Node Representations Research Articles

Rumor Detection With Hierarchical Representation on Bipartite Ad Hoc Event Trees.

The rapid growth of social media has caused tremendous effects on information propagation, raising extreme challenges in detecting rumors. Existing rumor detection methods typically exploit the reposting propagation of a rumor candidate for detection by regarding all reposts to a rumor candidate as a temporal sequence and learning semantics representations of the repost sequence. However, extracting informative support from the topological structure of propagation and the influence of reposting authors for debunking rumors is crucial, which generally has not been well addressed by existing methods. In this article, we organize a claim post in circulation as an ad hoc event tree, extract event elements, and convert it into bipartite ad hoc event trees in terms of both posts and authors, i.e., author tree and post tree. Accordingly, we propose a novel rumor detection model with hierarchical representation on the bipartite ad hoc event trees called BAET. Specifically, we introduce word embedding and feature encoder for the author and post tree, respectively, and design a root-aware attention module to perform node representation. Then we adopt the tree-like RNN model to capture the structural correlations and propose a tree-aware attention module to learn tree representation for the author tree and post tree, respectively. Extensive experimental results on two public Twitter datasets demonstrate the effectiveness of BAET in exploring and exploiting the rumor propagation structure and the superior detection performance of BAET over state-of-the-art baseline methods.

A Controllability Reinforcement Learning Method for Pancreatic Cancer Biomarker Identification.

Pancreatic cancer is one of the most malignant cancers with rapid progression and poor prognosis. The use of transcriptional data can be effective in finding new biomarkers for pancreatic cancer. Many network-based methods used to identify cancer biomarkers are proposed, among which the combination of network controllability appears. However, most of the existing methods do not study RNA, rely on priori and mutations information, or can only achieve classification tasks. In this study, we propose a method combined Relational Graph Convolutional Network and Deep Q-Network called RDDriver to identify pancreatic cancer biomarkers based on multi-layer heterogeneous transcriptional regulation network. Firstly, we construct a regulation network containing long non-coding RNA, microRNA, and messenger RNA. Secondly, Relational Graph Convolutional Network is used to learn the node representation. Finally, we use the idea of Deep Q-Network to build a model, which score and prioritize each RNA with the Popov-Belevitch-Hautus criterion. We train RDDriver on three small simulated networks, and calculate the average score after applying the model parameters to the regulation networks separately. To demonstrate the effectiveness of the method, we perform experiments for comparison between RDDriver and other eight methods based on the approximate benchmark of three types cancer drivers RNAs.

Motif Graph Neural Network.

Graphs can model complicated interactions between entities, which naturally emerge in many important applications. These applications can often be cast into standard graph learning tasks, in which a crucial step is to learn low-dimensional graph representations. Graph neural networks (GNNs) are currently the most popular model in graph embedding approaches. However, standard GNNs in the neighborhood aggregation paradigm suffer from limited discriminative power in distinguishing high-order graph structures as opposed to low-order structures. To capture high-order structures, researchers have resorted to motifs and developed motif-based GNNs. However, the existing motif-based GNNs still often suffer from less discriminative power on high-order structures. To overcome the above limitations, we propose motif GNN (MGNN), a novel framework to better capture high-order structures, hinging on our proposed motif redundancy minimization operator and injective motif combination. First, MGNN produces a set of node representations with respect to each motif. The next phase is our proposed redundancy minimization among motifs which compares the motifs with each other and distills the features unique to each motif. Finally, MGNN performs the updating of node representations by combining multiple representations from different motifs. In particular, to enhance the discriminative power, MGNN uses an injective function to combine the representations with respect to different motifs. We further show that our proposed architecture increases the expressive power of GNNs with a theoretical analysis. We demonstrate that MGNN outperforms state-of-the-art methods on seven public benchmarks on both the node classification and graph classification tasks.

Contrastive learning for fair graph representations via counterfactual graph augmentation

Graph neural networks (GNNs) have exhibited excellent performance in graph representation learning. However, GNNs might inherit biases from the data, leading to discriminatory predictions. Existing study mainly concentrates on attaining fairness through counterfactuals related to node attributes, overlooking the causal impact of bias in the graph structure. Herein, we introduce a novel framework called fair contrastive learning based on counterfactual graph augmentation (FCLCA), aimed at learning counterfactual fairness by mitigating graph structure bias. FCLCA first generates two counterfactual graphs through structural augmentation. Next, we maximize the consistency between representations produced by nodes in these two counterfactual graphs using contrastive learning. In addition, FCLCA uses adversarial debiasing learning to further reduce the influence of sensitive attributes on the learned node representations. Finally, an optimized training strategy is used for contrastive learning to enhance the learning of counterfactual fairness. Comprehensive experiments conducted on four real-world datasets proved the effectiveness of FCLCA in balancing classification performance and fairness.

Hierarchical Graph Neural Network: A Lightweight Image Matching Model with Enhanced Message Passing of Local and Global Information in Hierarchical Graph Neural Networks

Graph Neural Networks (GNNs) have gained popularity in image matching methods, proving useful for various computer vision tasks like Structure from Motion (SfM) and 3D reconstruction. A well-known example is SuperGlue. Lightweight variants, such as LightGlue, have been developed with a focus on stacking fewer GNN layers compared to SuperGlue. This paper proposes the h-GNN, a lightweight image matching model, with improvements in the two processing modules, the GNN and matching modules. After image features are detected and described as keypoint nodes of a base graph, the GNN module, which primarily aims at increasing the h-GNN’s depth, creates successive hierarchies of compressed-size graphs from the base graph through a clustering technique termed SC+PCA. SC+PCA combines Principal Component Analysis (PCA) with Spectral Clustering (SC) to enrich nodes with local and global information during graph clustering. A dual non-contrastive clustering loss is used to optimize graph clustering. Additionally, four message-passing mechanisms have been proposed to only update node representations within a graph cluster at the same hierarchical level or to update node representations across graph clusters at different hierarchical levels. The matching module performs iterative pairwise matching on the enriched node representations to obtain a scoring matrix. This matrix comprises scores indicating potential correct matches between the image keypoint nodes. The score matrix is refined with a ‘dustbin’ to further suppress unmatched features. There is a reprojection loss used to optimize keypoint match positions. The Sinkhorn algorithm generates a final partial assignment from the refined score matrix. Experimental results demonstrate the performance of the proposed h-GNN against competing state-of-the-art (SOTA) GNN-based methods on several image matching tasks under homography, estimation, indoor and outdoor camera pose estimation, and 3D reconstruction on multiple datasets. Experiments also demonstrate improved computational memory and runtime, approximately 38.1% and 26.14% lower than SuperGlue, and an average of about 6.8% and 7.1% lower than LightGlue. Future research will explore the effects of integrating more recent simplicial message-passing mechanisms, which concurrently update both node and edge representations, into our proposed model.

VulTR: Software vulnerability detection model based on multi-layer key feature enhancement

Software vulnerabilities pose a huge threat to current network security, which continues to lead to data leaks and system damage. In order to effectively identify and patch these vulnerabilities, researchers have proposed automated detection methods based on deep learning. However, most of the existing methods only rely on single-dimensional data representation and fail to fully explore the composite characteristics of the code. Among them, the sequence embedding method fails to effectively capture the structural characteristics of the code, while the graph embedding method focuses more on the global characteristics of the overall graph structure and is still insufficient in optimizing the representation of nodes. In view of this, this paper constructs the VulTR model, which incorporates an importance assessment mechanism to strengthen the key syntax levels of the source code (from lexical elements to nodes and graph-level structures), significantly improving the importance of key vulnerability features in classification decisions. At the same time, a relationship connection diagram is constructed to describe the spatial characteristics of the correlations between functions. Experimentally verified, VulTR's F1 scores on both synthetic and real data sets exceed those of the compared models (VulDeePecker, SySeVR, Devign, VulCNN, IVDetect, and mVulPreter).

A hierarchical GNN across semantic and topological domains for predicting circRNA-microRNA interactions

Identifying circRNA-microRNA interactions (CMI) is a significant biomedical issue in recent years. This problem provides insights into using circRNA as biomarkers, developing cancer therapies and producing cancer vaccines. Using computational methods for identification is a more time-efficient and cost-effective approach. In computational methods, using graphs to represent and explore the CMI is a mainstream approach. However, existing relevant methods do not achieve optimal results by utilizing both the semantic information extracted from sequences and the topological information extracted from graph structures. To address this issue, we propose HGLMALLM, a graph contrastive learning method that learns node representation crossing both the semantic domain generated via motif-aware pre-trained LLMs and the topological domain extracted from hierarchical graph structures. Our method effectively addresses the issue in existing Message Passing Neural Network (MPNN) method that edge components losing heterogeneity after multiple iterations. Moreover, this method utilizes the heterogeneity of graph which is extended from the traditional bipartite graph to heterogeneous through the semantic domain. Two commonly used datasets were partitioned based on the distribution of node degrees. Then, we benchmarked our method against existing methods. In the independent testing set evaluation, it achieved a 3 % and 1 % improvement on two datasets. Our method demonstrated the best stability in ten-fold cross-validation on the training set. A test conducted on the peripheral components reveals robust performance of our model. A dataset collected from real scenarios was used to demonstrate the strong predictive ability of our method for identifying unidentified CMI.

Gating-Enhanced Hierarchical Structure Learning in Hyperbolic Space and Multi-scale Neighbor Topology Learning in Euclidean Space for Prediction of Microbe-Drug Associations.

Identifying drug-related microbes may help us explore how the microbes affect the functions of drugs by promoting or inhibiting their effects. Most previous methods for the prediction of microbe-drug associations focused on integrating the attributes and topologies of microbe and drug nodes in Euclidean space. The heterogeneous network composed of microbes and drugs has a hierarchical structure, and the hyperbolic space is helpful for reflecting the structure. However, the previous methods did not fully exploit the structure. We propose a multi-space feature learning enhanced microbe-drug association prediction method, MFLP, to fuse the hierarchical structure of microbe and drug nodes in hyperbolic space and the multiscale neighbor topologies in Euclidean space. First, we project the nodes of the microbe-drug heterogeneous network on the sphere in hyperbolic space and then construct a topology which implies hierarchical structure and forms a hierarchical attribute embedding. The node information from multiple types of neighbor nodes with the new topological structure in the tangent plane space of a sphere is aggregated by the designed gating-enhanced hyperbolic graph neural network. Second, the gate at the node feature level is constructed to adaptively fuse the hierarchical features of microbe and drug nodes from two adjacent graph neural encoding layers. Third, multiple neighbor topological embeddings for each microbe and drug node are formed by neighborhood random walks on the microbe-drug heterogeneous network, and they cover neighborhood topologies with multiple scales, respectively. Finally, as each scale of topological embedding contains its specific neighborhood topology, we establish an independent graph convolutional neural network for the topology and form the topological representations of microbe and drug nodes in Euclidean space. The comparison experiments based on cross validation showed that MFLP outperformed several advanced prediction methods, and the ablation experiments verified the effectiveness of MFLP's major innovations. The case studies on three drugs further demonstrated MFLP's ability in being applied to discover potential candidate microbes for the given drugs.

Decoupled semantic graph neural network for knowledge graph embedding

Knowledge graph embedding (KGE) learns an embedding space for a more accurate representation of entities and relations. Although the KGE model has proliferated, it often fails to fully capture the rich semantics in knowledge graphs. Some studies attempt to apply decoupling methods to decompose node representations, yet they neglect the semantic noise generated during the decoupled process. To address these issues, we introduce the decoupled semantic graph network for KGE (named DSGNet), which employs a novel approach that combines semantic decoupling with structural awareness. DSGNet begins by projecting the knowledge graph into distinct semantic spaces, ensuring minimal correlation between them to achieve effective semantic decoupling. To mitigate semantic noise, a top-k sampling technique is applied to the decoupled graphs. DSGNet then aggregates these different semantic graphs using a relation-aware aggregation module, followed by a multi-layer aggregation process for enhanced node representation. The extensive experiments demonstrate that DSGNet performs competitively on two popular datasets. We make our code publicly available at https://github.com/HubuKG/DSGNet.

SiGNN: A spike-induced graph neural network for dynamic graph representation learning

In the domain of dynamic graph representation learning (DGRL), capturing the temporal evolution within real-world networks is of paramount importance. Spiking Neural Networks (SNNs), renowned for their temporal dynamics and low-power characteristics, provide an efficient solution for temporal processing in DGRL tasks. However, due to the spike-based information encoding mechanism of SNNs, existing DGRL methods that employ SNNs face significant limitations in their representational capacity. To address this issue, we propose an innovative framework named Spike-induced Graph Neural Network (SiGNN) for learning enhanced spatiotemporal representations on dynamic graphs. Specifically, we achieve a harmonious integration of SNNs and GNNs through a novel Temporal Activation (TA) mechanism. This mechanism enables SiGNN to effectively harness the temporal dynamics of SNNs while circumventing the representational constraints imposed by the binary nature of spikes. Furthermore, leveraging the inherent adaptability of SNNs, SiGNN incorporates an in-depth analysis of evolutionary patterns within graphs across multiple time granularities, thereby facilitating the acquisition of multiscale temporal node representations. Extensive experiments on various real-world dynamic graph datasets demonstrate the superior performance of SiGNN in the node classification task. Our code is publicly available at https://github.com/CharliedoD/SiGNN.

Adaptive denoising graph contrastive learning with memory graph attention for recommendation

Graph contrastive learning has emerged as a powerful technique for dealing with graph noise and mining latent information in networks, that has been widely applied in GNN-based collaborative filtering. Traditional graph contrastive learning methods commonly generate multiple augmented views, and then learn node representations by maximizing the consistency between these views. However, on one hand, manual view construction methods necessitate expert knowledge and a trial-and-error process. On the other hand, adaptive view construction methods require decoders which results in increased training costs. To address the aforementioned limitations, in this paper, we propose the Adaptive Denoising Graph Contrastive Learning with Memory Graph Attention for Recommendation (ADGA) framework. Firstly, we introduce the memory graph attention mechanism to capture node attention during multi-hop information aggregation. Then, unlike previous methods that required additional node representations to generate views, ADGA proposes, for the first time, directly using attention to adaptively generate structure-aware contrastive learning views. It reduces the training cost of the model and improves the cross-view consistency of node representations, that offers a new paradigm for adaptive graph contrastive learning. Experimental results on three real-world datasets demonstrate that ADGA achieves state-of-the-art performance in recommendation tasks. The code is available at https://github.com/Andrewsama/ADGA.

Graph pooling in graph neural networks: methods and their applications in omics studies

Graph neural networks (GNNs) process the graph-structured data using neural networks and have proven successful in various graph processing tasks. Currently, graph pooling operators have emerged as crucial components that bridge the gap between node representation learning and diverse graph-level tasks by transforming node representations into graph representations. Given the rapid growth and widespread adoption of graph pooling, this review aims to summarize the existing graph pooling operators for GNNs and their representative applications in omics. Specifically, we first present a comprehensive taxonomy of existing graph pooling algorithms, expanding the categorization for both global and hierarchical pooling operators, and for the first time reviewing the inverse operation of graph pooling, named unpooling. Next, we describe the general evaluation framework for graph pooling operators, encompassing three fundamental aspects: experimental setup, ablation analysis, and model interpretation. We also discuss open issues that significantly influence the design of graph pooling operators, including complexity, connectivity, adaptability, additional loss, and attention mechanisms. Finally, we summarize bioinformatics applications of graph pooling operators in omics, including graphs of gene interaction, medical images, and protein structures for drug discovery and disease diagnosis. Furthermore, we showcase the impact of graph pooling operators on research in specific real-world domains, with a focus on prediction performance and model interpretability. This review provides methodological insights in machine learning based graph modeling and related omics research, as well as an ongoing resource by gathering related papers and code in a dedicated GitHub repository (https://github.com/Hou-WJ/Graph-Pooling-Operators-and-Bioinformatics-Applications).

Knowledge-driven hierarchical intents modeling for recommendation

Previous studies on user–item interaction graphs have typically concentrated on simple interactions, often overlooking the significant role of user intent in shaping these interactions. While some recent research has explored intent relationships to enhance modeling, these approaches mainly focus on user preferences derived from interactions, ignoring the knowledge information with good interpretation in knowledge graphs. In addition, some recent work usually utilize undenoised graph structure information to learn the node representations, which introduces plenty of noise and impedes the well learning of users’ preference. In this paper, we utilize the rich interpretable knowledge information in the knowledge graph to design a novel knowledge-driven hierarchical intent modeling framework called KHIM. The focus is on designing a hierarchical user intent modeling process and an intent-based multi-view contrastive learning mechanism. The former extracts both the popular and personalized preferences of users from attribute tuples within the knowledge graph at the global-level and local-level, respectively. For global level, we capture the user’s true intents from positive view and augment the positive intent representation with negative view. While the latter generates high-quality user and item representations through multi-level cross-view contrastive learning. Two data augmentation strategies are designed during the contrastive process to mitigate the effects of noise in the learning process. Additionally, we also designed a neighbor filtering strategy based on semantic view to obtain more neighbor semantic information of user and item nodes, so as to further improve the recommendation performance. Experimental results on three benchmark datasets demonstrate that KHIM significantly outperforms various state-of-the-art approaches, highlighting its effectiveness in leveraging knowledge graph information for better recommendations.

Unsupervised multi-view graph representation learning with dual weight-net

Unsupervised multi-view graph representation learning (UMGRL) aims to capture the complex relationships in the multi-view graph without human annotations, so it has been widely applied in real-world applications. However, existing UMGRL methods still face the issues as follows: (i) Previous UMGRL methods tend to overlook the importance of nodes with different influences and the importance of graphs with different relationships, so that they may lose discriminative information in nodes with large influences and graphs with important relationships. (ii) Previous UMGRL methods generally ignore the heterophilic edges in the multi-view graph to possibly introduce noise from different classes into node representations. To address these issues, we propose a novel bi-level optimization UMGRL framework with dual weight-net. Specifically, the lower-level optimizes the parameters of encoders to obtain node representations of different graphs, while the upper-level optimizes the parameters of the dual weight-net to adaptively and dynamically capture the importance of node level, graph level, and edge level, thus obtaining discriminative fused representations for downstream tasks. Moreover, theoretical analysis demonstrates that the proposed method shows a better generalization ability on downstream tasks, compared to previous UMGRL methods. Extensive experimental results verify the effectiveness of the proposed method on public datasets, in terms of different downstream tasks, compared to numerous comparison methods.

SFGCN: Synergetic fusion-based graph convolutional networks approach for link prediction in social networks

Accurate Link Prediction (LP) in Social Networks (SNs) is crucial for various practical applications, such as recommendation systems and network security. However, traditional techniques often struggle to capture the intricate and multidimensional nature of these networks. This paper presents a novel approach, the Synergetic Fusion-based Graph Convolutional Networks (SFGCN), designed to enhance LP accuracy in SNs. The SFGCN model utilizes a fusion architecture that combines structural features and other attribute data through early, intermediate, and late fusion mechanisms to create improved node and edge representations. We thoroughly evaluate our SFGCN model on seven real-world datasets, encompassing citation networks, co-purchase networks, and academic publication domains. The results demonstrate its superiority over baseline GCN architectures and other selected LP methods, achieving a 6.88 % improvement in accuracy. The experiments show that our model captures the complex interactions and dependencies within SNs, providing a comprehensive understanding of their underlying dynamics. The approach mentioned can be effectively applied in the domain of SN analysis to enhance the accuracy of LP results. This method not only improves the precision of predictions but also enhances the adaptability of the model in diverse SN scenarios.

Temporal Graph Network for continuous-time dynamic event sequence

Continuous-Time Dynamic Graph (CTDG) methods have shown their superior ability in learning representations for dynamic graph-structured data, the methods split the sequential updating process into discrete batches to reduce the computation costs, as a result, the message constructor in existing CTDG methods cannot be optimized by gradient descent and is designed to be parameter-free. In particular, this layer fails to embed complex event subgraphs and ignores the structure information, while most real-world events are structured and complex. For example, a paper publication event in an academic graph contains different relations like authorship and citations. Furthermore, the corresponding nodes could not receive position-wise messages to make precise representation updates. To tackle this issue, we propose a new method called Temporal Graph Network for continuous-time dynamic Event sequence (TGNE) with a structure-aware message constructor to update node representation with complex event subgraph, by treating message construction and delivery as a message-passing process, in this way, the message constructor can be formalized as a graph neural network layer. TGNE extends the input of CTDG methods to subgraphs with complex structures and preserves more information in message delivery. Extensive experiments demonstrate that the proposed method can achieve competitive performance on traditional tasks on bipartite graphs and event sequence learning tasks on heterogeneous graphs.

Masked contrastive graph representation learning for age estimation

Age estimation of face images is a crucial task with various practical applications in areas such as video surveillance and Internet access control. While deep learning-based age estimation methods like Convolutional Neural Networks (CNN), Multilayer Perceptrons (MLP), and Transformers have shown remarkable performance, they have limitations when modeling complex or irregular objects, and often contain redundant information. Although existing graph neural networks (GNNs) can alleviate this issue, they only considers the structural information of the image and ignores the semantic features, thus the feature representation capability of graph nodes is limited. To address these issues, this paper proposes a novel Masked Contrastive Graph Representation Learning (MCGRL) method for accurate age estimation of face images. Our approach leverages CNN to extract semantic features of the image, which are then partitioned into patches that serve as nodes in the graph. We use a masked graph convolutional network (GCN) to derive image-based node representations that capture rich structural information. Finally, we incorporate multiple losses to explore the complementary relationship between structural information and semantic features, which improves the feature representation capability of GCN. Experimental results on real-world face image datasets demonstrate the superiority of our proposed method over other state-of-the-art age estimation approaches. Our code is available at https://github.com/yuntaoshou/MCGRL

Graph Representation Learning Based on Specific Subgraphs for Biomedical Interaction Prediction.

Discovering the novel associations of biomedical entities is of great significance and can facilitate not only the identification of network biomarkers of disease but also the search for putative drug targets.Graph representation learning (GRL) has incredible potential to efficiently predict the interactions from biomedical networks by modeling the robust representation for each node.> However, the current GRL-based methods learn the representation of nodes by aggregating the features of their neighbors with equal weights. Furthermore, they also fail to identify which features of higher-order neighbors are integrated into the representation of the central node. In this work, we propose a novel graph representation learning framework: a multi-order graph neural network based on reconstructed specific subgraphs (MGRS) for biomedical interaction prediction. In the MGRS, we apply the multi-order graph aggregation module (MOGA) to learn the wide-view representation by integrating the multi-hop neighbor features. Besides, we propose a subgraph selection module (SGSM) to reconstruct the specific subgraph with adaptive edge weights for each node. SGSM can clearly explore the dependency of the node representation on the neighbor features and learn the subgraph-based representation based on the reconstructed weighted subgraphs. Extensive experimental results on four public biomedical networks demonstrate that the MGRS performs better and is more robust than the latest baselines.

Demystifying and Mitigating Bias for Node Representation Learning.

Node representation learning has attracted increasing attention due to its efficacy for various applications on graphs. However, fairness is a largely under-explored territory within the field, although it is shown that the use of graph structure in learning amplifies bias. To this end, this work theoretically explains the sources of bias in node representations obtained via graph neural networks (GNNs). It is revealed that both nodal features and graph structure lead to bias in the obtained representations. Building upon the analysis, fairness-aware data augmentation frameworks are developed to reduce the intrinsic bias. Our theoretical analysis and proposed schemes can be readily employed in understanding and mitigating bias for various GNN-based learning mechanisms. Extensive experiments on node classification and link prediction over multiple real networks are carried out, and it is shown that the proposed augmentation strategies can improve fairness while providing comparable utility to state-of-the-art methods.

Learning Joint 2-D and 3-D Graph Diffusion Models for Complete Molecule Generation.

Designing new molecules is essential for drug discovery and material science. Recently, deep generative models that aim to model molecule distribution have made promising progress in narrowing down the chemical research space and generating high-fidelity molecules. However, current generative models only focus on modeling 2-D bonding graphs or 3-D geometries, which are two complementary descriptors for molecules. The lack of ability to jointly model them limits the improvement of generation quality and further downstream applications. In this article, we propose a joint 2-D and 3-D graph diffusion model (JODO) that generates geometric graphs representing complete molecules with atom types, formal charges, bond information, and 3-D coordinates. To capture the correlation between 2-D molecular graphs and 3-D geometries in the diffusion process, we develop a diffusion graph transformer (DGT) to parameterize the data prediction model that recovers the original data from noisy data. The DGT uses a relational attention mechanism that enhances the interaction between node and edge representations. This mechanism operates concurrently with the propagation and update of scalar attributes and geometric vectors. Our model can also be extended for inverse molecular design targeting single or multiple quantum properties. In our comprehensive evaluation pipeline for unconditional joint generation, the experimental results show that JODO remarkably outperforms the baselines on the QM9 and GEOM-Drugs datasets. Furthermore, our model excels in few-step fast sampling, as well as in inverse molecule design and molecular graph generation. Our code is provided in https://github.com/GRAPH-0/JODO.