and Frossard [2021], Gómez-Bombarelli et al. [2018], Kwon et al. [2020], Wu et al. [2024]. VAEs comprise an encoder that maps the input graph into a 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 et al. [2018], Popova et 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], Luo et al. [2021b], Lippe and Gavves [2021], Shi* et al. [2020]. 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 Kipf [2018], Łukasz Maziarka et al. [2019], Martinkus et al. [2022a]. GANs include a generator, which learns to create realistic molecular 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 a Markov chain for the diffusion process Vignac et al. [2023], Xu et al. [2022], Liu et al. [2024b], Xu et al. [2024a], Liu et al. [2024a], Hoogeboom et al. [2022], 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 Luo 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 Jo et al. [2022a], Lee et al. [2023], Huang et al. [2023], Bao et al. [2023]. Furthermore, studies have shown that different noise schedules may significantly affect molecule generation quality Shi et al. [2025], Nichol and Dhariwal [2021]. Comprehensive surveys on the subjects include Mengchun Zhang et al. [2023], 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 a Markov process, where each node and edge label evolves independently of the others. This assumption mirrors the approach in image-based diffusion models. Denoising is performed using a graph transformer network, trained by

minimizing the cross-entropy loss between the predicted probabilities for nodes and edges and the ground-truth graph. The transformer network reconstructs a denoised graph from a noisy input. To generate new graphs, a 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-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 τ -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-based Methods. Flow Matching (FM) has recently emerged as an alternative to diffusion models for generative tasks Lipman et al. [2023], Dao et al. [2023]. 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 a 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-performs on key biochemical motif molecular property metrics such as QED.

3 DMol Diffusion

3.1 Review of Standard Graph Diffusion 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 an undirected graph G 0 , encoded as z 0 , resulting in a sequence of increasingly noisy data points ( z 1 , . . . , z T ) . The noise addition process is Markovian, which translates to q ( z 1 , . . . , z T | z 0 ) = ∏ T t =1 q ( z t | z t -1 ) .

More precisely, the encoding of a 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 and is represented as a one-hot vector in R a . The same is true for edges, which are assumed to belong to one of b classes (with one class indicating the absence of the edge). We follow the notation from Vignac et al. [2023], where x i is used to denote the one-hot encoding of the class of node i . The one-hot encodings are arranged in a node matrix X of dimensions n × a . Again, in a similar manner, edges are encoded into a tensor E of dimension n × n × 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 l stands for the class (state) of node l , while e ls stands for the class (state) of edge ls .

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 1 X , . . . , Q T X ) , where each entry [ Q t X ] ij defines the probability of an arbitrary node transitioning from state i to state j at time step t . A similar model is used for edges, where the transition matrices ( Q 1 E , . . . , Q T E ) describe the probabilities [ Q t E ] ij of an arbitrary edge transitioning from state i to state j . Formally, [ Q t X ] ij = q ( x t = j | x t -1 = i ) and [ Q t E ] ij = q ( e t = j | e t -1 = i ) , where x and e correspond to a generic node and edge, while the superscript indicates the time stamp.

Consequently, the transition probability at time t is given by q ( z t | z t -1 ) = z t -1 Q t , where z stands for either x or e and the subscript of Q is set accordingly. Adding noise to a graph G t -1 at

time t -1 results in a new graph G t = ( X t , E t ) , and involves sampling each node and edge class from a categorical distribution succinctly described as q ( G t | G t -1 ) = ( X t -1 Q t X , E t -1 Q t E ) and q ( G t | G 0 ) = ( X 0 Q t X , E 0 Q t E ) , where Q t X = Q 1 X . . . Q t X and Q t E = Q 1 E . . . Q t E . The state transition matrices are invertible, which ensures the reversibility of the noise model: the graphs at time t and time 0 can be transformed into each other using the state transition matrix. Reversibility guarantees consistency between the forward and backward processes and simplifies backward denoising.

A common issue with most known diffusion models is the large number of steps needed for convergence Hang et al. [2023], Wang et al. [2024]. For large training sets the computational complexity may be overwhelming (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 t , we randomly select N ( t ) nodes and M ( t ) edges from G 0 contained in the complete subgraph induced by the selected nodes (note that the absence of an edge is represented by a special label so that we are effectively dealing with a complete graph), and change their classes according to fixed state transition matrices Q X and Q E , respectively, to obtain the graph G t . Here, N ( t ) and M ( t ) are deterministic functions of the time step index t . 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.

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(a) At each time step t , we randomly select a predetermined deterministic number of nodes and add noise only to the class labels of these nodes and a fixed number of edges in the subgraph induced by the nodes. Both selection numbers are controlled by the time index t . The top row depicts how the number of selected nodes N ( t ) and edges M ( t ) of the selected subgraph evolves with t (we set n = 6 , k = 2 , r = 0 . 2 ).

(b) DMol diffusion relies on randomly selecting N ( t ) nodes and M ( t ) edges from the subgraph induced by the N ( t ) nodes at each time step t , and diffusing their labels to generate a noisy graph G t . The denoising network ϕ θ learns to predict the denoised graph from G t . During inference, the denoised graph is combined with q ( ˆ G t -1 | ˆ G t , ˆ G 0 ) to predict ˆ G t -1 .

For N ( t ) and M ( t ) , we adopt cosine schedules, N ( t ) = ⌊ (1 -α ) n ⌋ and M ( t ) = ⌊ (1 -α ) N ( t ) ( N ( t ) -1) 0 . 5 r ⌋ , where α = cos 2 (0 . 5 π ( t/T + c ) / (1 + c )) , and n stands for the the number of nodes in the graph, r is a hyperparameter and c is a small positive constant. The total number of diffusion steps is fixed at T = k n , where k is a small hyperparameter (typically set to 1 or 2 so as to keep the number of diffusion steps as small as possible; details behind the selection of appropriate values of k 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.

Additionally, our forward process constrains the class changes of edges to the subgraph induced by the nodes that underwent a 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. [2023] have shown that making the

transition probability from class i 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 to when using uniform transitions. We follow the same procedure and define the marginal probabilities of node classes in the training dataset as m x ∈ R 1 × a and those of edge classes as m e ∈ R 1 × b , respectively. Hence,

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During the forward process, we compute the distributions of all node and edge classes after state transitions. However, we only focus on the distributions corresponding to the selected N ( t ) nodes and M ( t ) edges, sampling from these distributions to update their states. We preserve the states of the remaining nodes from the previous step. Algorithm 1 and Figure 1b illustrate our forward process.

Algorithm 1 Forward Process

Input: G 0 = ( X 0 , E 0 ) , state transition matrices Q X & Q E , node&edge selection functions N ( · )& M ( · ) .

Process:

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Set G t = G 0 and replace K x X t with K x ˜ X t , and K e E t with K e ˜ E t .

3.3 Denoising/Sampling Process

During the denoising process, we first sample a random graph based on the marginal distributions of nodes and edges at t = T . This random graph is then used as the input to the denoising network ϕ θ to predict a denoised graph. Based on the prediction, we use the noise model to obtain the noisy graph at t = T -1 . Subsequently, this noisy graph is fed into the denoising network, and the above process is repeated iteratively until a new graph is sampled.

To start the sampling process, we need to compute the marginal distributions of nodes and edges at t = T . At time t = T , a node class i at t = T is a result of transitions from some other node classes at t = 0 . As a result, the marginal probability of the i -th node class at time t = T equals

where p x i and p x j are the marginal probabilities of node classes i and j at time t = 0 , respectively. For edges, we similarly have that the marginal probabilities of the i -th edge class at t = T equals

where p e i and p e l are the marginal probabilities of edge classes i and l at t = 0 .

After sampling the random graph, we use the denoising model to predict the clean graph and continue this process iteratively. At time step t , the input to the denoising model includes not only the noisy graph and the time t but also two additional functions: R x ( t ) = N ( t ) /n and

Algorithm 2 Sampling Process

Input: Denoising Network ϕ θ , node&edge class distribution m T x = [ p x,T i ] & m T e = [ p e,T k ] , state transition matrices Q X & Q E .

Process:

Sample ˆ G T from m T x × m T e .

for z ← f ( ˆ G t , t ) {Structural and spectral features}.

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R e ( t ) = M ( t ) / (0 . 5 n ( n -1)) , which describe the proportion of nodes and edges in the noisy graph at time t 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 N ( t ) nodes and M ( t ) edges, thereby helping the network make more accurate predictions. The sampling steps are listed in Algorithm 2. Figure 8 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 ( ˆ G t -1 | ˆ G t , ϕ θ ) , factorizes as q ( ˆ G t -1 | ˆ G 0 ) q θ ( ˆ G 0 | ˆ G t ) . 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,

where ˆ p G = (ˆ p X , ˆ p E ) denotes the predicted probability distributions, while x i and e ij 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 (ˆ p X ) equals the class index of highest probability for each node, while argmax ( X ) 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 N ( t ) , M ( 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,

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 l MSE (ˆ p G , G ) ,

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 Efficiency of DMol

To explain the sampling efficiency of DMol, we analyze the minimum time step increment δt required to perturb exactly one node (i.e., the smallest discrete noise unit) during the diffusion process. We compare DMol with DiGress; however, it 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

where p x i denotes the probability of node type i (for derivations, see Appendix D.2). The inequality demonstrates that DMol requires fewer steps to perturb a 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 D.

4 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.

Specifically, we convert a small predefined number ( 3 -15 ) of most frequently occurring subgraphs/motifs (which may differ for different training sets) that can 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 are directly 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 a chemically valid manner.

Although superficially similar to part of the JTVAE Jin et al. [2019] approach, motif compression is significantly different. The JT approach uses not just motifs but a large number (roughly several hundreds) of submolecular structures, and requires finding a minimum spanning tree during the encoding process; furthermore, V AEs do not directly generate graphs, and one has to perform complicated decoding. The main issue is that each compressed molecular substructure can form many different types of bonds with different atoms during the reconstruction process. The options are ranked according to a special scoring function and the overall process is computationally demanding. The motifs used in DMol for different datasets are available in the Appendix, Figures 3, 4, 5.

5 Experiments

We present next the results of running DMol on several benchmarking datasets. For experiments pertaining to other graph models (e.g., SBMs and planar graphs), the reader is referred to Appendix G.1.

Benchmarks. We compared DMol to several graph generation methods. The benchmarking models used in the experiments include GraphVAE Simonovsky and Komodakis [2018], GT-VAE Mitton et al. [2021], Set2GraphVAE Vignac and Frossard [2021], SPECTRE Martinkus et al. [2022b], GraphNVP Madhawa et al. [2019], GDSS Jo et al. [2022b], GruM Jo et al. [2024], DiGress Vignac et 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 Polykovskiy et al. [2020], and GUACAMOL Brown et al. [2019] datasets. QM9 is a small dataset: we used 100 K molecules for training, 20 K for validation, and 13 K for testing. Both MOSES

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 a 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 , r = 0 . 2 , λ 1 = 5 , λ 2 = 1 , λ 3 = 1 . The number of training epochs is set to 500 . The experiments were conducted using NVIDIA H100 GPUs. The QM9 runs were executed on a single GPU, while the MOSES and GUACAMOL experiments were processed in parallel using 4 GPUs.

Evaluation Metrics. 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 generated 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 introduce their own set of metrics for reporting results. In addition to V, U, and N, MOSES incorporates a filter score, the Frechet ChemNet Distance (FCD) Preuer et al. [2018], SNN, and scaffold similarity scores, while GUACAMOL uses FCD and KL divergence. The definition of these metrics are in Appendix E. The metrics mentioned above still have certain limitations. Since these metrics assess the overall quality of the generation pro-

Table 3: Classical validity, uniqueness and novelty performance comparison on GUACAMOL, for comparable generation complexity.

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 compared to DeFoG.

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 introduced Shingle distance (SD) (which quantifies the divergence between two molecular motif (shingle) distributions, see the Appendix F). The CLscore is calculated by considering how many substructures in a molecule also occur in the drug-like dataset ChEMBL Gaulton et al. [2016]. QED evaluates drug-likeness 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 a batch of 64 molecules.

Results. The experimental results are presented in Tables 1, 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 metrics.

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.

6 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 Probabilities. For each motif, we compute the probability that a molecule contains that specific motif. The results (detailed in Appendix Table 10) 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 motifs (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 to verify that our model maintains realistic structural relationships. For example, the co-occurrence probability of benzene (c1ccccc1) and pyridine (c1ccncc1) is 0 . 11 in the training set 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 11). 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 (c1ccccc1, 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 N ( 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.

7 Conditional Generation Based on Scaffolds and Docking Results

To avoid issues associated with direct conditional generation, we instead proposed a scaffold-informed pipeline. The key ideas are to cluster the training set molecules according to either their RDKit features, or embeddings generated by Mol2Vec or Grover Jaeger 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 I.

8 Conclusion

We present DMol, a highly computationally efficient motif-preserving graph diffusion for small molecule drug generation. DMol couples node and edge noise through a combination of deterministic and stochastic mechanisms, adapts the loss function and includes motif compression. On biochemically 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 S. Krafczyk from the National Center for Supercomputing Applications (NCSA) at the University of Illinois Urbana-Champaign for his valuable assistance in resolving technical simulation issues encountered during our experiments. Furthermore, we thank Jiwoong Jung and Ziheng Qi for useful discussion.

References

Fan Bao, Min Zhao, Zhongkai Hao, Peiyao Li, Chongxuan Li, and Jun Zhu. Equivariant energyguided SDE for inverse molecular design. In The Eleventh International Conference on Learning Representations , 2023. URL https://openreview.net/forum?id=r0otLtOwYW .

Richard J. Bickerton, Gaia V. Paolini, Jérémy Besnard, Sorel Muresan, and Andrew L. Hopkins. Quantifying the chemical beauty of drugs. Nature chemistry , 4 2:90-8, 2012. URL https: //api.semanticscholar.org/CorpusID:205289650 .

Nathan Brown, Marco Fiscato, Marwin H.S. Segler, and Alain C. Vaucher. Guacamol: Benchmarking models for de novo molecular design. Journal of Chemical Information and Modeling , 59 (3):1096-1108, March 2019. ISSN 1549-960X. doi: 10.1021/acs.jcim.8b00839. URL http: //dx.doi.org/10.1021/acs.jcim.8b00839 .

Sven Bühlmann and Jean-Louis Reymond. Chembl-likeness score and database gdbchembl. Frontiers in Chemistry , 8, 2020. ISSN 2296-2646. doi: 10.3389/fchem.2020.00046. URL https://www. frontiersin.org/journals/chemistry/articles/10.3389/fchem.2020.00046 .

Hongyang Chen, Can Xu, Lingyu Zheng, Qiang Zhang, and Xuemin Lin. Diffusion-based graph generative methods, 2024. URL https://arxiv.org/abs/2401.15617 .

Xiaohui Chen, Yukun Li, Aonan Zhang, and Li-Ping Liu. Nvdiff: Graph generation through the diffusion of node vectors, 2023. URL https://arxiv.org/abs/2211.10794 .

François R J Cornet, Grigory Bartosh, Mikkel N. Schmidt, and Christian A. Naesseth. Equivariant neural diffusion for molecule generation. In ICML 2024 AI for Science Workshop , 2024. URL https://openreview.net/forum?id=3iih8PGAH7 .

Quan Dao, Hao Phung, Binh Nguyen, and Anh Tran. Flow matching in latent space. arXiv preprint arXiv:2307.08698 , 2023.

Nicola De Cao and Thomas Kipf. MolGAN: An implicit generative model for small molecular graphs. ICML 2018 workshop on Theoretical Foundations and Applications of Deep Generative Models , 2018.

Prafulla Dhariwal and Alex Nichol. Diffusion models beat gans on image synthesis. CoRR , abs/2105.05233, 2021. URL https://arxiv.org/abs/2105.05233 .

Yuanqi Du, Tianfan Fu, Jimeng Sun, and Shengchao Liu. Molgensurvey: A systematic survey in machine learning models for molecule design, 2022. URL https://arxiv.org/abs/2203. 14500 .

Wenqi Fan, Chengyi Liu, Yunqing Liu, Jiatong Li, Hang Li, Hui Liu, Jiliang Tang, and Qing Li. Generative diffusion models on graphs: Methods and applications. arXiv preprint arXiv:2302.02591 , 2023.

Anuththari Gamage, Eli Chien, Jianhao Peng, and Olgica Milenkovic. Multi-motifgan (mmgan): Motif-targeted graph generation and prediction. In ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , pages 4182-4186, 2020. doi: 10.1109/ICASSP40776.2020.9053451.

Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky T. Q. Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching, 2024. URL https://arxiv.org/abs/2407.15595 .

Anna Gaulton, Anne Hersey, Michał Nowotka, A. Patrícia Bento, Jon Chambers, David Mendez, Prudence Mutowo, Francis Atkinson, Louisa J. Bellis, Elena Cibrián-Uhalte, Mark Davies, Nathan Dedman, Anneli Karlsson, María Paula Magariños, John P. Overington, George Papadatos, Ines Smit, and Andrew R. Leach. The chembl database in 2017. Nucleic Acids Research , 45(D1): D945-D954, 11 2016. ISSN 0305-1048. doi: 10.1093/nar/gkw1074. URL https://doi.org/ 10.1093/nar/gkw1074 .

Rafael Gómez-Bombarelli, Jennifer N Wei, David Duvenaud, José Miguel Hernández-Lobato, Benjamín Sánchez-Lengeling, Dennis Sheberla, Jorge Aguilera-Iparraguirre, Timothy D Hirzel, Ryan P Adams, and Alán Aspuru-Guzik. Automatic chemical design using a data-driven continuous representation of molecules. ACS central science , 4(2):268-276, 2018.

D S Goodsell and A J Olson. Automated docking of substrates to proteins by simulated annealing. Proteins , 8(3):195-202, 1990.

Kilian Konstantin Haefeli, Karolis Martinkus, Nathanaël Perraudin, and Roger Wattenhofer. Diffusion models for graphs benefit from discrete state spaces, 2023. URL https://arxiv.org/abs/ 2210.01549 .

Tiankai Hang, Shuyang Gu, Chen Li, Jianmin Bao, Dong Chen, Han Hu, Xin Geng, and Baining Guo. Efficient diffusion training via min-snr weighting strategy. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 7441-7451, 2023.

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems , volume 33, pages 6840-6851. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper_files/paper/2020/file/ 4c5bcfec8584af0d967f1ab10179ca4b-Paper.pdf .

Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J. Fleet. Video diffusion models, 2022. URL https://arxiv.org/abs/2204.03458 .

Emiel Hoogeboom, Victor Garcia Satorras, Clément Vignac, and Max Welling. Equivariant diffusion for molecule generation in 3d, 2022. URL https://arxiv.org/abs/2203.17003 .

Han Huang, Leilei Sun, Bowen Du, Yanjie Fu, and Weifeng Lv. Graphgdp: Generative diffusion processes for permutation invariant graph generation, 2022. URL https://arxiv.org/abs/ 2212.01842 .

Han Huang, Leilei Sun, Bowen Du, and Weifeng Lv. Conditional diffusion based on discrete graph structures for molecular graph generation, 2023. URL https://arxiv.org/abs/2301.00427 .

Sabrina Jaeger, Simone Fulle, and Samo Turk. Mol2vec: unsupervised machine learning approach with chemical intuition. Journal of chemical information and modeling , 58(1):27-35, 2018.

Jan H. Jensen. A graph-based genetic algorithm and generative model/monte carlo tree search for the exploration of chemical space. Chem. Sci. , 10:3567-3572, 2019. doi: 10.1039/C8SC05372C. URL http://dx.doi.org/10.1039/C8SC05372C .

Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction tree variational autoencoder for molecular graph generation. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning , volume 80 of Proceedings of Machine Learning Research , pages 2323-2332. PMLR, 10-15 Jul 2018. URL https://proceedings.mlr.press/ v80/jin18a.html .

Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction tree variational autoencoder for molecular graph generation, 2019. URL https://arxiv.org/abs/1802.04364 .

Jaehyeong Jo, Seul Lee, and Sung Ju Hwang. Score-based generative modeling of graphs via the system of stochastic differential equations. arXiv:2202.02514 , 2022a. URL https://arxiv. org/abs/2202.02514 .

Jaehyeong Jo, Seul Lee, and Sung Ju Hwang. Score-based generative modeling of graphs via the system of stochastic differential equations. In International conference on machine learning , pages 10362-10383. PMLR, 2022b.

Jaehyeong Jo, Dongki Kim, and Sung Ju Hwang. Graph generation with diffusion mixture, 2024. URL https://arxiv.org/abs/2302.03596 .

Lingkai Kong, Jiaming Cui, Haotian Sun, Yuchen Zhuang, B. Aditya Prakash, and Chao Zhang. Autoregressive diffusion model for graph generation, 2023. URL https://openreview.net/ forum?id=98J48HZXxd5 .

Youngchun Kwon, Dongseon Lee, Youn-Suk Choi, Kyoham Shin, and Seokho Kang. Compressed graph representation for scalable molecular graph generation. Journal of Cheminformatics , 12:1-8, 2020.

Greg Landrum et al. Rdkit: Open-source cheminformatics, 2006.

Seul Lee, Jaehyeong Jo, and Sung Ju Hwang. Exploring chemical space with score-based out-ofdistribution generation. Proceedings of the 40th International Conference on Machine Learning , 2023.

Renjie Liao, Yujia Li, Yang Song, Shenlong Wang, Will Hamilton, David K Duvenaud, Raquel Urtasun, and Richard Zemel. Efficient graph generation with graph recurrent attention networks. Advances in neural information processing systems , 32, 2019.

Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling, 2023. URL https://arxiv.org/abs/2210.02747 .

Phillip Lippe and Efstratios Gavves. Categorical normalizing flows via continuous transformations. In International Conference on Learning Representations , 2021. URL https://openreview. net/forum?id=-GLNZeVDuik .

Gang Liu, Jiaxin Xu, Tengfei Luo, and Meng Jiang. Inverse molecular design with multi-conditional diffusion guidance. CoRR , abs/2401.13858, 2024a. URL https://doi.org/10.48550/arXiv. 2401.13858 .

Gang Liu, Jiaxin Xu, Tengfei Luo, and Meng Jiang. Graph diffusion transformers for multi-conditional molecular generation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems , 2024b. URL https://openreview.net/forum?id=cfrDLD1wfO .

Meng Liu, Keqiang Yan, Bora Oztekin, and Shuiwang Ji. GraphEBM: Molecular graph generation with energy-based models. In Energy Based Models Workshop - ICLR 2021 , 2021. URL https: //openreview.net/forum?id=Gc51PtL_zYw .

Yijing Liu, Chao Du, Tianyu Pang, Chongxuan Li, Wei Chen, and Min Lin. Graph diffusion policy optimization. CoRR , abs/2402.16302, 2024c. URL https://doi.org/10.48550/arXiv.2402. 16302 .

Shitong Luo, Chence Shi, Minkai Xu, and Jian Tang. Predicting molecular conformation via dynamic graph score matching. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems , 2021a. URL https: //openreview.net/forum?id=hMY6nm9lld .

Youzhi Luo, Keqiang Yan, and Shuiwang Ji. Graphdf: A discrete flow model for molecular graph generation. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning , volume 139 of Proceedings of Machine Learning Research , pages 7192-7203. PMLR, 18-24 Jul 2021b. URL https://proceedings.mlr.press/v139/ luo21a.html .

Kaushalya Madhawa, Katushiko Ishiguro, Kosuke Nakago, and Motoki Abe. Graphnvp: An invertible flow model for generating molecular graphs, 2019. URL https://arxiv.org/abs/1905. 11600 .

Karolis Martinkus, Andreas Loukas, Nathanaël Perraudin, and Roger Wattenhofer. Spectre: Spectral conditioning helps to overcome the expressivity limits of one-shot graph generators. In International Conference on Machine Learning , pages 15159-15179. PMLR, 2022a.

Karolis Martinkus, Andreas Loukas, Nathanaël Perraudin, and Roger Wattenhofer. Spectre: Spectral conditioning helps to overcome the expressivity limits of one-shot graph generators, 2022b. URL https://arxiv.org/abs/2204.01613 .

Mengchun Zhang, Maryam Qamar, Taegoo Kang, Yuna Jung, Chenshuang Zhang, Sung-Ho Bae, and Chaoning Zhang. A survey on graph diffusion models: Generative ai in science for molecule, protein and material. 2023. doi: 10.13140/RG.2.2.26493.64480. URL https://rgdoi.net/10. 13140/RG.2.2.26493.64480 .

Rocío Mercado, Tobias Rastemo, Edvard Lindelöf, Günter Klambauer, Ola Engkvist, Hongming Chen, and Esben Jannik Bjerrum. Graph networks for molecular design. Machine Learning: Science and Technology , 2(2):025023, 2021.

Joshua Mitton, Hans M Senn, Klaas Wynne, and Roderick Murray-Smith. A graph vae and graph transformer approach to generating molecular graphs. arXiv preprint arXiv:2104.04345 , 2021.

Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning , volume 139 of Proceedings of Machine Learning Research , pages 8162-8171. PMLR, 18-24 Jul 2021. URL https://proceedings.mlr.press/v139/nichol21a.html .

Chenhao Niu, Yang Song, Jiaming Song, Shengjia Zhao, Aditya Grover, and Stefano Ermon. Permutation invariant graph generation via score-based generative modeling. In International Conference on Artificial Intelligence and Statistics , pages 4474-4484. PMLR, 2020a.

Chenhao Niu, Yang Song, Jiaming Song, Shengjia Zhao, Aditya Grover, and Stefano Ermon. Permutation invariant graph generation via score-based generative modeling, 2020b. URL https://arxiv.org/abs/2003.00638 .

Daniil Polykovskiy, Alexander Zhebrak, Benjamin Sanchez-Lengeling, Sergey Golovanov, Oktai Tatanov, Stanislav Belyaev, Rauf Kurbanov, Aleksey Artamonov, Vladimir Aladinskiy, Mark Veselov, Artur Kadurin, Simon Johansson, Hongming Chen, Sergey Nikolenko, Alan AspuruGuzik, and Alex Zhavoronkov. Molecular sets (moses): A benchmarking platform for molecular generation models, 2020. URL https://arxiv.org/abs/1811.12823 .

Mariya Popova, Mykhailo Shvets, Junier Oliva, and Olexandr Isayev. Molecularrnn: Generating realistic molecular graphs with optimized properties, 2019. URL https://arxiv.org/abs/ 1905.13372 .

Kristina Preuer, Philipp Renz, Thomas Unterthiner, Sepp Hochreiter, and Günter Klambauer. Fréchet chemnet distance: A metric for generative models for molecules in drug discovery, 2018. URL https://arxiv.org/abs/1803.09518 .

Yiming Qin, Manuel Madeira, Dorina Thanou, and Pascal Frossard. Defog: Discrete flow matching for graph generation, 2025. URL https://arxiv.org/abs/2410.04263 .

Yu Rong, Yatao Bian, Tingyang Xu, Weiyang Xie, Ying Wei, Wenbing Huang, and Junzhou Huang. Grover: Self-supervised message passing transformer on large-scale molecular data. arXiv preprint arXiv:2007.02835 , 2(3):17, 2020.

Marwin HS Segler, Thierry Kogej, Christian Tyrchan, and Mark P Waller. Generating focused molecule libraries for drug discovery with recurrent neural networks. ACS central science , 4(1): 120-131, 2018.

Chence Shi*, Minkai Xu*, Zhaocheng Zhu, Weinan Zhang, Ming Zhang, and Jian Tang. Graphaf: a flow-based autoregressive model for molecular graph generation. In International Conference on Learning Representations , 2020. URL https://openreview.net/forum?id=S1esMkHYPr .

Chence Shi, Shitong Luo, Minkai Xu, and Jian Tang. Learning gradient fields for molecular conformation generation, 2021. URL https://arxiv.org/abs/2105.03902 .

Jiaxin Shi, Kehang Han, Zhe Wang, Arnaud Doucet, and Michalis K. Titsias. Simplified and generalized masked diffusion for discrete data, 2025. URL https://arxiv.org/abs/2406. 04329 .

Martin Simonovsky and Nikos Komodakis. GraphVAE: Towards generation of small graphs using variational autoencoders, 2018. URL https://openreview.net/forum?id=SJlhPMWAW .

Michael A Skinnider. Invalid smiles are beneficial rather than detrimental to chemical language models. Nature Machine Intelligence , 6(4):437-448, 2024.

Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learning , pages 2256-2265. PMLR, 2015.

Xiangru Tang, Howard Dai, Elizabeth Knight, Fang Wu, Yunyang Li, Tianxiao Li, and Mark Gerstein. Asurvey of generative ai for de novo drug design: New frontiers in molecule and protein generation, 2024. URL https://arxiv.org/abs/2402.08703 .

Brian L. Trippe, Jason Yim, Doug Tischer, David Baker, Tamara Broderick, Regina Barzilay, and Tommi S. Jaakkola. Diffusion probabilistic modeling of protein backbones in 3d for the motifscaffolding problem. In The Eleventh International Conference on Learning Representations , 2023. URL https://openreview.net/forum?id=6TxBxqNME1Y .

Jui-Yi Tsai, Ya-Wen Teng, Ho Chiok Yew, De-Nian Yang, and Lydia Y. Chen. Cdgraph: Dual conditional social graph synthesizing via diffusion model, 2023. URL https://arxiv.org/ abs/2311.01729 .

Fedor Velikonivtsev, Mikhail Mironov, and Liudmila Prokhorenkova. Challenges of generating structurally diverse graphs, 2024. URL https://arxiv.org/abs/2409.18859 .

Clément Vignac and Pascal Frossard. Top-n: Equivariant set and graph generation without exchangeability. CoRR , abs/2110.02096, 2021. URL https://arxiv.org/abs/2110.02096 .

Clement Vignac, Igor Krawczuk, Antoine Siraudin, Bohan Wang, Volkan Cevher, and Pascal Frossard. Digress: Discrete denoising diffusion for graph generation. In The Eleventh International Conference on Learning Representations , 2023. URL https://openreview.net/ forum?id=UaAD-Nu86WX .

Lilapati Waikhom and Ripon Patgiri. Graph neural networks: Methods, applications, and opportunities, 2021. URL https://arxiv.org/abs/2108.10733 .

Kai Wang, Mingjia Shi, Yukun Zhou, Zekai Li, Zhihang Yuan, Yuzhang Shang, Xiaojiang Peng, Hanwang Zhang, and Yang You. A closer look at time steps is worthy of triple speed-up for diffusion model training. arXiv preprint arXiv:2405.17403 , 2024.

Yuyang Wang, Zijie Li, and Amir Barati Farimani. Graph Neural Networks for Molecules , page 21-66. Springer International Publishing, 2023. ISBN 9783031371967. doi: 10.1007/978-3-031-37196-7_ 2. URL http://dx.doi.org/10.1007/978-3-031-37196-7_2 .

Huaijin Wu, Xinyu Ye, and Junchi Yan. QVAE-mole: The quantum VAE with spherical latent variable learning for 3-d molecule generation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems , 2024. URL https://openreview.net/forum?id=RqvesBxqDo .

Lemeng Wu, Chengyue Gong, Xingchao Liu, Mao Ye, and qiang liu. Diffusion-based molecule generation with informative prior bridges. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems , 2022. URL https://openreview.net/forum?id=TJUNtiZiTKE .

Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl Leswing, and Vijay Pande. Moleculenet: a benchmark for molecular machine learning. Chemical science , 9(2):513-530, 2018.

Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems , 32(1):4-24, 2021. doi: 10.1109/TNNLS.2020.2978386.

Minkai Xu, Lantao Yu, Yang Song, Chence Shi, Stefano Ermon, and Jian Tang. Geodiff: A geometric diffusion model for molecular conformation generation. In International Conference on Learning Representations , 2022. URL https://openreview.net/forum?id=PzcvxEMzvQC .

Zhe Xu, Ruizhong Qiu, Yuzhong Chen, Huiyuan Chen, Xiran Fan, Menghai Pan, Zhichen Zeng, Mahashweta Das, and Hanghang Tong. Discrete-state continuous-time diffusion for graph generation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems , 2024a. URL https://openreview.net/forum?id=YkSKZEhIYt .

Zhe Xu, Ruizhong Qiu, Yuzhong Chen, Huiyuan Chen, Xiran Fan, Menghai Pan, Zhichen Zeng, Mahashweta Das, and Hanghang Tong. Discrete-state continuous-time diffusion for graph generation, 2024b. URL https://arxiv.org/abs/2405.11416 .

Nianzu Yang, Huaijin Wu, Kaipeng Zeng, Yang Li, and Junchi Yan. Molecule generation for drug design: a graph learning perspective, 2024. URL https://arxiv.org/abs/2202.09212 .

Jiaxuan You, Rex Ying, Xiang Ren, William Hamilton, and Jure Leskovec. Graphrnn: Generating realistic graphs with deep auto-regressive models. In International conference on machine learning , pages 5708-5717. PMLR, 2018.

Chengxi Zang and Fei Wang. Moflow: an invertible flow model for generating molecular graphs. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining , pages 617-626, 2020.

He Zhang, Bang Wu, Xingliang Yuan, Shirui Pan, Hanghang Tong, and Jian Pei. Trustworthy graph neural networks: Aspects, methods, and trends. Proceedings of the IEEE , 112(2):97-139, 2024. doi: 10.1109/JPROC.2024.3369017.

Łukasz Maziarka, Agnieszka Pocha, Jan Kaczmarczyk, and Michał Warchoł. Mol-cycleGAN - a generative model for molecular optimization, 2019. URL https://openreview.net/forum? id=BklKFo09YX .

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A Motif Compression

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Since each dataset has a different size of graph and complexity, we choose a different number of motifs, K to be converted to supernodes. We select 3 motifs for QM9 3, 15 motifs for MOSES 4, and 15 motifs for GUACAMOL 5.

B DiGress vs DMol

We now further compare DiGress and DMol. First, we observe that DiGress also constructs state transition matrices based on marginal distributions, with the key difference with respect to DMol being that its transition matrices vary over time as

where α t and β t are time-dependent scaling parameters, while Q t X and Q t E are the state transition matrices used to transition from G t -1 to G t . However, since the forward noise adding process is Markovian, DiGress directly utilizes Q t X and Q t E to achieve the transition from G 0 to G t , where Q t X = Q 1 X . . . Q t X and Q t E = Q 1 E . . . Q t E .

Since ( 1m ) 2 = 1m , we have

where α t = ∏ t τ =1 α τ and β t = 1 -α t . Additionally, DiGress also adopts a cosine schedule for α t : α t = cos 2 (0 . 5 π ( t/T + c ) / (1 + c )) , where c > 0 is a constant.

In DMol, for a graph dataset with n nodes, the number of nodes undergoing state changes at time step t equals N ( t ) = (1 -α ) n . At the same time step in Digress, the expected number of nodes undergoing state changes equals ∑ i np x i (1 -α )(1 -p x i ) , where p x i represents the marginal probability of node class i . Therefore, at the same time step, the number of nodes modified by DMol is 1 ∑ i p x i (1 -p x i ) 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 i is p x i , whereas in DiGress, this proportion equals p x i (1 -p x i ) . The ratio of these fractions equals 1 -p x i .

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 our noise model and that modified by Digress equals r ∑ j p e j (1 -p e j ) , where r is the hyperparameter mentioned in Section 3.2. At each time step, this ratio needs to 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 j differs from the same in DiGress by a constant factor of 1 -p e j , where p e j equals the marginal probability of edge class j . As a 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 to DMol to obtain Digress:

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 at each step by a factor of ∑ i p x i (1 -p x i ) .

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.

Scale the number of edges to be changed at each step by a factor of ∑ j p e j (1 -p e j ) r .

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 x i , and multiply the random scores for edges by 1 -p e j .

These modifications are straightforward to 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 selection. This framework extends traditional hierarchical variational inference structures by incorporating a crucial dependency relationship: the selection of edges to perturb is conditioned on the selection of nodes at each timestep.

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For a molecular graph G , the model likelihood must account for this conditional relationship, according to:

Note the key difference in the conditional probability p θ ( G t -1 | G t , K x ( t ) , K e ( t )) , which explicitly accounts for the selection matrices K x ( t ) and K e ( t ) that designate which nodes and edges receive perturbations at time t .

Following established variational inference principles, we derive a tractable evidence lower bound (ELBO) adapted for this conditional structure:

Where the diffusion component L t now explicitly models the conditional relationship between node and edge noise:

This can be further decomposed to highlight the conditional structure:

Note the critical difference in the edge term, where the predicted probability p θ ( e t -1 ij | G t , K x ( t -1)) is explicitly conditioned on the node selection K x ( t -1) . This formulation captures the key aspect of DMol's noise process: edges are 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 ( n G ) from the empirical node count distributions across the molecular corpus. The prior term involves categorical KL 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 1 .

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 m X . This is easy to see because p 0 = m X , so p 1 = α 1 p 0 + β 1 m X = m X , and by induction, p t = m X for all t , where p t is the node type distribution at step t . In DMol, the node type distribution changes over time. For a node with type i selected for perturbation, the probability of transitioning to type j = i is P ( j | i selected ) = p X ( j ) 1 -p X ( i ) . This leads to a different recurrence relation for the probability distribution over time.

D.2 Efficiency

To compare the efficiency of DMol and DiGress, we compare the time step increment required to change exactly one node(minimal noise) under both settings. In DMol, the number of nodes changing at time t is N ( t ) = (1 -α t ) · n . To find δt DMol such that δN ( t ) = 1 , we need

The number of nodes that change at time t + δt equals

For DMol, we use T = kn and the cosine schedule for α ,

Taking the derivative with respect to t and using a small angle approximation for small δt , we obtain:

For small δt , we can approximate:

Therefore,

Setting this equal to 1 and solving for δt , we can have the value of δt DMol :

For DiGress, we use a similar approach to find δt Digress . First, the expected number of nodes changed by DiGress at time t equals:

Following a similar derivation as for DMol, we arrive at

Then, the ratio of the minimum time steps of interest equals

Because ∑ i p x i · (1 -p x i ) < 1 for any probability distribution, we have that

This proves that δt Digress > δt DMol , or, in words, that DMol requires a smaller time step increment to change exactly one node. As a 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.

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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 G 0 and the graph at time t , G t , is approximately d H ( G 0 , G t ) ∼ β t · n . This Hamming distance captures the difference between the number of node/edge types in the two graphs. For DiGress, the expected Hamming distance is E [ d H ( G 0 , G t )] ∼ β t · n · ∑ i ∈C X p X ( i ) · (1 -p X ( i )) . Since ∑ i ∈C X p X ( i ) · (1 -p X ( i )) < 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 of various physicochemical descriptors.

F ChEMB Likeness, QED Scores, and Shingle Distance

The ChEMBL-Likeness Score (CLscore) Bühlmann and Reymond [2020] is a metric used to assess the drug-likeness of a molecule. It is determined by analyzing how many of the molecule's substructures are present in the ChEMBL datasetGaulton et al. [2016], a database of bioactive molecules with drug-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 to assess how 'drug-like" a compound is. It combines eight physicochemical properties, including molecular mass ( M r ), 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 1 (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 a reference dataset. It is computed by comparing the frequencies of important, predefined molecular drug substructures (motifs), known in RdKit as shingles, according to the ChEMBL dataset Gaulton et al. [2016]. The same set of shingles is used in the computation of the ChEMBL-Likeness 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-relevant motifs, SD offers a 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 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 4 nodes), measured by the relative squared Maximum Mean Discrepancy (MMD).

MMDmeasures the discrepancy between two sets of distributions. The relative squared MMD is defined as follows:

where G gen , G train , and G test are the sets of generated graphs, training graphs, and test graphs.

In our experiments, we 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 5. 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 use of special subgraph selection procedures. The results are summarized in Table 6. The model we used is 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 is because the MSE loss penalizes discrepancies between the number of altered nodes and edges and the noise added during the forward process, resulting in 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.

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G.3 Selection of the parameter k

The hyperparameter k determines the total number of diffusion steps since we set T = kn , where n is the number of nodes. In practice, k is typically set to 1 or 2 to reduce the number of diffusion steps and enhance performance (as explained in Figure 7). The latter observation can be intuitively explained as 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 k , while the y-axis remains unchanged (Figure 7). Consequently, the number of diffusion steps required to modify a single node increases. This leads to a 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 a 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 k . As shown in Table 7, the sample time is approximately proportional to k . Moreover, setting k = 2 yields the highest validity.

G.4 Complete Experimental Results for the QM9 Dataset

For complete performance comparison on the QM9 dataset, please refer to Table 8.

G.5 Forward Process Schedule

To examine the impact of the cosine schedule on different molecular scaffolds and topologies, we conducted an ablation study comparing different values of the hyperparameter c in the cosine schedule α = cos 2 (0 . 5 π ( t/T + c ) / (1+ c )) , as well as comparing cosine versus linear schedules on the MOSES dataset. The results are presented in Table 9.

The experimental results demonstrate that varying the value of c has a relatively small impact on crucial performance metrics, with an optimal performance achieved at c = 0 . 008 , which is the setting used in our experiments. This finding aligns with the findings by DiGress, which also adopts c = 0 . 008 as the default value.

When comparing cosine and linear schedules, we see that the cosine schedule clearly outperforms 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 intermediate stages. This distribution better aligns with the learning characteristics of 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.

G.6 Scaffold-Constrained and Conditional Generation

To avoid issues associated with direct conditional generation, we instead proposed a scaffold-informed pipeline. The key ideas are to cluster the training set molecules according to either their RDKit features, or embeddings generated by Mol2Vec or Grover Jaeger et al. [2018], Rong et 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 Mol2Vec or Grover to 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 a companion paper.

G.7 Motif Probability Distribution Analysis

Table 10 presents the complete per-motif probabilities for the top 30 most frequent motifs in the MOSES dataset. Table 11 shows the marginal distributions of node and supernode categories.

H Examples of Generated Molecules.

Figure 8 illustrates the sampling process.

N

N

t=T

t=3T/4

t=T/2

t=T/4

0 =1

HO

NH

NH

HO

OH

NH

NH

NH

NH

NH

HN

HO

NH

NH

NH

NH

NH

NF

NH

NH

NH

NH

OH

NH

NH

NH

NH

HN

I Sample Docking Results

We used AutoDock Vina version 1.2.0 Goodsell and Olson [1990] to visualize the docking of two of the top-scoring generated molecules (with respect to the ChEMBL likeness) that also exhibit strong biding affinities on the 1PXH 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 1PXH represents the protein tyrosine phosphatase 1B with bidentate inhibitor compound 2. The free energy results are shown in Figures 12 and 13, and they indicate that the generated molecules can significantly outperform or underperform existing reference drugs.

(c) Generated drug 2: Free energy = -6 . 0 kcal/mol

8

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, we acknowledge that molecular generation technologies could potentially be misused for designing harmful substances if deployed without appropriate safeguards. Additionally, as with many other AI 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 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 metrics for the generated molecules, since an overwhelming number of created samples cannot be synthesized or do not dock on 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 to 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.

Peer reviews:
comments: 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.

STRENGTHS AND WEAKNESSES:
pros:
* 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.
* 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.
* 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.

cons:
* None

QUESTIONS:
* How is DMol compared to FragFM[1]?

[1]Lee et al.,  FragFM: Hierarchical Framework for Efficient Molecule Generation via Fragment-Level Discrete Flow Matching

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nWeakness:\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.\n\nQUESTIONS:\n- 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\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\nOn 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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \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.\n\nQUESTIONS:\nPlease refer the review in weakness.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-21"}