Graph Space Research Articles

Efficient generation of grids and traversal graphs in compositional spaces towards exploration and path planning

AbstractDiverse disciplines across science and engineering deal with problems related to compositions, which exist in non-Euclidean simplex spaces, rendering many standard tools inaccurate or inefficient. This work explores such spaces conceptually in the context of materials discovery, quantifies their computational feasibility, and implements several essential methods specific to simplex spaces through a new high-performance open-source library . Most significantly, we derive and implement an algorithm for constructing a novel n-dimensional simplex graph data structure, containing all discretized compositions and possible neighbor-to-neighbor transitions. Critically, no distance or neighborhood calculations are performed, instead leveraging pure combinatorics and order in procedurally generated simplex grids, keeping the algorithm $${\mathcal{O}}(N)$$ O ( N ) , with minimal memory, enabling rapid construction of graphs with billions of transitions in seconds. Additionally, we demonstrate how such graph representations can be combined to homogeneously express complex path-planning problems, while facilitating efficient deployment of existing high-performance gradient descent, graph traversal, and other optimization algorithms.

Intrinsically Hölder Sections in Metric Spaces

We introduce the notion of intrinsically Holder graphs in metric spaces that generalized the one of ¨ intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to Holder maps but ¨ is profoundly different from them. We prove some relevant results as the Ascoli-Arzela compactness Theorem, ` Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over R or C, a convex set and an equivalence relation for intrinsically Holder graphs. These last three properties are new also in the Lipschitz case.

Ensemble-Enhanced Semi-Supervised Learning with Optimized Graph Construction for High-Dimensional Data.

Graph-based methods have demonstrated exceptional performance in semi-supervised classification. However, existing graph-based methods typically construct either a predefined graph in the original space or an adaptive graph within the output space, which often limits their ability to fully utilize prior information and capture the optimal intrinsic data distribution, particularly in high-dimensional data with abundant redundant and noisy features. This paper introduces a novel approach: Semi-Supervised Classification with Optimized Graph Construction (SSC-OGC). SSC-OGC leverages both predefined and adaptive graphs to explore intrinsic data distribution and effectively employ prior information. Additionally, a graph constraint regularization term (GCR) and a collaborative constraint regularization term (CCR) are incorporated to further enhance the quality of the adaptive graph structure and the learned subspace, respectively. To eliminate the negative effect of constructing a predefined graph in the original data space, we further propose a Hybrid Subspace Ensemble-enhanced framework based on the proposed Optimized Graph Construction method (HSE-OGC). Specifically, we construct multiple hybrid subspaces, which consist of meticulously chosen features from the original data to achieve high-quality and diverse space representations. Then, HSE-OGC constructs multiple predefined graphs within hybrid subspaces and trains multiple SSC-OGC classifiers to complement each other, significantly improving the overall performance. Experimental results conducted on various high-dimensional datasets demonstrate that HSE-OGC exhibits outstanding performance.

RNA fold prediction by Monte Carlo in graph space and the statistical mechanics of tertiary interactions.

Using a graph representation of RNA structures, we have studied the ensembles of secondary and tertiary graphs two sets of RNA with Monte Carlo simulations. The first consisted of 91 target ribozyme and riboswitch sequences of moderate lengths (< 150 nt) having a variety of secondary, H-type pseudoknots and kissing loop interactions. The second set consisted of 71 more diverse sequences across many RNA families. Using a simple empirical energy model for tertiary interactions and only sequence information for each target as input, the simulations examined how tertiary interactions impact the statistical mechanics of the fold ensembles. The results show that the graphs proliferate enormously when tertiary interactions are possible, producing an entropic driving force for the ensemble to access folds having tertiary structures even though they are overall energetically unfavorable in the energy model. For each of the targets in the two test sets, we assessed the quality of the model and the simulations by examining how well the simulated structures were able to predict the native fold and compared the results to fold predictions from ViennaRNA. Our model generated good or excellent predictions in a large majority of the targets. Overall, this method was able to produce predictions of comparable quality to Vienna, but it outperformed Vienna for structures with H-type pseudoknots. The results suggest that while tertiary interactions are predicated on real-space contacts, their impacts on the folded structure of RNA can be captured by graph space information for sequences of moderate lengths, using a simple tertiary energy model for the loops, the base pairs and base stacks.

Remote Sensing LiDAR and Hyperspectral Classification with Multi-Scale Graph Encoder–Decoder Network

The rapid development of sensor technology has made multi-modal remote sensing data valuable for land cover classification due to its diverse and complementary information. Many feature extraction methods for multi-modal data, combining light detection and ranging (LiDAR) and hyperspectral imaging (HSI), have recognized the importance of incorporating multiple spatial scales. However, effectively capturing both long-range global correlations and short-range local features simultaneously on different scales remains a challenge, particularly in large-scale, complex ground scenes. To address this limitation, we propose a multi-scale graph encoder–decoder network (MGEN) for multi-modal data classification. The MGEN adopts a graph model that maintains global sample correlations to fuse multi-scale features, enabling simultaneous extraction of local and global information. The graph encoder maps multi-modal data from different scales to the graph space and completes feature extraction in the graph space. The graph decoder maps the features of multiple scales back to the original data space and completes multi-scale feature fusion and classification. Experimental results on three HSI-LiDAR datasets demonstrate that the proposed MGEN achieves considerable classification accuracies and outperforms state-of-the-art methods.

Enhanced tensor based embedding anchor learning for multi-view clustering

Existing anchor graph based multi-view clustering methods can overcome the problem of high computational cost in traditional multi-view clustering methods. However, the anchor points selected from high-dimensional data often contain irrelevant noise and outliers, which would affect the clustering performance. To address this issue, we propose an embedding anchor based multi-view clustering method, called enhanced tensor based embedding anchor learning (ETEAL). Specifically, we unify the learning process of latent embedding space, anchor points, and anchor graphs into a common framework, which eliminates noise and redundant data in the original space and enhances the synergistic optimization between the components. Meanwhile, we employ an enhanced tensor strategy to constrain the embedding anchor graphs, which exploits the higher-order relationships between views and recovers the global low-rank property of the embedding anchor graphs. Finally, we develop an anchor graph fusion strategy, which significantly reduces the huge overhead of traditional graph fusion that requires the construction of complete graphs. Experimental results on eight benchmark datasets show that the proposed method significantly outperforms other state-of-the-art methods in terms of scalability and clustering accuracy.

S2MRI-ADNet: an interpretable deep learning framework integrating Euclidean-graph representations of Alzheimer's disease solely from structural MRI.

To establish a multi-dimensional representation solely on structural MRI (sMRI) for early diagnosis of AD. A total of 3377 participants' sMRI from four independent databases were retrospectively identified to construct an interpretable deep learning model that integrated multi-dimensional representations of AD solely on sMRI (called s2MRI-ADNet) by a dual-channel learning strategy of gray matter volume (GMV) from Euclidean space and the regional radiomics similarity network (R2SN) from graph space. Specifically, the GMV feature map learning channel (called GMV-Channel) was to take into consideration spatial information of both long-range spatial relations and detailed localization information, while the node feature and connectivity strength learning channel (called NFCS-Channel) was to characterize the graph-structured R2SN network by a separable learning strategy. The s2MRI-ADNet achieved a superior classification accuracy of 92.1% and 91.4% under intra-database and inter-database cross-validation. The GMV-Channel and NFCS-Channel captured complementary group-discriminative brain regions, revealing a complementary interpretation of the multi-dimensional representation of brain structure in Euclidean and graph spaces respectively. Besides, the generalizable and reproducible interpretation of the multi-dimensional representation in capturing complementary group-discriminative brain regions revealed a significant correlation between the four independent databases (p < 0.05). Significant associations (p < 0.05) between attention scores and brain abnormality, between classification scores and clinical measure of cognitive ability, CSF biomarker, metabolism, and genetic risk score also provided solid neurobiological interpretation. The s2MRI-ADNet solely on sMRI could leverage the complementary multi-dimensional representations of AD in Euclidean and graph spaces, and achieved superior performance in the early diagnosis of AD, facilitating its potential in both clinical translation and popularization.

Curvature estimates for spacelike stationary graphs in the 4-dimensional Lorentz space

Curvature estimates for spacelike stationary graphs in the 4-dimensional Lorentz space

Parallel Sampling of Decomposable Graphs Using Markov Chains on Junction Trees

Bayesian inference for undirected graphical models is mostly restricted to the class of decomposable graphs, as they enjoy a rich set of properties making them amenable to high-dimensional problems. While parameter inference is straightforward in this setup, inferring the underlying graph is a challenge driven by the computational difficulty in exploring the space of decomposable graphs. This work makes two contributions to address this problem. First, we provide sufficient and necessary conditions for when multi-edge perturbations maintain decomposability of the graph. Using these, we characterize a simple class of partitions that efficiently classify all edge perturbations by whether they maintain decomposability. Second, we propose a novel parallel nonreversible Markov chain Monte Carlo sampler for distributions over junction tree representations of the graph. At every step, the parallel sampler executes simultaneously all edge perturbations within a partition. Through simulations, we demonstrate the efficiency of our new edge perturbation conditions and class of partitions. We find that our parallel sampler yields improved mixing properties in comparison to the single-move variate, and outperforms current state-of-the-art methods in terms of accuracy and computational efficiency. The implementation of our work is available in the Python package parallelDG. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Robertson’s conjecture and universal finite generation in the homology of graph braid groups

We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest.

SGNN-T: Space graph neural network coupled transformer for molecular property prediction

Molecular properties play a crucial role in material discovery, protein interaction and drug development. The appearance of Graph Neural Network (GNN) significantly improved the performance of molecular property prediction. However, nodes in GNN only update the features of neighbor nodes, resulting in insufficient ability to encode global feature information. The self- attention mechanism in transformer can encode the global information except for local information of molecules, while its spatial information is insufficient. Since molecules are three-dimensional spatial structures, spatial geometry information is an important attribute for molecules properties. To consider these factors, a network model of Space Graph Neural Network coupled Transformer (SGNN-T) is proposed in this paper which can combine global and local molecule information with three-dimensional spatial structures for molecular properties prediction. In this model, Graph neural network Geometric Feature Fusion Module (GGFF) and Transformer Spatial Geometric Feature Enhancement Module (TSGFE) are included to enhance the spatial geometry learning ability of the network. The GGFF module constructs a parallel graph neural network by thinking over atoms, bonds and bond angles at the same time which effectively complements the spatial information of the network by leading into bond angles than normal GNN. The TSGFE module introduces the coordinates and centrality degree features coupled with the features by GGFF into transformer to further enhance the geometric expression ability of the module. Through these two parts, SGNN-T model can encode local and global information of molecules at the same time. Property prediction experiments are executed on the QM9, OMDB and MEGNet dataset. The results of MAE show the proposed model has the best performance than the popular models.

Distributed Ishikawa algorithms for seeking the fixed points of multi-agent global operators over time-varying communication graphs

In this article, the problem of seeking fixed points for global operators over the time-varying graphs in a real Hilbert space is studied. The global operator is a linear combination of local operators, each local operator being accessed privately by one agent for less resource consumption. All agents form a network and they need to cooperate to solve problems. To this end, on the basis of the centralized Ishikawa iteration, the distributed Ishikawa algorithm (D-I) is first proposed. In the sequel, to predigest the calculational complexity, further considering the situation that only the random part of each operator coordinate is calculated in each iteration, the distributed block coordinate Ishikawa algorithm (D-BI) is also designed. The results indicate that the proposed D-I and D-BI algorithms can weakly converge to a fixed point of the multi-agent global operator. Eventually, we give a few numerical examples to illustrate practical benefits of the proposed algorithms.

Strongly 𝑫𝜶𝒑 −closed graphs in bitopological spaces

In this paper, we introduce the concepts of D_α^p-open, D_α^p-closed subsets, pairwise-α-closed, pairwise-g-closed subsets, pairwise-strongly α-closed graph G(f) and strongly D_α^p-closed graph of bitopological spaces. We showed that each closed graph is D_α^p-closed. In addition, the concepts of D_α^p-continuous, open, and closed functions are defined, and the relations between τ_p-α, τ_p-g, and D_α^p-continuous functions are clarified. The fact that strongly D_α^p-closed graph is D_α^p-closed is illustrated. We studied when the graph G(f) is p-strongly closed and p-D_α-closed subsets of the bitopological space (X,τ₁,τ₂). Moreover, the notions of D_α^i-interior of a subset of X and D_α^i-closure are defined.

GMMDA: Gaussian mixture modeling of graph in latent space for graph data augmentation

Graph data augmentation (GDA), which manipulates graph structure and/or attributes, has been demonstrated as an effective method for improving the generalization of graph neural networks on semi-supervised node classification. As a data augmentation technique, label preservation is critical, that is, node labels should not change after data manipulation. However, most existing methods overlook the label preservation requirements. Determining the label-preserving nature of a GDA method is highly challenging, owing to the non-Euclidean nature of the graph structure. In this study, for the first time, we formulate a label-preserving problem (LPP) in the context of GDA. The LPP is formulated as an optimization problem in which, given a fixed augmentation budget, the objective is to find an augmented graph with minimal difference in data distribution compared to the original graph. To solve the LPP problem, we propose GMMDA, a generative data augmentation (DA) method based on Gaussian mixture modeling (GMM) of a graph in a latent space. We designed a novel learning objective that jointly learns a low-dimensional graph representation and estimates the GMM. The learning is followed by sampling from the GMM, and the samples are converted back to the graph as additional nodes. To uphold label preservation, we designed a minimum description length (MDL)-based method to select a set of samples that produces the minimum shift in the data distribution captured by the GMM. Through experiments, we demonstrate that GMMDA can improve the performance of graph convolutional network on Cora, Citeseer and Pubmed by as much as 7.75%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$7.75\\%$$\\end{document}, 8.75%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8.75\\%$$\\end{document} and 5.87%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5.87\\%$$\\end{document}, respectively, significantly outperforming the state-of-the-art methods.

SPORT: A Subgraph Perspective on Graph Classification with Label Noise

Graph neural networks (GNNs) have achieved great success recently on graph classification tasks using supervised end-to-end training. Unfortunately, extensive noisy graph labels could exist in the real world because of the complicated processes of manual graph data annotations, which may significantly degrade the performance of GNNs. Therefore, we investigate the problem of graph classification with label noise, which is demanding because of the complex graph representation learning issue and serious memorization of noisy samples. In this work, we present a novel approach called S ubgra p h Set Netw or k with Sample Selection and Consis t ency Learning (SPORT) for this problem. To release the overfitting of GNNs, SPORT proposes to characterize each graph as a set of subgraphs generated by certain predefined stratagems, which can be viewed as samples from its underlying semantic distribution in graph space. Then we develop an equivariant network to encode the subgraph set with the consideration of the symmetry group. To further release the influences of noisy examples, we leverage the predictions of subgraphs to measure the likelihood of a sample being clean or noisy, followed by effective label updating. In addition, we propose a joint loss to advance the model generalizability by introducing consistency regularization. Comprehensive experiments on a wide range of graph classification datasets demonstrate the effectiveness of our SPORT. Specifically, SPORT outperforms the most competing baseline by up to 6.4%.

Exploring the space of graphs with fixed discrete curvatures

Abstract Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. In this work, we consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. We address this problem in two ways: first, we develop an evolutionary algorithm to sample graphs with discrete curvatures close to a given set. We use this algorithm to explore how other network statistics vary when constrained by the discrete curvatures in the network. Second, we solve the exact reconstruction problem for the specific case of Forman–Ricci curvature. By leveraging the theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature.

GeoExplainer: Interpreting Graph Convolutional Networks with geometric masking

For Graph Convolutional Network (GCN), the inherent geometric topology and relationships within spatio-temporal graphs are critical to factors affect its predictive performance. Understanding the decision-making process of GCN remains a challenging task. Existing perturbation-based explanation methods are primarily tailored for Graph Neural Network (GNN) models and focus on how the information of nodes or subgraphs in static graphs impacts GNN predictions. Due to the geometric properties of spatio-temporal graphs, screw theory serves as crucial mathematical tools for describing the geometric relationships and patterns of motion within the geometric space of spatio-temporal graphs. Therefore,we propose GeoExplainer, which utilizes screw theory to geometric masking for spatio-temporal graphs with strong geometric correlations, generating explanations for GCN. Our method perturbs spatio-temporal graphs using geometric masking, and ultimately generating explanations in the form of geometric masks serve as importance mappings for model prediction results. Firstly, we design a screw motion vector model with geometric constraints to describe the spatio-temporal graph in screw system. Then, we design a geometric screw perturbation to generate smooth geometric masks. Finally, we propose a geometric masking approach to optimize and update the geometric mixture mask. We utilize pointwise mutual information (PMI) to maximize the correlation between the model predictions after geometric perturbation and the original predictions, obtaining the optimal geometric mask as an explanation for GCN. The interpretable experiments of ST-GCN, HD-GCN, GCN and GIN models on four datasets with geometric relationships demonstrate that GeoExplainer outperforms eight competitive interpretation methods in quantitative and visualization analysis.

Theoretical analysis and implementation of mandatory operations-based accelerated search in graph space for hybrid flow shop scheduling

For the hybrid flow shop scheduling problem (HFSP), this study addresses its high computational burden associated with insertion operations due to over-much objective function calculations as the number of jobs increases. Unfortunately, the search solution space and existing accelerated local search methods exhibit limitations. We find that there are many operations on the critical path that do not affect the completion time, especially when there are multiple critical paths. However, deleting mandatory operations must make the original makespan value smaller. Moreover, only specific insertion positions have the potential to decrease the makespan of the changed sequence. In view of this, we emphasize the need for improved accelerated local search techniques tailored to overcome these limitations, and propose the theoretical analysis and methods of an accelerated local search based on effective insertion of mandatory operations in the graph for the HFSP with makespan criterion. More specifically, first, we represent a solution using a solution graph, and introduce four definitions related to mandatory operations in the graph space. Second, we propose five theorems and their corresponding proofs. Third, we present a mandatory operations-based accelerated iterated greedy algorithm (MOAIG) in the graph space according to the above theorems. Finally, we conduct many experiments on two well-known benchmarks (a total of 576 instances). Through statistical analysis, it becomes evident that the computational time required for calculating the makespan associated with operations-based insertion has decreased significantly. The development of the mandatory operations-based effective insertion accelerated local search is a promising approach to significantly mitigate the time complexity by reducing the numbers of operations perturbed and insertion positions involved in the calculation, and makes up for gaps in existing accelerated insertion methods for HFSP.

Consensus local graph for multiple kernel clustering

Due to the ability to represent the relationship among data, graph has received extensive attention in clustering field. As illustrated by existing studies, learning graph in kernel space is an effective way to capture the structure of data. Among kernel-based methods, multiple kernel learning (MKL) is increasingly popular since it can automatically utilize the complementary information contained in base kernels. Although many existing MKL-based graph learning algorithms have promising learning abilities, we observe that they (1) put too much attention on learning the consensus kernel which probably contains lots of redundant information and loses the diversity of base kernels; (2) ignore the local structure of the representations in multiple kernel spaces. To eliminate the bad effect of these issues, we propose a novel graph learning method, termed consensus local graph based on MKL (CLGMKL), for clustering. In CLGMKL, a low-rank kernel matrix is used to extract the discriminative information of each base kernel and a local graph is constructed to capture the local structure contained in each new kernel. Then CLGMKL combines these local graphs in a self-weighed way. Since the above processes are associated with each other, we jointly optimize them to obtain the overall optimality. An effective learning scheme with proved convergence is developed to optimize the combined objective function. Finally, extensive experiments on some popular datasets are conducted to test the effectiveness of the presented method. As illustrated, CLGMKL is more competitive than the state-of-the-art algorithms.

Boosting the performance of molecular property prediction via graph–text alignment and multi-granularity representation enhancement

Deep learning is playing an increasingly important role in accurate prediction of molecular properties. Prior to being processed by a deep learning model, a molecule is typically represented in the form of a text or a graph. While some methods attempt to integrate these two forms of molecular representations, the misalignment of graph and text embeddings presents a significant challenge to fuse two modalities. To solve this problem, we propose a method that aligns and fuses graph and text features in the embedding space by using contrastive loss and cross attentions. Additionally, we enhance the molecular representation by incorporating multi-granularity information of molecules on the levels of atoms, functional groups, and molecules. Extensive experiments show that our model outperforms state-of-the-art models in downstream tasks of molecular property prediction, achieving superior performance with less pretraining data. The source codes and data are available at https://github.com/zzr624663649/multimodal_molecular_property.