Graph Representation Learning

The quality of graph-structured data is fundamental to the success of modern graph analysis techniques such as Graph Neural Networks (GNNs). However, real-world graph data is often suboptimal, suffering from issues such as noise and incomplete connections. Graph Structure Learning (GSL) has emerged as a promising technique that adaptively optimizes node connections. However, we observe that the effectiveness of GSL often comes at the cost of a dramatic expansion in edge count, resulting in significant storage and computational overhead. In this work, we reveal that this limitation stems from the prevalent use of similarity-based edge construction, which predominantly connects highly similar neighbors based on their embeddings, introducing substantial structure redundancy. To address this, we propose a novel Informative Graph Structure Learning method (InGSL), which jointly considers both similarity and diversity in edge construction by incorporating a mutual-information-guided learning strategy. Notably, InGSL serves as a plug-in module that can be seamlessly integrated into existing GSL frameworks. Through extensive experiments on six representative GSL methods, we demonstrate that InGSL achieves significant performance improvements at a reduced number of edges.

Security knowledge graphs can provide computable external memory for security agents, but constructing them from long-form cyber threat intelligence (CTI) remains difficult: LLMs often lack grounded security-domain knowledge, and end-to-end document-to-graph training is hard to supervise with cheap, stable rewards. We present GRID (Graph Representation of Intelligence Data), an end-to-end framework for security text knowledge graph construction. GRID first builds security-domain supervision from CTI articles by creating traceable article-graph alignments through graph extraction and knowledge-graph-conditioned text revision. It then turns document-to-graph learning into a scripted task bank combining four-option multi-select questions with triple-level regex matching targets, yielding more stable task-specific rewards than repeatedly scoring full graph outputs with an LLM judge. Using this supervision pipeline, we train two Qwen3-4B-Instruct-2507-based 4B extractors: a primary Task-bank Reward model and a secondary End2End Reward model with LLM-as-judge precision/recall rewards. On 249 CTI articles from GRID, CASIE, CTINexus, MalKG, and SecureNLP, the Task-bank Reward model with the ontology-guided GRID extraction pipeline reaches 84.62% source-averaged precision, 64.91% source-averaged recall, and 68.53% Avg F1, achieving the best source-averaged recall and near-top Avg F1 with lower token usage and deployment cost. The End2End Reward model reaches 76.91% precision, 53.85% recall, and 58.06% Avg F1. Further analyses show that task-bank rewards can be built once offline and reused across later post-training runs, outperforming online End2End LLM-as-judge reward and weaker alternatives such as Choice-only Reward and End2End SFT without RL.

Graph Neural Networks (GNNs) have become the dominant framework for inductive graph-level learning. Yet most benchmarks focus on the regime $n \gg p$, where the number of graphs $n$ greatly exceeds the number of nodes per graph $p$. This overlooks biological domains such as omics, which operate in the opposite $n \ll p$ regime, characterized by large graphs of genes, transcripts, or proteins across few patient samples. This raises the question: \textit{how do GNNs perform in this low-sample, high-node omics setting?} We introduce \texttt{OgBench} (Omics-Graph Bench), the first benchmarking platform for graph-level prediction in the $n \ll p$ regime characteristic of omics data. We provide a standardized, end-to-end modular infrastructure from raw omics data to families of featured graphs with varied structural properties. We benchmark classical GNNs, as well as GNNs designed for large graphs and omics applications, alongside MLPs and machine learning baselines to establish reference performances. Our results show that widely used GNNs often do not outperform simple MLPs and classical baselines. These findings challenge the prevailing assumption that graph structure inherently adds value in this domain, fostering a critical reassessment of current learning paradigms. Ultimately, by exposing these limitations, OgBench provides the open-source ecosystem necessary for the community to develop and validate novel architectures explicitly tailored for biological graphs. The code is available at https://github.com/geometric-intelligence/ogbench.

We introduce an empowered transposed Fully Connected Weighted (t-FCW) graph representation to embed point clouds into a metric space. While original t-FCW has shown promising results for point cloud classification, the reasons behind its effectiveness and its broader applicability remained unclear. In this work, we analyze the properties that make the empowered and original t-FCW effective and design a network that uses the empowered t-FCW exclusively as feature extractors. From an interpretability perspective, we build memory banks for classification, part segmentation, and semantic segmentation using the empowered t-FCW. Our analysis reveals that the empowered t-FCW inherits robustness from surface descriptors, provides interpretability through dimension-wise relations. These properties enable a highly efficient and interpretable network, which processes the ModelNet40 classification problem in approximately 7 seconds on an NVIDIA RTX A5000 GPU. Importantly, empowered t-FCW can function both as a lightweight standalone baseline and as a complementary plug-in to existing deep models.

Graph neural networks are widely used for node classification, but they remain vulnerable to out-of-distribution (OOD) shifts in node features and graph structure. Prior work established that methods trained with standard supervised learning (SL) objectives tend to capture spurious signals from either features and/or structure, leaving the model fragile under distributional changes. To address this, we propose TIDE, a novel and effective Tri-Component Information Decomposition framework that explicitly decomposes information into feature-specific, structure-specific and joint components. TIDE aims to preserve only the label-relevant part of the joint information while filtering out spurious feature- and structure-specific information, thereby enhancing the separation between in-distribution (ID) and OOD nodes. Beyond the framework, we provide theoretical and empirical analyses showing that an information bottleneck objective is preferable to standard SL for graph OOD detection, with higher ID confidence and a greater entropy gap between ID and OOD data. Extensive experiments across seven datasets confirm the efficacy of TIDE, achieving up to a 34% improvement in FPR95 over strong baselines while maintaining competitive ID accuracy.

Graph coarsening is a graph dimensionality reduction technique that aims to construct a smaller and more tractable graph while preserving the essential structural and semantic properties of the original graph. However, most existing methods rely on pair-wise similarity matching, where each node independently searches for its best partner based on global information. This selfishness matching paradigm incurs substantial computational and memory overhead. To address this problem, we shift to a non-selfishness principle that prioritizes the collective interference of neighborhood in coarsening, and propose an efficient method named NOPE, which achieves linear memory consumption and near-linear computational complexity in the number of nodes. Furthermore, we derive a faster variant NOPE*, which reduces O(δ\dot d) interference evaluation to O(d) based on the local isotropy assumption, and consequently alleviates the computational bottleneck for high-degree nodes. Experimental results show that NOPE* achieves 1.8-10\times speedup over NOPE and surpass almost all baselines with 1-3 orders of magnitude acceleration. Meanwhile, learning on coarsened graphs yields comparable performance to original graphs, and can even show superior performance over LLM-based graph reasoning owing to compact graph information. The code can be available at https://github.com/dazonglian/NOPE-main.

Continual graph learning (CGL) aims to learn from dynamically evolving graphs while mitigating catastrophic forgetting. Existing CGL approaches typically adopt a task-based formulation, where the data stream is partitioned into a sequence of discrete tasks with pre-defined boundaries. However, such assumptions rarely hold in real-world environments, where data distributions evolve continuously and task identity is often unavailable. To better reflect realistic non-stationary environments, we revisit continual graph learning from a task-free perspective. We propose a unified formulation that models the data stream as a time-varying mixture of latent task distributions, enabling continuous modeling of distribution drift. Based on this formulation, we construct \emph{DRIFT}, a benchmark that spans a spectrum of transition dynamics ranging from hard task switches to smooth distributional drift through a Gaussian parameterization. We evaluate representative continual learning methods under this task-free setting and observe substantial performance degradation compared to traditional task-based protocols. Our findings indicate that many existing approaches implicitly rely on task boundary information and struggle under realistic task-free graph streams. This work highlights the importance of studying continual graph learning under realistic non-stationary conditions and provides a benchmark for future research in this direction. Our code is available at https://github.com/UConn-DSIS/DRIFT.

Financial transaction fraud prevention faces challenges such as complex relationship structures, concealed behavioral patterns, and dynamically changing data distribution. Discrimination models relying solely on independent sample features are insufficient to fully characterize the risks of group collaboration and chain transfers within transaction networks. This paper proposes a graph neural network representation learning and risk discrimination framework for financial transaction fraud prevention. It integrates transaction records and identity information into node attributes and constructs a transaction graph based on shared attributes and interaction consistency to explicitly model inter-transaction relationships. In model design, a multi-layer message passing mechanism is employed to aggregate neighborhood information, learn node embedding representations containing structural context semantics, and output transaction-level fraud probability and risk scores through a lightweight risk discrimination head. A weighted supervision objective is introduced to mitigate training bias caused by class imbalance, and structural consistency regularization constraints are combined to suppress the impact of noisy edges on representation drift, thereby improving the stability and usability of risk characterization. Experiments are conducted on a publicly available financial transaction dataset, comparing various methods in the same direction and comprehensively evaluating them under a unified evaluation protocol. The results show that the proposed method outperforms other methods in risk ranking and probability calibration quality, validating the effectiveness of graph structure modeling and representation learning collaboration in financial transaction fraud prevention.

Graph Self-Supervised Learning (GSSL) has emerged as a powerful paradigm for generating high-quality representations for graph-structured data. While multi-scale graph contrastive learning has received increasing attention, many existing methods still predominantly focus on a single graph abstraction level. To address this limitation, we propose a unified contrastive framework that can target node-level, proximity-level, cluster-level, and graph-level information and integrate them through a linear combination of similarity scores on positive pairs and dissimilarity scores (i.e., similarity scores on negative pairs). Furthermore, current approaches typically assign uniform penalty strengths to all examples, which reduces optimization flexibility and leads to ambiguous convergence status. To overcome this, we introduce a novel parameter-free fine-grained self-weighting mechanism that adaptively assigns weights to individual similarity and dissimilarity scores. The proposed mechanism emphasizes the scores that deviate significantly from their target values. Our approach not only enhances optimization flexibility but also eliminates the computational overhead of hyperparameter tuning in conventional multi-task GSSL methods. Comprehensive experiments on real-world datasets show that our methods consistently outperform state-of-the-art approaches across downstream tasks, including classification, clustering, and link prediction, in both single-level and multi-level scenarios.

Segmenting small and sparse structures in large-scale images is fundamentally constrained by voxel-level, lattice-bound computation and extreme class imbalance -- dense, full-resolution inference scales poorly and forces most pipelines to rely on fixed regionization or downsampling, coupling computational cost to image resolution and attenuating boundary evidence precisely where minority structures are most informative. We introduce SEMIR (Semantic Minor-Induced Representation Learning), a representation framework that decouples inference from the native grid by learning a task-adapted, topology-preserving latent graph representation with exact decoding. SEMIR transforms the underlying grid graph into a compact, boundary-aligned graph minor through parameterized edge contraction, node deletion, and edge deletion, while preserving an exact lifting map from minor predictions to lattice labels. Minor construction is formalized as a few-shot structure learning problem that replaces hand-tuned preprocessing with a boundary-alignment objective: minor parameters are learned by maximizing agreement between predicted boundary elements and target-specific semantic edges under a boundary Dice criterion, and the induced minor is annotated with scale- and rotation-robust geometric and intensity descriptors and supports efficient region-level inference via message passing on a graph neural network (GNN) with relational edge features. We benchmark SEMIR on three tumor segmentation datasets -- BraTS 2021, KiTS23, and LiTS -- where targets exhibit high structural variability and distributional uncertainty. SEMIR yields consistent improvements in minority-structure Dice at practical runtime. More broadly, SEMIR establishes a framework for learning task-adapted, topology-preserving latent representations with exact decoding for high-resolution structured visual data.

Message-passing based approaches form the default backbone of most learning architectures on graph-structured data. However, the rapid progress of modern deep learning architectures in other domains, particularly sequence modeling, raises the question of how graph learning can benefit from these advances. We introduce Linearized Graph Sequence Models, a framework that recasts message-passing graph computation from the perspective of sequence modeling to simplify architectural choices. Our approach systematically separates the computational processing depth from the information propagation depth, allowing core graph architectural decisions to be treated as sequence modeling choices. Specifically, we analyze, both empirically and theoretically, what sequence properties make methods effective for learning and preserving the graph inductive bias. In particular, we validate our findings, demonstrating improved performance on long-range information tasks in graphs. Our findings provide a principled way to integrate modern sequence modeling advances into message-passing based graph learning. Beyond this, our work demonstrates how the separation of processing and information depth can recast central architectural questions as input modeling choices.

Graph-based machine learning has emerged as a promising approach for manufacturability analysis by learning directly from CAD models represented as Boundary Representations (B-reps), exploiting both surface geometry and topological connectivity. However, purely geometric representations often lack the process-specific semantics required for accurate manufacturability prediction: many manufacturing factors, such as surface roles or bend intent, are not explicitly encoded in shape alone and are difficult for data-driven models to infer reliably. We propose a hybrid approach that addresses this challenge by enriching B-rep attributed adjacency graphs with manufacturing features recognized through a rule-based module. Applied to sheet metal bending, recognized features, such as bend characteristics, flange lengths, and surface roles are integrated as node attributes, concentrating the learning signal on process-relevant geometric patterns. Experiments on both a large-scale synthetic manufacturability benchmark and a real-world industrial dataset with measured bending times, one of the first such validations on genuine production data, demonstrate that combining domain knowledge with graph-based learning improves prediction accuracy across both tasks. The results demonstrate that hybrid modeling offers a feasible and effective path toward deployable tools for manufacturability assessment and effort estimation in industrial CAD environments.

Weakly supervised graph anomaly detection aims to unveil unusual graph instances, e.g., nodes, whose behaviors significantly differ from normal ones, given only a limited number of annotated anomalies and abundant unlabeled samples. A major challenge is to learn a meaningful latent feature representation that reduces intra-class variance among normal data while remaining highly sensitive to anomalies. Although recent works have applied self-supervised feature learning for graph anomaly detection, their strategies are not specifically tailored to its unique requirements, motivating our exploration of a more domain-specific approach. In this paper, we introduce a weakly supervised graph anomaly detection method that leverages a feature learning strategy tailored for graph anomalies. Our approach is built upon a multi-task learning scheme that extracts robust feature representations through synthesized anomalies. We generate synthetic anomalies by perturbing the normal graph in various ways and assign a dedicated detection head to each anomaly type, ensuring that learned features are sensitive to potential deviations from normal patterns. Although synthetic anomalies may not perfectly replicate real-world patterns, they provide valuable auxiliary data for effective feature learnin, much like features learned from ImageNet classification transfer to downstream vision tasks. Additionally, we adopt a two-phase learning strategy: an initial warm-up phase using only synthetic samples, followed by a full-training phase integrating both tasks, to balance the influence of synthetic and real data. Extensive experiments on public datasets demonstrate the superior performance of our method over its competitors. Code is available at https://github.com/yj-zhou/SAWGAD.

Graph representation learning has become a standard approach for analyzing networked data, with latent embeddings widely used for link prediction, community detection, and related tasks. Yet a basic design choice, the latent dimension, is still treated as a brittle hyperparameter, fixed before training and tuned by held-out performance. Learned factors are also identifiable only up to rotation and rescaling, so the nominal rank rarely coincides with the quantity that governs model behavior. We propose Spectral Prefix Extraction and Capacity-Targeted Representation Analysis (Spectra), which replaces rank as the unit of analysis with the spectrum of a learned positive semidefinite kernel, trace-normalized so that spectra are comparable across fits. The normalized eigenvalues form a distribution on the simplex, and their Shannon effective rank acts both as a summary of learned capacity and as a controllable training-time coordinate: a single scalar shapes this realized dimension during training, and bisection targets any desired value within the rank cap. To theoretically support that, we show local regularity and monotonicity of the realized-dimension profile. Across collaboration, social, biological, and infrastructure networks, Spectra traces performance--capacity frontiers that make the trade-off between predictive accuracy and realized dimension visible. It performs competitively with strong link-prediction baselines, yields aligned lower-capacity views of the same fitted model through spectral prefixes, and provides a principled handle on capacity in the overparameterized regime. Capacity thus becomes a property of the fitted model rather than a hyperparameter of the training.

Biomedical knowledge graphs underwrite drug repurposing and clinical decision support, yet the upstream ontologies they depend on update on independent cycles that add millions of edges and deprecate hundreds of thousands more between releases. Yet existing continual graph learning has been studied almost exclusively on synthetic random splits of static, generic KGs, a regime that cannot reproduce the asynchronous, structured evolution real biomedical KGs undergo. To this end, we introduce PrimeKG-CL, a CGL benchmark built from nine authoritative biomedical databases (129K+ nodes, 8.1M+ edges, 10 node types, 30 relation types) with two genuine temporal snapshots (June 2021, July 2023; 5.83M edges added, 889K removed, 7.21M persistent), 10 entity-type-grouped tasks, multimodal node features, and a per-task persistent/added/removed test stratification. On three tasks (biomedical relationship prediction, entity classification, KGQA), we evaluate six CL strategies across four KGE decoders, plus LKGE, an LLM-RAG agent, and CMKL. We find that decoder choice and continual learning strategy interact strongly: no single strategy performs best across all decoders, and mismatched combinations can significantly degrade performance. Moreover, only DistMult exhibits a clear separation between persistent and deprecated knowledge, indicating that standard metrics conflate retention of still-valid facts with failure to forget outdated ones; this effect is absent under RotatE. In addition, multimodal features improve entity-level tasks by up to 60%, and a recent CKGE framework (IncDE) failed to scale to our 5.67M-triple base task across five attempts up to 350GB RAM. Data, pipeline, baselines, and the stratified split are released openly. Dataset:huggingface.co/datasets/yradwan147/PrimeKGCL|Code:github.com/yradwan147/primekg-cl-neurips2026

Protein-protein interactions (PPIs) are fundamental to cellular function and disease mechanisms. Current learning-based PPI predictors focus on learning powerful protein representations but neglect designing specialized classification heads. They mainly rely on generic aggregating methods like concatenation or dot products, which lack biological insight. Motivated by the biological "L3 rule", where multiple length-3 paths between a pair of proteins indicate their interaction likelihood, our study addresses this gap by designing a biologically informed PPI classifier. In this paper, we provide empirical evidence that popular PPI datasets strongly support the L3 rule. We propose an L3-path-regularized graph prompt learning method called L3-PPI, which can generate a prompt graph with virtual L3 paths based on protein representations and controls the number of paths. L3-PPI reformulates the classification of protein embedding pairs into a graph-level classification task over the generated prompt graph. This lightweight module seamlessly integrates with PPI predictors as a plug-and-play component, injecting the interaction prior of complementarity to enhance performance. Extensive experiments show that L3-PPI achieves superior performance enhancements over advanced competitors.

Graph learning research has increasingly shifted toward continual graph learning (CGL), which better reflects real-world scenarios where graphs evolve over time. However, existing CGL methods largely assume clean supervision and overlook a critical challenge: the newly arriving portions of the graph are often noisy, due to annotation errors or adversarial corruption. This mismatch limits their applicability in practice. In this work, we study robust continual graph learning, where models must simultaneously handle catastrophic forgetting and noisy supervision in evolving graph data. We show that label noise introduces a new failure mode, catastrophic remembering, where models persistently reinforce corrupted knowledge across tasks. To address these challenges, we propose a Unified Flow-Oriented framework (UFO). First, UFO models conditional feature distributions via flow-based generative modeling and produces replay representations, mitigating forgetting without storing historical data. Second, UFO estimates instance-level reliability scores to distinguish clean from noisy nodes, reducing the impact of corrupted supervision and alleviating catastrophic remembering. Extensive experiments on four benchmark graph datasets under varying noise ratios demonstrate that UFO consistently outperforms existing methods in both accuracy and forgetting metrics. Code is available at: https://anonymous.4open.science/r/UFO.

Graph-level anomaly detection (GLAD) is crucial for ensuring the reliability of graph-driven applications by identifying abnormal graphs that deviate from the majority. Considering the privacy concerns in distributed scenarios, federated graph-level anomaly detection (FedGLAD) has emerged as a promising solution to enable collaborative detection without sharing raw data. However, existing methods suffer from poor generalization due to the reliance on unrealistic synthetic anomalies and insufficient personalization capabilities under data heterogeneity. To address these challenges, we propose a novel Federated graph-level anomaly detection approach with Cluster-adaptIve GAted Reconstruction (FedCIGAR). Specifically, we design a reconstruction-based paradigm trained on normal graphs to avoid synthetic data. Furthermore, we introduce a client-side node contribution gating mechanism and a server-side sliding window-based clustering strategy to tackle data heterogeneity. Extensive experiments demonstrate that FedCIGAR achieves superior performance and robustness in contrast to state-of-the-art methods.

Bilevel graph structure learning is widely understood to improve graph neural networks by jointly optimizing model parameters and a learned graph structure, with the resulting performance gain attributed to the rewired adjacency. We find that this attribution may be overstated: training-dynamics effects in the inner loop, rather than the rewiring itself, capture a substantial share of the gain. To establish this, we introduce frozen-$φ$, a control that freezes the graph while retaining the inner-loop training schedule. This decomposes the bilevel gain into an inner channel of $T$-step training dynamics with implicit gradient regularization and a graph channel of the graph rewiring itself. On spatio-temporal flow forecasting the inner channel matches or exceeds the full bilevel pipeline, accounting for 78-101% of the gain; on node classification it accounts for 37-44% under a Bernoulli edge-level parameterization. We also verify that classical spectral diagnostics can dissociate from task gain. We propose frozen-$φ$ as a standardized diagnostic for bilevel graph structure learning, with graph distillation as a method-agnostic complement. A three-precondition framework further predicts the sign of the bilevel gain on all six benchmarks.

Reasoning-acting frameworks enhance large language models (LLMs) by interleaving reasoning with actions for dynamic information acquisition. However, extending this paradigm to graph learning remains underexplored. Graph data is inherently structured, with information distributed across nodes and edges and encoded through both topology and latent representations. As a result, effective reasoning over graphs requires not only retrieving informative evidence from the graph, but also progressively refining the accumulated context during multi-step inference. In this work, we propose GraphReAct, a graph reasoning-acting framework that enables step-by-step inference over graph-structured data. Specifically, we design a graph-based action space with two complementary retrieval actions: topological retrieval, which captures local structural dependencies, and semantic retrieval, which accesses non-local but relevant evidence in the representation space. These actions dynamically expand the reasoning context. To further support multi-step reasoning, we introduce another type of action, context refinement, which distills and reorganizes accumulated information into a compact representation. By interleaving reasoning with both retrieval and refinement actions, our framework enables a progressive transition from context expansion to compression. Extensive experiments on six benchmark datasets demonstrate that GraphReAct consistently outperforms state-of-the-art methods, validating the effectiveness of reasoning-acting for graph learning.

Graph Neural Networks (GNNs) achieve high performance but can be opaque to humans, making it difficult to understand and compare the many proposed architectures. While existing explainability methods attribute individual predictions to nodes, edges, or features, they do not provide architectural transparency or explain the fundamental performance gap between simple and more complex models. To address this limitation, we introduce Model-to-Data (M2D) distillation, a new framework that increases transparency by transferring model complexity into the data space. M2D distills the teacher model into an augmented graph with enriched features and structure, enabling a simple student to match the teacher's performance. By materializing model behavior in the data, our approach allows humans to inspect architectural advantages directly. We show that M2D reveals underlying mechanisms such as fairness objectives and attention-based aggregation in an interpretable way, enhancing GNN transparency while preserving performance.

Graph representation learning (GRL) has evolved from topology-only graph embeddings to task-specific supervised GNNs, and more recently to reusable representations and graph foundation models (GFMs). However, existing evaluations mainly measure clean transfer, adaptation, and task coverage. It remains unclear whether GRL methods stay reliable when deployment stresses affect graph signals, graph contexts, label support, structural groups, or predictive evidence. We introduce GRL-Safety, a multi-axis safety evaluation benchmark for GRL. GRL-Safety evaluates twelve representative methods, spanning topology-only embedding methods, supervised GNNs, self-supervised graph models, and GFMs, on twenty-five graph datasets under standardized evaluation conditions while preserving method-native adaptation. The evaluation covers five safety axes: corruption robustness, OOD generalization, class imbalance, fairness, and interpretation, with per-axis and sub-condition reporting rather than a single aggregate score. Our analysis yields three cross-axis insights that can inspire future research. First, safety behavior is shaped by the interaction between representation design and the stressed graph factor, rather than by method family alone. Second, foundation-era methods show axis-specific strengths rather than broad safety dominance. Third, several deployment regimes remain difficult even for the best evaluated method, revealing capability gaps that require new robustness, adaptation, or training objectives beyond model selection. The benchmark, evaluation protocols, and code are available at: https://github.com/GXG-CS/GRL-Safety.

Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging. Unsupervised tasks in particular require interpretable, scalable, and reliable size-aware graph representations. Our work addresses these issues by tracking the structural diversity of a graph across coarsening levels. The resulting graph embeddings, which we denote diversity curves, are interpretable by construction, efficient, and directly comparable across coarsening hierarchies. Specifically, we track the spread of graphs, a novel isometry invariant that is inherently well-suited for encoding the metric diversity and geometry of graphs. We utilise edge contraction coarsening and prove that this improves expressivity, thus leading to more powerful graph-level representations than structural descriptors alone. Demonstrating their utility over a range of baseline methods in practice, we use diversity curves to (i) cluster and visualise simulated graphs across varying sizes, (ii) distinguish the geometry of single-cell graphs, (iii) compare the structure of molecular graph datasets, and (iv) characterise geometric shapes.

Graph-based learning on functional magnetic resonance imaging (fMRI) has shown strong potential for brain network analysis. However, existing methods degrade under cross-site out-of-distribution (OOD) settings because site-conditioned confounders induce non-pathological shortcuts, while functional connectivity constructed by temporal averaging obscures transient neurodynamics, limiting generalization to unseen sites. In this paper, we propose Cross-site OOD Robust brain nEtwork (CORE), a unified framework for brain network learning across unseen sites. CORE first performs site-aware confounder decoupling to mitigate site-conditioned bias and extract a cross-site population scaffold of reproducible diagnostic connectivity edges. It then profiles transient pathway dynamics over this scaffold using lightweight temporal descriptors and organizes scaffold edges into a line graph for transferable pathway-level modeling. Finally, CORE introduces a prior-guided subject-adaptive gating mechanism that leverages scaffold-derived population priors while preserving subject-specific connectivity variability. Extensive experiments under leave-one-site-out evaluation on real-world datasets (ABIDE, REST-meta-MDD, SRPBS, and ABCD) show that CORE consistently outperforms state-of-the-art baselines, with up to 6.7% relative gain. Furthermore, CORE remains robust to atlas variations, maintaining performance gains across different brain parcellation schemes.

Regularization is often used in high-dimensional regression settings to generate a sparse model, which can save tremendous computing resources and identify predictors that are most strongly associated with the response. When the predictors can be represented by a Gaussian graphical model, the structure of the predictor graph can be exploited during regularization. Our proposed model exploits this underlying predictor graph structure by decomposing the estimated coefficient vector into a sum of latent variables that correspond to the sum of each node contribution to the coefficient vector. Regularization is then performed on the latent variables rather than on the coefficient vector directly. We use a penalty function that permits a clear user-defined trade-off between the L1 and L2 penalties and propose a novel proximal projection during optimization. Further, our implementation computes the projection operator for the intersection of selected groups, which conserves more computing resources compared to predictor duplication methods, especially for high-dimensional data. Through simulation, we evaluate the performance of our approach under different graph structures and node counts, and present results on real-world data. Results suggest that our method exhibits stable performance relative to other singly or doubly sparse graphical regression models.

Federated Graph Learning (FGL) enables collaborative learning over distributed graph data, yet existing approaches largely rely on a closed-world assumption, limiting their applicability in dynamic environments where novel categories continuously emerge. To bridge this gap, we target the practical scenario of Federated Graph Generalized Category Discovery (FGGCD), aiming to collaboratively discover novel categories across decentralized graph clients while retaining knowledge of known categories. We observe that FGGCD introduces two fundamental challenges: (1) the Neighborhood Absorption Effect, where structural fragmentation leads to biased neighborhood aggregation, causing novel nodes to be misclassified as known categories; and (2) Global Semantic Inconsistency, where the aforementioned local biases propagate to the server and are amplified by heterogeneous subgraph distributions, hindering cross-client knowledge integration. To address these issues, we propose GCD-FGL, an FGL framework for GCD that integrates a client-side Topology-Reliable Semantic Alignment and Discovery process to mitigate the neighborhood absorption effect, and a server-side Hierarchical Prototype Alignment strategy to resolve global semantic inconsistency. Extensive experiments on five real-world graph datasets demonstrate that GCD-FGL consistently outperforms state-of-the-art baselines, achieving an average absolute gain of +4.86 in HRScore.

We develop graph-based methods for semi-supervised learning based on label propagation on a data similarity graph. When data is abundant or arrive in a stream, the problems of computation and data storage arise for any graph-based method. We propose a fast approximate online algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local representative points that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We also present graph-based methods for detecting conditional anomalies and apply them to the identification of unusual clinical actions in hospitals. Our hypothesis is that patient-management actions that are unusual with respect to the past patients may be due to errors and that it is worthwhile to raise an alert if such a condition is encountered. Conditional anomaly detection extends standard unconditional anomaly framework but also faces new problems known as fringe and isolated points. We devise novel nonparametric graph-based methods to tackle these problems. Our methods rely on graph connectivity analysis and soft harmonic solution. Finally, we conduct an extensive human evaluation study of our conditional anomaly methods by 15 experts in critical care.

Graph contrastive learning (GCL) has become a central paradigm for self-supervised representation learning in computational intelligence, with applications spanning recommendation, anomaly detection, and personalization. A key limitation of existing methods is their reliance on static negative sampling, which fails to account for the dynamic informativeness and computational cost of negatives during training. We propose AdNGCL, an adaptive negative scheduling framework with a hardness-aware scheduler (HANS) that formulates negative selection as a loss-gated, budget-constrained process across hard, intermediate, and easy strata. The scheduler dynamically adjusts step sizes based on contrastive loss trends under both global and per-category budgets, while periodically refreshing samples to maintain diversity without exceeding compute constraints. Experiments on nine benchmark graph datasets demonstrate that AdNGCL consistently advances state-of-the-art performance, achieving the best accuracy on seven datasets and second-best on the remaining two, while offering explicit control over computational cost. These results highlight the value of budget-aware, loss-sensitive scheduling as a general strategy for improving the robustness and efficiency of representation learning in emerging computational intelligence applications.

Graph federated learning (GFL) facilitates decentralized training on distributed graph data while keeping sensitive user information local, aligning with policies such as GDPR and CCPA that grant users the right to freely join or withdraw from learning systems. However, even decentralized, user information can persist after quitting, potentially propagating to central servers and then redistributing to malicious clients. This privacy leakage during user withdrawal, despite its importance, has received seldom attention in GFL. To fill the gap, we explore the potential of machine unlearning (MU) to thoroughly remove user information. However, classical MU methods are known to degrade overall performance, a problem that is exacerbated in GFL due to local message passing and global model collaboration. To this end, we make two adjustments to mitigate this challenge for GFL. First, we ensure unlearning updates that minimally affect overall performance, steering them in directions orthogonal to the gradients from learning other data. Second, we introduce virtual clients, maintained by the central server, to preserve graph topology and global embeddings without recovering information of removed entities. We conduct comprehensive experiments under a representative user-withdrawal scenario and propose a novel membership inference framework to rigorously evaluate and validate the reliability of our privacy preservation. The experimental results demonstrate the effectiveness of our approach, which also surpasses the performance of seven state-of-the-art baseline methods.

This paper introduces the use of per-class regularization hyperparameters in Gabriel graph-based binary classifiers. We demonstrate how the quality index used for regularization behaves both in the margin region and in the presence of outliers, and how incorporating this regularization flexibility can lead to solutions that effectively eliminate outliers while training the classifier. We also show how it can address class imbalance by generating higher and lower thresholds for the majority and minority classes, respectively. Thus, rather than having a single solution based on fixed thresholds, flexible thresholds expand the solution space and can be optimized through hyperparameter tuning algorithms. Friedman test shows that flexible thresholds are capable of improving Gabriel graph-based classifiers.

Graph self-supervised learning typically relies on large-scale unlabeled datasets, heavily inflating computational costs. However, empirical evidence suggests that these datasets contain substantial redundancy-our analysis reveals that uniformly subsampling 50% of graphs retains over 96% of downstream performance. To exploit this redundancy, we introduce GraphSculptor for pre-training coreset construction. Unlike methods dependent on additional training-time signals or limited solely to topological statistics, GraphSculptor provides a label-free solution that constructs coresets via two complementary perspectives: intrinsic structure and contextual semantics. Concretely, structural diversity is quantified using intrinsic graph statistics, yielding a structural feature vector for each graph, while semantic diversity is captured by utilizing a pre-trained language model to encode descriptions generated via graph-to-text. GraphSculptor integrates these signals into a unified metric space and performs cluster-aware selection to preserve joint structural-semantic diversity. We further derive a theoretical bound on the loss gap between coreset and full-data pre-training, offering theoretical motivation for our selection formulation. Extensive experiments demonstrate that GraphSculptor effectively sculpts the dataset: a 10% coreset achieves 99.6% of full-data performance while reducing pre-training time by nearly 90%, offering a scalable solution for data-efficient graph pre-training.

Representation learning is central to graph machine learning, powering tasks such as link prediction and node classification. However, most graph embeddings are hard to interpret, offering limited insight into how learned features relate to graph structure. Many networks naturally admit a role-mixture view, where nodes are best described as mixtures over latent archetypal factors. Motivated by this structure, we propose a compositional graph embedding framework grounded in Aitchison geometry, the canonical geometry for comparing mixtures. Nodes are represented as simplex-valued compositions and embedded via isometric log-ratio (ILR) coordinates, which preserve Aitchison distances while enabling unconstrained optimization in Euclidean space. This yields intrinsically interpretable embeddings whose geometry reflects relative trade-offs among archetypes and supports coherent behavior under component restriction; we consider both fixed and learnable ILR bases. Across node classification and link prediction, our method achieves competitive performance with strong baselines while providing explainability by construction rather than post-hoc. Finally, subcompositional coherence enables principled component restriction: removing and renormalizing subsets preserves a well-defined geometry, which we exploit via subcompositional dimensionality removal to probe how archetype groups influence representations and predictions.

We present \textit{CluProp}, a novel framework that reimagines varied-density clustering in high-dimensional spaces as a label propagation process over neighborhood graphs. Our approach formally bridges the gap between density-based clustering and graph connectivity, leveraging efficient propagation mechanisms from network science to mitigate the parameter sensitivity inherent in traditional density-based methods. Specifically, we introduce a deterministic density-based propagation strategy to ensure scalable neighborhood identification. The framework is agnostic to the choice of distance metric and exhibits superior performance on large-scale data, processing millions of points in minutes while consistently outperforming existing baselines in accuracy.

Electroencephalogram (EEG) signals are vital for automated seizure detection, but their inherent noise makes robust representation learning challenging. Existing graph construction methods, whether correlation-based or learning-based, often generate redundant or irrelevant edges due to the noisy nature of EEG data. This significantly impairs the quality of graph representation and limits downstream task performance. Motivated by the remarkable reasoning and contextual understanding capabilities of large language models (LLMs), we explore the idea of using LLMs as graph edge refiners. Specifically, we propose a two-stage framework: we first verify that LLM-based edge refinement can effectively identify and remove redundant connections, leading to significant improvements in seizure detection accuracy and more meaningful graph structures. Building on this insight, we further develop a robust solution where the initial graph is constructed using a Transformer-based edge predictor and multilayer perceptron, assigning probability scores to potential edges and applying a threshold to determine their existence. The LLM then acts as an edge set refiner, making informed decisions based on both textual and statistical features of node pairs to validate the remaining connections. Extensive experiments on TUSZ dataset demonstrate that our LLM-refined graph learning framework not only enhances task performance but also yields cleaner and more interpretable graph representations.

Terrestrial network limitations drive the integration of non-terrestrial networks (NTNs), notably mega-constellations comprising thousands of low Earth orbit (LEO) satellites. While these satellites act as interconnected network switches via inter-satellite links (ISLs), their massive scale creates severe bottlenecks for network management. To address this, we propose a scalable, hierarchical software-defined networking (SDN) framework. Our architecture leverages graph neural networks (GNNs) to compactly represent the constellation topology, and Koopman theory to linearize nonlinear dynamics. Specifically, a Graph Koopman Autoencoder (GKAE) forecasts spatio-temporal behavior within a linear subspace for each orbital shell. A central SDN controller then aggregates these shell-level predictions for globally coordinated control. Simulations on the Starlink constellation demonstrate that our approach achieves at least a 42.8% improvement in spatial compression and a 10.81% improvement in temporal forecasting compared to established baselines, all while utilizing a significantly smaller model footprint.

Graph few-shot learning, which focuses on effectively learning from only a small number of labeled nodes to quickly adapt to new tasks, has garnered significant research attention. Despite recent advances in graph few-shot learning that have demonstrated promising performance, existing methods still suffer from several key limitations. First, during the meta-training phase, these methods typically perform node representation learning in Euclidean space, which often fails to capture the inherently hierarchical structure existing in real-world graph data. Second, during the meta-testing phase, they usually fit an empirical target distribution derived from only a few support samples, even when this distribution significantly deviates from the true underlying distribution. To address these issues, we propose IMPRESS, a novel framework that IMproves graPh few-shot learning with hypeRbolic spacE and denoiSing diffuSion. Specifically, our model learns node representations in a hyperbolic space and enriches the support distribution through denoising diffusion mechanisms. Theoretically, IMPRESS achieves a tighter generalization bound. Empirically, IMPRESS consistently outperforms competitive baselines across multiple benchmark datasets.

Heterogeneous graphs with heterophily have emerged as a powerful abstraction for modeling complex real-world systems, where nodes of different types and labels interact in diverse and often non-homophilous ways. Despite recent advances, robust representation learning for such graphs remains largely unexplored, particularly in the presence of noisy or misleading connectivity. In this work, we investigate this problem and identify structural noise as a critical challenge that significantly degrades model performance. To address this issue, we propose a unified framework, Heterogeneous Graph Unified Learning (HGUL), which jointly handles heterophily and noisy graph structures. The framework consists of three complementary modules: a kNN-based graph construction module that recovers reliable local neighborhoods, a graph structure learning module that adaptively refines the adjacency by filtering noisy edges, and a heterogeneous affinity learning module that captures class-level relationships via an extended affinity matrix derived from a polynomial graph kernel. Extensive experiments on multiple datasets demonstrate that HGUL consistently outperforms existing methods on clean graphs and maintains strong robustness under varying levels of structural noise. The results further underscore the importance of jointly modeling heterophily and noise in heterogeneous graph learning.

This paper proposes a novel algorithm for semisupervised learning. This algorithm learns graph cuts that maximize the margin with respect to the labels induced by the harmonic function solution. We motivate the approach, compare it to existing work, and prove a bound on its generalization error. The quality of our solutions is evaluated on a synthetic problem and three UCI ML repository datasets. In most cases, we outperform manifold regularization of support vector machines, which is a state-of-the-art approach to semi-supervised max-margin learning.

Graph Contrastive Learning (GCL) has emerged as a prominent framework for unsupervised graph representation learning. However, relying on augmentation design alone to define the invariances learned by GCL can be brittle under structural perturbations. To address this issue, we propose Cheeger--Hodge Contrastive Learning (CHCL), a framework that aligns a perturbation-stable Cheeger--Hodge joint signature across augmented views for robust graph representation learning. The proposed signature combines a Cheeger-inspired connectivity signature derived from the algebraic connectivity $λ_2$ with the low-frequency spectrum of the 1-Hodge Laplacian, thereby capturing both global connectivity and higher-order structural information. By aligning encoder representations with the proposed Cheeger--Hodge joint signature across augmented views, CHCL learns graph embeddings that are robust to local structural perturbations. Extensive experiments on standard benchmarks, transfer settings demonstrate that CHCL consistently improves performance, robustness, and generalization.

This paper addresses the problem of anomaly detection in accounting subject association structures, proposing a structured modeling and unsupervised discriminant framework based on graph neural networks. This framework is used to mine stable correspondences between subjects and identify structural deviations from general ledger details and voucher entries. The method first abstracts accounting subjects as graph nodes, and the co-occurrence and debit/credit correspondence of subjects in the same business record are abstracted as weighted edges. The edge weights are characterized by statistical measures such as co-occurrence frequency or amount aggregation, thus forming a period-level accounting subject association graph. In the representation learning stage, a message passing mechanism is used to fuse the node's own attributes and neighborhood context to obtain node embeddings containing structural information. In the anomaly detection stage, the rationality of subject pair connections is estimated through a relation reconstruction decoder, and edge-level anomaly scores are defined based on the degree of deviation in reconstruction probabilities. These scores are then aggregated to obtain node-level risk ranking and local anomaly localization. This framework can simultaneously capture local substructure anomalies and cross-community anomaly connections without relying on anomaly labeling, outputting traceable subject pair risk clues. Comparative experiments demonstrate more stable comprehensive discriminant capabilities and higher top-ranking accuracy.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this work, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node of an undirected graph and its expected rating is similar to the one of its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose three algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of node evaluations.

Graph neural Ordinary Differential Equations (ODE) combine neural ODE with the message passing mechanism of Graph Neural Networks (GNN), providing a continuous-time modeling method for graph representation learning. However, in dynamic graph scenarios, existing graph neural ODEs typically employ a unified message passing mechanism, assuming that inter-node interactions share the same message passing function at any time, which makes it challenging to capture the diversity and time-varying nature of inter-node interaction patterns. To address this, we propose Time-varying Interaction Graph Ordinary Differential Equations (TI-ODE). The core idea of TI-ODE is to decompose the evolution function of a graph ODE into a set of learnable interaction basis functions, where each basis function corresponds to a distinct type of inter-node interaction. These basis functions are dynamically combined through time-dependent learnable weights, enabling inter-node interaction patterns to adaptively evolve over time. Experimental results on six dynamic graph datasets demonstrate that TI-ODE consistently outperforms existing methods and achieves state-of-the-art performance on attribute prediction tasks, and experiments on the \textit{Covid} dataset further verify the interpretability and generalizability of our TI-ODE. Furthermore, we demonstrate both theoretically and empirically that TI-ODE exhibits superior robustness compared to models utilizing a unified message-passing mechanism.

Motivation: Multi-omics integration can improve cancer subtyping, but modality informativeness and noise vary across cancer types and patients. Existing graph-based methods optimize modality weights jointly with the classification objective and therefore lack independent reliability estimates, so low-quality omics distort patient similarity graphs and amplify noise through message passing. Results: We propose CMGL, a two-stage framework that estimates per-sample modality reliability through evidential deep learning and uses the frozen confidence scores to guide cross-omics fusion and graph construction. On four MLOmics cancer-subtype tasks and the 32-class pan-cancer task, CMGL consistently improves over the strongest baseline, surpassing it by 4.03% in average accuracy on the four single-cancer tasks. Its representations recover the PAM50 intrinsic subtypes of breast invasive carcinoma (BRCA), and the BRCA-trained model transfers without fine-tuning to kidney renal clear cell carcinoma (KIRC), stratifying patients into prognostically distinct groups.

We propose PathBoost, a gradient tree boosting method for graph-level classification and regression that learns discriminative path-based features directly from the input graph structure. Building on a previous work, which was tailored to a specific chemistry application, PathBoost introduces three key extensions: (i) adaptation to binary classification through gradient boosting with a logistic loss, (ii) incorporation of multiple node and edge attributes into the path feature space via a prefix-based decomposition, and (iii) automatic anchor node selection based on categorical attribute diversity, eliminating the need for the user to specify the starting point of the considered path features. We compared PathBoost to graph neural networks and graph kernel approaches on several benchmark datasets, obtaining better results in half of them, and comparable results in the rest. PathBoost shows better performances on graphs with larger average node counts. Overall, the results demonstrate that path-based boosting methods can be competitive with more complex black-box approaches.

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.

Graph neural networks achieve strong node-classification accuracy, but learned message passing entangles ego attributes, neighborhood smoothing, high-pass graph differences, class geometry, and classifier-boundary effects inside opaque representations. This obscures why nodes are classified as they are and which graph-learning mechanisms a dataset requires. We propose WG-SRC, a white-box signal-subspace probe for prediction and graph dataset diagnosis. WG-SRC replaces learned message passing with a fixed, named graph-signal dictionary containing raw features, row- and symmetric-normalized low-pass propagation, and high-pass graph differences. It combines Fisher coordinate selection, class-wise PCA subspaces, closed-form multi-alpha ridge classification, and validation-based score fusion, so prediction and analysis rely on explicit class subspaces, energy-controlled dimensions, and closed-form linear decisions. As a white-box graph-learning instrument, WG-SRC uses predictive performance to validate its diagnostics. Across six node-classification datasets, it remains competitive with reproduced baselines and achieves positive average gain under aligned splits. Its atlas decomposes behavior into raw-feature, low-pass, high-pass, class-geometric, and ridge-boundary components. The resulting fingerprints distinguish low-pass-dominated Amazon graphs, mixed high-pass and class-geometrically complex Chameleon behavior, and raw- or boundary-sensitive WebKB graphs. Aligned interventions show when high-pass blocks act as removable noise, when raw or graph-derived signals should be preserved, and when ridge correction matters. WG-SRC therefore serves both as a functioning white-box classifier and as a dataset-fingerprinting probe, enabling fingerprint-conditioned analysis of how black-box model components behave under different graph-signal conditions.

Graph Neural Networks (GNNs) excel on relational data, but standard benchmarks unrealistically assume the graph is centrally available. In practice, settings such as Federated Graph Learning, distributed systems, and privacy-sensitive applications involve graph data that are localized, fragmented, noisy, and privacy-leaking. We present a unified framework for this setting. We introduce LoGraB (Local Graph Benchmark), which decomposes standard datasets into fragmented benchmarks using three strategies and four controls: neighborhood radius $d$, spectral quality $k$, noise level $σ$, and coverage ratio $p$. LoGraB supports graph reconstruction, localized node classification, and inter-fragment link prediction, with Island Cohesion. We propose AFR (Adaptive Fidelity-driven Reconstruction), a method for noisy spectral fragments. AFR scores patch quality via a fidelity measure combining a gap-to-truncation stability ratio and structural entropy, then assembles fragments using RANSAC-Procrustes alignment, adaptive stitching, and Bundle Adjustment. Rather than forcing a single global graph, AFR recovers large faithful islands. We prove heat-kernel edge recovery under a separation condition, Davis--Kahan perturbation stability, and bounded alignment error. We establish a Spectral Leakage Proposition: under a spectral-gap assumption, polynomial-time Bayesian recovery is feasible once enough eigenvectors are shared, complementing AFR's deterministic guarantees. Experiments on nine benchmarks show that LoGraB reveals model strengths and weaknesses under fragmentation, AFR achieves the best F1 on 7/9 datasets, and under per-embedding $(ε,δ)$-Gaussian differential privacy, AFR retains 75% of its undefended F1 at $ε=2$. Our anonymous code is available at https://anonymous.4open.science/r/JMLR_submission

Graph-based techniques and spectral graph theory have enriched the field of machine learning with a variety of critical advances. A central object in the analysis is the graph Laplacian L, which encodes the structure of the graph. We consider the problem of learning over this Laplacian in a distributed streaming setting, where new edges of the graph are observed in real time by a network of workers. In this setting, it is hard to learn quickly or approximately while keeping a distributed representation of L. To address this challenge, we present a novel algorithm, GSQUEAK, which efficiently sparsifies the Laplacian by maintaining a small subset of effective resistances. We show that our algorithm produces sparsifiers with strong spectral approximation guarantees, all while processing edges in a single pass and in a distributed fashion.

Graph representation learning has achieved notable success in encoding graph-structured data into latent vector spaces, enabling a wide range of downstream tasks. However, these node representations remain opaque and difficult to interpret. Existing explainability methods primarily focus on supervised settings or on explaining individual representation dimensions, leaving a critical gap in explaining the overall structure of node representations. In this paper, we propose TACENR (Task-Agnostic Contrastive Explanations for Node Representations), a local explanation method that identifies not only attribute features but also proximity and structural ones that contribute the most in the representation space. TACENR builds on contrastive learning, through which we learn a similarity function in the representation space, revealing which are the features that play an important role in the representation of a node. While our focus is on task-agnostic explanations, TACENR can be applied to supervised scenarios as well. Experimental results demonstrate that proximity and structural features play a significant role in shaping node representations and that our supervised variant performs comparably to existing task-specific approaches in identifying the most impactful features.

Graph-theoretic approaches offer simplicity, interpretability, and low computational cost for molecular property prediction. Among these, the model proposed by Mukwembi and Nyabadza, based on the external activity $D(G)$ and internal activity $ζ(G)$ indices, achieved strong results on a small flavonoid dataset. However, its ability to generalize to larger and chemically diverse datasets has not been tested. This study evaluates the baseline $D(G)$-$ζ(G)$ polynomial model on five benchmark datasets from MoleculeNet, covering biological activity (BACE, 1,513 molecules), lipophilicity (LogP synthetic, 14,610 molecules; LogP experimental, 753 molecules), aqueous solubility (ESOL, 1,128 molecules), and hydration free energy (SAMPL, 642 molecules). The baseline model achieves an average $R^2 = 0.24$, confirming limited transferability. To address this, a systematic enhancement framework is proposed, progressively incorporating Ridge regularization, additional graph descriptors, physicochemical properties, ensemble learning with Gradient Boosting, Lasso feature selection, and a hybrid approach combining topological indices with Morgan fingerprints. The enhanced models raise the average best $R^2$ to 0.79, with individual improvements ranging from 165% to 274%. All improvements are statistically significant ($p < 0.001$). A direct comparison with a Graph Convolutional Network under identical experimental conditions shows that the enhanced classical models match or outperform deep learning on all five datasets. Comparison with the recent GNN+PGM hybrid of Djagba et al.\ further confirms competitiveness, with the enhanced models achieving the best results on two datasets and tying on one. The entire framework requires no GPU, trains in under five minutes, and uses only open-source tools, making it accessible for researchers in resource-limited settings.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

Small language models fine-tuned for graph property estimation have demonstrated strong in-distribution performance, yet their generalization capabilities beyond training conditions remain poorly understood. In this work, we systematically investigate the boundaries of structural inference in fine-tuned small language models along two generalization axes - graph size and graph family distribution - and assess domain-learning capability on real-world graph benchmarks. Using a controlled experimental setup with three instruction-tuned models in the 3-4B parameter class and two graph serialization formats, we evaluate performance on graphs substantially larger than the training range and across held-out random graph families. Our results show that fine-tuned models maintain strong ordinal consistency across structurally distinct graph families and continue to rank graphs by structural properties on inputs substantially larger than those seen during training, with distinct architecture-specific degradation profiles. These findings delineate where fine-tuned small language models generalize reliably, providing empirical grounding for their use in graph-based reasoning tasks.

Graph self-supervised learning can reduce the need for labeled graph data and has been widely used in recommendation, social networks, and other web applications. However, existing methods often underuse high-frequency signals and may overfit to specific local patterns, which limits representation quality and generalization. We propose Frequency-Corrupt Based Graph Self-Supervised Learning (FC-GSSL), a method that builds corrupted graphs biased toward high-frequency information by corrupting nodes and edges according to their low-frequency contributions. These corrupted graphs are used as inputs to an autoencoder, while low-frequency and general features are reconstructed as supervision targets, forcing the model to fuse information from multiple frequency bands. We further design multiple sampling strategies and generate diverse corrupted graphs from the intersections and unions of the sampling results. By aligning node representations from these views, the model can discover useful frequency combinations, reduce reliance on specific high-frequency components, and improve robustness. Experiments on 14 datasets across node classification, graph prediction, and transfer learning show that FC-GSSL consistently improves performance and generalization.

Spatial graphs provide a lightweight and elegant representation of curvilinear anatomical structures such as blood vessels, lung airways, and neuronal networks. Accurately modeling these graphs is crucial in clinical and (bio-)medical research. However, the high spatial resolution of large networks drastically increases their complexity, resulting in significant computational challenges. In this work, we aim to tackle these challenges by proposing VesselTok, a framework that approaches spatially dense graphs from a parametric shape perspective to learn latent representations (tokens). VesselTok leverages centerline points with a pseudo radius to effectively encode tubular geometry. Specifically, we learn a novel latent representation conditioned on centerline points to encode neural implicit representations of vessel-like, tubular structures. We demonstrate VesselTok's performance across diverse anatomies, including lung airways, lung vessels, and brain vessels, highlighting its ability to robustly encode complex topologies. To prove the effectiveness of VesselTok's learnt latent representations, we show that they (i) generalize to unseen anatomies, (ii) support generative modeling of plausible anatomical graphs, and (iii) transfer effectively to downstream inverse problems, such as link prediction.

Multivariate data often exhibit complex dependencies that violate the assumption of isotropic residual noise. For such cases, we introduce Graph Regularized PCA (GR-PCA). It is a graph-based regularization of PCA that incorporates the dependency structure of the data features by learning a sparse precision graph and biasing loadings toward the low-frequency Fourier modes of the corresponding graph Laplacian. Consequently, high-frequency signals are suppressed, while graph-coherent low-frequency ones are preserved, yielding interpretable principal components aligned with conditional relationships. We evaluate GR-PCA on synthetic data spanning diverse graph topologies, signal-to-noise ratios, and sparsity levels. Compared to mainstream alternatives, it concentrates variance on the intended support, produces loadings with lower graph-Laplacian energy, and remains competitive in out-of-sample reconstruction. When high-frequency signals are present, the graph Laplacian penalty prevents overfitting, reducing the reconstruction accuracy but improving structural fidelity. The advantage over PCA is most pronounced when high-frequency signals are graph-correlated, whereas PCA remains competitive when such signals are nearly rotationally invariant. The procedure is simple to implement, modular with respect to the precision estimator, and scalable, providing a practical route to structure-aware dimensionality reduction that improves structural fidelity without sacrificing predictive performance.

Predictive machine learning models generally excel on in-distribution data, but their performance degrades on out-of-distribution (OOD) inputs. Reliable deployment therefore requires robust OOD detection, yet this is particularly challenging for irregular 3D graphs that combine continuous geometry with categorical identities and are unordered by construction. Here, we present a probabilistic OOD detection framework for complex 3D graph data built on a diffusion model that learns a density of the training distribution in a fully unsupervised manner. A key ingredient we introduce is a unified continuous diffusion over both 3D coordinates and discrete features: categorical identities are embedded in a continuous space and trained with cross-entropy, while the corresponding diffusion score is obtained analytically via posterior-mean interpolation from predicted class probabilities. This yields a single self-consistent probability-flow ODE (PF-ODE) that produces per-sample log-likelihoods, providing a principled typicality score for distribution shift. We validate the approach on protein-ligand complexes and construct strict OOD datasets by withholding entire protein families from training. PF-ODE likelihoods identify held-out families as OOD and correlate strongly with prediction errors of an independent binding-affinity model (GEMS), enabling a priori reliability estimates on new complexes. Beyond scalar likelihoods, we show that multi-scale PF-ODE trajectory statistics - including path tortuosity, flow stiffness, and vector-field instability - provide complementary OOD information. Modeling the joint distribution of these trajectory features yields a practical, high-sensitivity detector that improves separation over likelihood-only baselines, offering a label-free OOD quantification workflow for geometric deep learning.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

We study the problem of learning an unknown graph via group queries on node subsets, where each query reports whether at least one edge is present among the queried nodes. In general, learning arbitrary graphs with $n$ nodes and $k$ edges is hard in the non-adaptive setting, requiring $Ω\big(\min\{k^2\log n,\,n^2\}\big)$ tests even when a small error probability is allowed. We focus on learning Erdős--Rényi (ER) graphs $G\sim\mathrm{ER}(n,q)$ in the non-adaptive setting, where the expected number of edges is $\bar{k}=q\binom{n}{2}$, and we aim to design an efficient testing--decoding scheme, namely, a non-adaptive test design together with a decoding algorithm, achieving asymptotically vanishing error probability. Prior work (Li--Fresacher--Scarlett, NeurIPS 2019) presents a testing--decoding scheme that attains an order-optimal number of tests $O(\bar{k}\log n)$ but incurs $Ω(n^2)$ decoding time, whereas their proposed sublinear-time algorithm incurs an extra $(\log \bar{k})(\log n)$ factor in the number of tests. We extend the binary splitting approach, recently developed for non-adaptive group testing, to the ER graph learning setting, and prove that the edge set can be recovered with high probability using $O(\bar{k}\log n)$ tests while attaining decoding time $O(\bar{k}^{1+δ}\log n)$ for any fixed $δ>0$.

Monocular 3D human pose estimation remains a fundamentally ill-posed inverse problem due to the inherent depth ambiguity in 2D-to-3D lifting. While contemporary video-based methods leverage temporal context to enhance spatial reasoning, they operate under a critical paradigm limitation: processing each sequence in isolation, thereby failing to exploit the strong structural regularities and repetitive motion patterns that pervade human movement across sequences. This work introduces the Pattern Reuse Graph Convolutional Network (PRGCN), a novel framework that formalizes pose estimation as a problem of pattern retrieval and adaptation. At its core, PRGCN features a graph memory bank that learns and stores a compact set of pose prototypes, encoded as relational graphs, which are dynamically retrieved via an attention mechanism to provide structured priors. These priors are adaptively fused with hard-coded anatomical constraints through a memory-driven graph convolution, ensuring geometrical plausibility. To underpin this retrieval process with robust spatiotemporal features, we design a dual-stream hybrid architecture that synergistically combines the linear-complexity, local temporal modeling of Mamba-based state-space models with the global relational capacity of self-attention. Extensive evaluations on Human3.6M and MPI-INF-3DHP benchmarks demonstrate that PRGCN establishes a new state-of-the-art, achieving an MPJPE of 37.1mm and 13.4mm, respectively, while exhibiting enhanced cross-domain generalization capability. Our work posits that the long-overlooked mechanism of cross-sequence pattern reuse is pivotal to advancing the field, shifting the paradigm from per-sequence optimization towards cumulative knowledge learning.

Learning on temporal graphs has become a central topic in graph representation learning, with numerous benchmarks indicating the strong performance of state-of-the-art models. However, recent work has raised concerns about the reliability of benchmark results, noting issues with commonly used evaluation protocols and the surprising competitiveness of simple heuristics. This contrast raises the question of which characteristics of the underlying graphs temporal graph learning models actually use to form their predictions. We address this by systematically evaluating eight models on their ability to capture eight fundamental characteristics related to the link structure of temporal graphs. These include structural characteristics such as density, temporal patterns such as recency, and edge formation mechanisms such as homophily. Using both synthetic and real-world datasets, we analyze how well models learn these characteristics. Our findings reveal a mixed picture: models capture some characteristics well but fail to reproduce others. With this, we expose important limitations. Overall, we believe that our results provide practical insights for the application of temporal graph learning models and motivate more interpretability-driven evaluations in graph learning research.