Graph Representation Learning

Single-cell RNA sequencing (scRNA-seq) clustering is essential for identifying cell types, but high dimensionality, sparsity, dropout, and technical noise hinder robust expression representation and cell graph construction. Existing masked autoencoders mainly use expression recovery for feature reconstruction, while graph clustering methods usually depend on fixed KNN graphs and do not feed recovered expression back into graph optimization. We propose scKDGM, a KAN-guided dynamic graph masked learning framework for scRNA-seq clustering. scKDGM uses graph-aware distribution preserving gene masking (GDP-Mask) to perturb cell identity, a KAN-based TAKGCN encoder to learn masked-view representations, mask-guided expression recovery to construct a dynamic graph, and cross-view contrastive learning to transfer recovery signals into topology updates. A ZINB loss models overdispersion and zero inflation. Experiments on 12 real scRNA-seq datasets show that scKDGM outperforms 10 baselines in average NMI and ARI.

Commodity futures can be represented hierarchically, with underlying assets at the upper level and individual futures contracts at the lower level. Entities at each level can be connected by edges reflecting inherent correlations, with cross-level edges capturing contract-to-underlying asset connections. Building on our observations of these structures, we propose a hierarchical graph learning approach for calendar spread (CS) strategies in commodity futures markets, addressing two significant gaps in the machine-learning literature: (i) the absence of learning-based methods for CS strategies in futures markets, and (ii) the lack of consideration of maturity-dependent interrelationships across commodity futures. We first establish the efficacy of CS strategies by analytically showing that CS strategies can possess higher risk-adjusted returns, measured by the information ratio, and lower risk, measured by variance and delta, than long-only strategies. We then introduce a method to convert learning-based predictions into CS positions. Next, we develop a hierarchical graph learning method that predicts futures price movements by utilizing the maturity-dependent interrelationships, thereby yielding a CS trading algorithm. Empirical results on commodity futures markets traded on the Chicago Mercantile Exchange Group demonstrate that our method outperforms benchmark models in both prediction and trading performance. We find that maturity-dependent interrelationships across commodity futures are instrumental in prediction and that CS trading based on hierarchical graph learning is effective for statistical arbitrage.

Federated Graph Learning facilitates collaborative graph modeling across distributed clients while preserving data privacy. However, real-world data categories frequently exhibit long-tailed distributions. Such statistical scarcity severely degrades performance in two ways: it biases the global model toward majority classes, and it structurally isolates minority nodes by submerging them in heterophilic, head-dominated neighborhoods. While existing methods attempt topology-agnostic statistical compensations, they often fail under data scarcity. Instead of recovering tail nodes, they overfit the structural noise from adjacent dominant classes, leading to representation degradation. To address these limitations, we propose FedEPD, a framework built on a dual decoupling paradigm that separates topological purification from semantic recalibration. Specifically, FedEPD utilizes distribution-aware Dirichlet energy pruning to filter spatial heterophilic edges. It then overcomes Non-IID distribution shifts by extracting robust global prototypes from topologically central nodes, which are incorporated into local representations via a spatial low-pass prototype injection. Furthermore, a two stage alternating optimization strategy strictly protects majority decision boundaries while improving minority accuracy. Extensive experiments demonstrate that FedEPD achieves state-of-the-art performance across diverse long-tailed benchmarks, yielding absolute improvements of up to 4.97% in Accuracy and 5.48% in Macro-F1.

Thin-structure segmentation--power lines, cracks, lane markings at 1-3 pixel width--requires preserving connectivity that standard representations preclude: patching severs continuous structures and conventional superpixels merge thin targets into background before classification. Topology-aware losses penalize connectivity breaks at the objective level but cannot recover what the representation has already destroyed. We propose SEMIR, a framework that replaces the pixel lattice with a parameterized graph minor whose contraction map preserves thin-structure connectivity under the contraction criterion. The minor collapses millions of pixels into tens or hundreds of boundary-aligned supernodes, enabling full-resolution inference without patching at scales demonstrated up to 21 MP in this paper; a lightweight GNN classifies the reduced graph and an exact map lifts predictions to pixel resolution. One pipeline--identical architecture, features, loss, and GNN hyperparameters across all dataset--matches or exceeds domain-specific baselines on TTPLA (power lines), CrackSeg9k (pavement cracks), and SkyScapes Lane (aerial markings) on Dice, IoU, and Boundary F1 while reducing mask fragmentation by at least 4.6x relative to SLIC at matched inference.

Attributed graph clustering partitions nodes by jointly exploiting node attributes and graph topology. It remains challenging due to attribute heterogeneity and representation degradation during graph learning. Real-world datasets often contain heterogeneous attributes, i.e., numerical and categorical attributes, complicating unified representation learning. This challenge becomes more complex in attributed graphs, where constructing a clustering-friendly graph structure from attributes and topology remains difficult. Under deep graph architectures, repeated graph propagation causes node embeddings to become overly similar, leading to the over-smoothing (OS) effect. Meanwhile, graph representation learning amplifies topological influence, making discriminative attribute information harder to exploit for clustering, an effect we refer to as over-dominating (OD). To bridge these gaps, an end-to-end framework, Any-type attributed Graph REpresentation lEarning (AGREE), is proposed. It unifies attributed graphs and any-type attributed data through multi-level alignment and similarity-based graph construction. Quaternion-based graph convolution strengthens attribute interaction to alleviate OD, while shallow graph architectures help relieve OS. The learned embeddings are jointly optimized for graph reconstruction and clustering, without requiring a predefined number of clusters during training. Experiments on diverse benchmarks show that AGREE achieves strong overall performance in accuracy, robustness, and adaptability.

Graph theory is inherently descriptive, capturing what relationships exist but not why they arise, because it treats edges as primitive constructs. This paper proposes a new explanatory framework for graph learning, where relationships emerge from latent continuous information entropy fields, and a graph becomes a discrete instantiation of an underlying field. To formalize this field, we introduce the Field-informed Graph Network (FGN). It learns a scalar field from node features and leverages it to modulate message passing. The information-theoretic objective balances structural fidelity with field smoothness, forming a self-reinforcing loop. In this loop, the field modulates information diffusion through field-modulated weighting, and the updated node representations iteratively refine the field. As a result, FGN learns by simulating its own co-evolution. Extensive experiments on node classification and graph classification benchmarks demonstrate superior performance, robustness to perturbations, and structurally coherent field representations.

We present an exploratory benchmark and quantum-inspired modeling prototype for fraud screening in dynamic financial transaction graphs. Coordinated fraud may not be visible from individual transactions alone, but may emerge as a multi-period relational pattern. We focus on sparse-ring fraud, a stylized pattern in which a completed directed cycle is distributed across several days, requiring models to integrate evidence across both time and graph structure. We study this problem using a synthetic transaction simulator with completed sparse-ring injections and broken-ring decoys. Daily directed transaction graphs are aggregated into rolling windows and represented using raw graph features, persistent-homology summaries, or hybrid feature vectors that combine both. We compare a gated recurrent unit (GRU) baseline with quantum-inspired Contextual Machine Learning (CML) as sequence-level classifiers. Because the benchmark uses synthetic data, a modest sample size, and sequence-level labels, the results are exploratory. Within this scope, topology-only summaries are too compressed to solve the supervised ring-completion task by themselves, largely because they remove account-pair identity and edge direction. The strongest results come from hybrid representations that combine identity-preserving graph features with topological summaries. These findings suggest that topology is most useful as a contextual layer over dynamic graph features, and that CML is a promising candidate model for fraud patterns whose evidence is distributed across temporal and relational context.

Graph embedding maps graph nodes into low-dimensional vectors to support applications such as recommendation, fraud detection, and graph-based retrieval-augmented generation (GraphRAG). As graphs scale to billions of edges, scalable and efficient graph embedding has become increasingly important. Existing frameworks commonly adopt a sampling-training paradigm, in which mini-batches are constructed by sampling nodes and their neighbors. However, sampling is typically decoupled from evolving embedding quality, causing redundant exploration of well-trained regions while under-sampling undertrained nodes. At the system level, such decoupling further leads to excessive communication, serialized execution, and low resource utilization in distributed environments. We present FeLoG, a feedback loop-driven system for scalable distributed graph embedding. (1) FeLoG introduces feedback-coupled sampling and training, dynamically prioritizing undertrained nodes according to real-time embedding-quality feedback, thereby reducing redundant computation and accelerating convergence. (2) It employs activity-aware communication that compresses frequently occurring node sequences to reduce intra-machine PCIe traffic and selectively synchronizes frequently updated embeddings to reduce inter-machine communication. (3) It adopts a round-interleaved pipeline that overlaps next-round sampling with current-round training to improve CPU-GPU utilization. Experiments against six state-of-the-art baselines on large-scale graphs show that FeLoG achieves an average speedup of 27.9x, reduces communication cost by more than 53.1%, and sustains over 80% CPU-GPU utilization.

Semi-supervised learning (SSL) on data streams is challenging due to the continuous evolution of high-volume data and the scarcity of labels. Existing methods are limited in leveraging the intrinsic relationships among samples because they typically rely on fixed similarity measures or static graph structures, which cannot capture how relationships evolve over time. We propose SLeDGe, an SSL method for data streams that jointly learns a predictive model and an adaptive graph structure under strict memory and label constraints. SLeDGe maintains compact labeled and unlabeled memories using distinct update strategies, balancing rapid adaptation to novel features with the retention of historical consistency. In addition, by encouraging sparsity in the relational graph, SLeDGe filters out spurious connections and enables effective propagation of label supervision. Across 12 datasets, SLeDGe outperforms state-of-the-art competitors, achieving average relative accuracy gains of 31.7% with 0.1% labels and 14.8% with 1% labels.

Origin-Destination (OD) demand prediction is fundamental to intelligent transportation systems, yet real-world OD flows are often dynamically sparse, long-tailed, and characterized by heterogeneous zero-flow patterns. These properties make it difficult to distinguish whether an OD connection is active from how much demand it generates once activated. Many existing methods primarily treat OD prediction as a single flow regression task, which limits their ability to model low-frequency, intermittent, and long-tailed OD interactions. To address these challenges, we propose SAGMTL, a Structure-Aware Graph Multi-Task Learning framework for dynamic sparse OD demand prediction. SAGMTL decomposes OD prediction into structural state modeling and flow intensity estimation, jointly learning regional activity states, OD connection activity, and edge-level flow intensity within a unified framework. Specifically, a node-edge collaborative representation module captures regional semantics, temporal dynamics, and spatial priors through interactive node-edge updates, producing structure-aware representations for dynamic OD interactions. Based on these representations, SAGMTL estimates OD flows by jointly modeling stable demand patterns and short-term fluctuations. A multi-constraint objective further improves sparsity awareness and structural consistency. Experiments on three real-world urban mobility datasets from Beijing, Chengdu, and Nanjing show that SAGMTL achieves superior overall performance compared with state-of-the-art baselines. Further analysis demonstrates that explicitly modeling regional activity, connection states, and flow intensity improves the robustness of dynamic sparse OD demand prediction.

Automated skin cancer classification from dermoscopic images remains challenging due to heterogeneous lesion structure, strong intra-class variability, and subtle visual differences between benign and malignant cases. Existing CNN/ViT pipelines typically rely on global or patch-level features and often combine patient metadata via late fusion, which limits spatially grounded multimodal reasoning. We present a novel region-based graph learning framework that explicitly models lesions as graphs of spatially coherent superpixel regions represented as frozen CNN features. To capture fine-grained lesion arrangements, we encode inter-regional geometry as edge attributes and introduce a dedicated metadata context node connected to all regions, providing structured integration of demographic/clinical variables within the same relational space. Node representations are updated using our edge-aware graph transformer followed by attention-driven propagation, and a final graph-level embedding for benign-malignant classification. Experiments on four public benchmarks demonstrate that explicit region-level relational modeling and graph-native multimodal fusion yield consistent gains over the state-of-the-art. Consequently, we establish a new graph-centric perspective in which CNN features are modeled as relational nodes and improved through contextual integration, yielding more expressive and robust classifications.

Map generalization remains one of the fundamental tasks in cartography, especially for the simplification and aggregation of complex building footprints. This study presents the first exploratory application of graph-based deep learning to both tasks, reformulating simplification as node movement prediction and aggregation as link prediction within a unified graph learning framework. We evaluate representative graph neural network architectures (GCN, GAT, and GraphSAGE) on multi-scale building datasets, showing that GraphSAGE demonstrates relative strengths in link prediction accuracy, while also revealing persistent challenges in precise node movement prediction. Beyond quantitative performance, the results highlight that aggregation poses greater complexity and challenges than simplification, underscoring the difficulty of capturing higher-level spatial relationships in map generalization with current deep learning approaches. Although limitations such as data imbalance and the need for post-processing remain, the study provides valuable insights and methodological directions for advancing automated map generalization with deep learning approaches.

We propose an information-theoretic framework for graph novelty generation, which aims to generate data that are distinct from existing patterns while preserving global structural consistency. Our approach embeds data into a latent space, models the latent distribution using finite mixture models, and generates novel samples by imposing explicit novelty and reliability conditions formulated in terms of description length. Specifically, novelty is enforced by requiring generated samples to be poorly explained by all existing mixture components, while reliability constrains their impact on the overall mixture structure under the Minimum Description Length (MDL) principle. We provide a theoretical analysis showing that, with appropriate threshold choices, the probabilities of misclassifying non-novel or unreliable samples converge to zero with explicit rates. Experiments on synthetic and benchmark graph datasets demonstrate that the proposed method enables principled novelty generation with quantifiable risk.

The Traveling Salesman Problem (TSP) is a cornerstone of combinatorial optimization and arises in many practical scenarios. Although graph-based learning approaches have been explored for TSP, the question of how to exploit graph structure more effectively remains open. We present the Anisotropic Graph Diffusion Network (AGDN), a new Graph Neural Network framework designed to solve TSP. Our method tackles two central difficulties: (1) the lack of informative topological prior in fully connected TSP graphs, and (2) losing connected nodes in the optimal solution after the commonly used graph sparsification techniques. To overcome these issues, we construct a MixScore transition matrix that merges node similarity with pairwise distance, and we develop an anisotropic graph diffusion strategy that supports efficient information exchange across multiple hops. Comprehensive experiments spanning diverse instance sizes and node distributions show that AGDN consistently outperforms existing methods while keeping computation time competitive. Furthermore, AGDN generalizes well to problem sizes and distributions beyond those seen during training. The implementation is publicly available at: https://github.com/LabRAI/AGDN.

Graph-based retrieval at billion-node scale requires jointly solving three tightly coupled problems -- graph construction, representation learning, and real-time serving -- yet existing work addresses each in isolation. We present RankGraph-2, a framework deployed at Meta that co-designs all three lifecycle stages for similarity-based retrieval (U2U2I and U2I2I), where each stage's requirements shape the others. Serving requires a co-learned cluster index to avoid expensive online KNN -- this pushes index co-training into the training objective. Training benefits from the observation that similarity-based retrieval tolerates pre-computed neighborhoods, eliminating online graph infrastructure -- this requires construction to produce self-contained data. Construction must also support hour-level refresh for item coverage. Acting on these cascading requirements, RankGraph-2 reduces hundreds of trillions of edges to hundreds of billions via subsampling with popularity bias correction, pre-computes multi-hop neighborhoods via personalized PageRank, and co-learns a residual-quantization cluster index that reduces serving computational cost by 83%. This lifecycle co-design enables a simple architecture to achieve 3.8 x higher recall than a GAT + Deep Graph Infomax model on a bipartite graph and 2.1 x higher than PyTorch-BigGraph on item retrieval. RankGraph-2 delivers up to +0.96% CTR and +2.75% CVR, and has powered 20+ retrieval launches across major surfaces.

This paper explores the problem of finding the minimum zero-forcing set on undirected graphs and proposes an adapted machine-learning framework to solve the problem. The minimum zero-forcing set problem is a graph coloring problem where the color of an initial set of nodes propagates throughout a network. The set of nodes is zero-forcing if it forces all uncolored nodes to change color under the constraint of the color-change rule. There are several applications to this problem across different domains such as network science, network control, and designing logical circuits. Finding the minimum zero-forcing set is shown to be NP-hard. We propose a reinforcement learning framework, SD-ZFS, that adapts the S2V-DQN architecture to the ZFS problem. We train several models on this adapted framework and analyze the performance across graph datasets that have varying structures. We evaluate how the models trained on the framework generalize, scale, and transfer to different network types. The results demonstrate the effectiveness of the framework when compared against the optimal solution and greedy heuristic. We provide further insight into how the ZFS problem can be solved through machine-learning and the influence of network structure on the problem.

In recent years, the rapid development of foundation models and graph pre-training technologies has spurred increasing interest in constructing a universal pre-trained graph model or Graph Foundation Model (GFM). However, a significant challenge is that existing models are unable to address feature heterogeneity in graph data without textual information, which hinders the transferability of graph models across different datasets. To bridge this gap, we propose the concept of learnable graph patches, which we regard as the smallest semantic units of any graph data. We decompose the graph into learnable graph patches by unfolding the node features and constructing corresponding patch structures separately. We then design a framework that mines transferable information from graph data across domains. Specifically, after extracting graph patches, we propose a patch encoder to extract knowledge from each unit and a patch aggregator to learn how the units are combined into a whole. Due to its domain-agnostic nature, the model can be applied to downstream data across different domains. Furthermore, we analyze the connection between our method and existing graph models, as well as the transferability of the node embeddings it generates. Empirically, our method not only achieves the capability to use multi-domain graphs for pre-training, but also shows enhanced performance across various downstream datasets and tasks. Moreover, we observe consistent improvement in downstream performance as the volume of pre-training data increases.

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Link prediction in knowledge graphs fundamentally depends on the quality of learned embeddings for entities and relations. However, most existing methods derive these embeddings by aggregating only the local neighborhood of each entity, neglecting the global structure of the knowledge graph. This limited view prevents models from capturing higher-level structural patterns that are essential for accurate and generalizable link prediction. To address these limitations, we introduce Model Graph Inductive Learning (\textbf{MGIL}), a framework that constructs a model graph by clustering entities based on the similarity of their incoming and outgoing relational structures or their entity types. A GNN is then applied to this model graph to produce embeddings that capture the global view of the knowledge graph. These embeddings subsequently serve as high-quality initial features %embeddings for the original knowledge graph, replacing random initialization and leading to more stable and expressive representations. Extensive experiments on standard and recently proposed inductive benchmarks demonstrate that MGIL achieves state-of-the-art or highly competitive performance in inductive link prediction, highlighting its effectiveness across diverse graph settings.

The rapid growth of modern training datasets has significantly increased computational cost, motivating dataset pruning~(DP) methods which retain only a subset of informative samples to reduce training cost. Existing pruning criteria typically rely on either intrinsic signals that assess samples independently or extrinsic signals that promote diversity via pairwise relations. While effective in their own specific regimes, each captures only one aspect of sample utility and lacks robustness across different pruning ratios or data distribution. In this work, we present a unified graph-based DP framework. By modeling the dataset as a weighted graph, where node weights encode intrinsic value and edge weights encode extrinsic value, DP can be cast as a Maximum Weight Clique Problem (MWCP). Although MWCP is NP-hard, its structure admits a principled greedy solution based on sample-wise marginal gains. Under a few mild conditions, we further prove that this unified objective enjoys a formal approximation guarantee, which applies to a broad family of importance metrics and provides practical design guidelines. Extensive experiments show that our method outperforms existing DP methods while substantially reducing training cost, reducing training time by over 40% without sacrificing accuracy on ImageNet-1k with ResNet-50.

Information lattice learning (ILL) learns interpretable rules of a signal by alternately projecting the signal onto a partition lattice that encodes a hierarchy of abstractions and lifting selected rules back to the signal domain. When the signal is a probability mass function, we show the probabilistic rules learned by ILL admit a natural probabilistic graphical model (PGM) interpretation and develop this interpretation in detail. A partition in ILL induces a deterministic quotient variable, and a rule is the marginal law of that quotient variable. A rule set is therefore a collection of marginal constraints over interpretable abstractions. General lifting is the feasible family of all joint distributions satisfying those constraints, while special lifting chooses a maximum-ignorance reconstruction, implemented in ILL by an L2 uniformity principle closely related to maximum entropy. Under a Shannon-entropy lifting, the same constraints yield a log-linear factor graph whose factors are indexed by learned abstractions. The information lattice itself, however, is not a Bayesian network: its edges encode refinement and coarsening of abstractions, not conditional dependence. Thus ILL is best viewed as structure learning for interpretable constraint-based factor graphs over quotient variables. This view clarifies how ILL relates to graphical models and maximum entropy models, while suggesting new directions for inference, identifiability, and hybrid symbolic-probabilistic learning.

Neural relational inference (NRI) methods discover interaction graphs from trajectories through variational reasoning on discrete potential edges. However, these methods typically rely on oversimplified, factorized graph priors. Such priors, typically nearing uniform distributions, treat edges as independent entities. This systemic misalignment does not match the real-world systems and yields diffuse and indecisive edge posteriors limiting the reliability of structural discovery. To address this, we propose \textit{Diff-prior}, a diffusion-parameterized adaptive prior used to calibrate latent graph distribution rather than generate graphs. Our core insight is to reframe prior integration as a learnable denoising-style calibration that organizes scattered, uncertain edge posteriors into a more reliable overall structure which can be trained by the diffusion model. Diff-prior learns an adaptive structure prior that performs structured calibration on the edge posteriors during inference, guiding it towards a distribution closer to the underlying structure. The diff-prior operates before structural sampling and acts as a denoising calibrator directly on the encoder edge distribution, which provides a generic training paradigm over structured variables. Experiments on standard benchmarks validated our framework, and the results indicate that Diff-prior improves the performance of structure inference and generates more decisive edge posteriors across multiple NRI-family architectures. The code is available on https://github.com/Hardy158118/Diffprior.

Large Language Models (LLMs) have advanced rapidly, but their limitations in structured and multi-hop reasoning underscore the need for graph-native, synergistic artificial intelligence (AI) systems. Graph-structured data underpins critical applications across social, biological, financial, transportation, web, and knowledge domains, making it essential to understand how LLMs can leverage graph computation for grounded, context-rich inference. Three complementary synergies are emerging: LLMs augmented with graph computation for retrieval and reasoning; bidirectional integration between LLMs and knowledge graphs (KGs), where LLMs support KG construction and curation while KGs enforce semantic constraints and factual consistency; and AI agents strengthened by graph algorithms for planning, decision making, and multi-step reasoning. In parallel, LLMs introduce new capabilities for graph data management and graph machine learning (ML) through natural language interfaces and hybrid LLM-graph neural network (GNN) pipelines. This tutorial synthesizes the algorithms, systems, and design principles driving these converging directions, offering data science and data mining researchers a unified perspective on integrating LLMs, graph data management, graph mining, graph ML, and agentic computation into next-generation graph-native AI systems.

Learning Bayesian network (BN) structure from sparse discrete data is hard: when each instance records only a few variables, most variable pairs lack the joint observations needed for reliable scoring, and data-only methods recover little structure. Imperfect domain knowledge, expressible as a weighted directed knowledge graph (KG), is often available. KG-SoftMAP encodes such a KG as a finite-strength, confidence-weighted edge prior and maximizes a MAP objective that adds this logit-form prior to the BDeu score. With an informative but imperfect KG, KG-SoftMAP recovers partial directed structure even at observation rate rho=0.05, with directed F1 (DF1) of 0.19-0.32 across benchmarks. At higher observation rates within this sparse grid, DF1 reaches 0.44-0.66 at rho=0.20 and 0.46-0.64 at rho=0.40. Across the same three rates, KG-SoftMAP without the KG prior averages DF1 0.00, 0.19, and 0.21. Stress tests that corrupt, remove, or blur the KG signal, together with checks on LLM-extracted graphs beyond canonical benchmarks, show that recovery rises and falls with KG quality. On three real sparse educational datasets without ground-truth DAGs, we evaluate prediction, calibration, and KG-consistency. On Short Answer Feedback (SAF), KG-SoftMAP+VE reaches Fail-class F1 0.75 versus 0.78 for logistic regression while also providing an inspectable concept graph, calibrated Fail probabilities, and posterior queries from partially observed concept evidence. The remaining datasets sharpen the operating picture: weak heuristic KG signal leaves prediction unchanged, while an independent expert ontology moves the learned graph toward expert relatedness.

Graph Contrastive Learning (GCL), which trains graph encoders by maximizing similarity between positive samples and minimizing it between negative ones, has emerged as a mainstream graph pre-training paradigm. It is widely recognized that positive samples are essential in GCLs. Ideally, maximizing the similarity of positive samples enables graph encoders to capture intrinsic semantic and patterns of graph data. However, we discover an interesting phenomenon: GCLs can achieve competitive performance even without positive samples. This motivates us to revisit the fundamental mechanism of positive samples in GCLs. From the perspective of Dirichlet energy, we theoretically finds that message passing, a key mechanism in graph encoders, trivializes the maximization of positive samples, preventing GCLs from effectively learning from positive samples. To address this, we propose SPGCL to mitigate the trivialization caused by message passing and restore the learning efficacy of positive samples. Specifically, we find that high Dirichlet energy features help positive samples provide effective learning signals while low Dirichlet energy features contribute little to positive learning signal but is useful for positive sampling. Based on this, SPGCL propagates only high Dirichlet energy features and uses low energy features to construct a probability matrix for reliable positive sampling. Extensive experiments demonstrate the effectiveness of SPGCL.

Relational deep learning (RDL) converts relational databases (RDBs) into heterogeneous graphs, but graphs derived directly from database schemas are often not well suited for how graph neural networks (GNNs) perform relational reasoning. We study what makes a relational graph suitable for deep learning and show that schema-derived graphs suffer from two systematic failures: information overload and semantic fragmentation. Our empirical analysis reveals that the desired graph is not the raw schema, but a result of controlled structural adaptation. Performance depends on balancing two operations: mitigating information overload via filtering, and repairing semantic fragmentation via injection. Specifically, filtering serves as a bias-variance knob with non-monotonic effects, while injection improves performance only when it explicitly restores the relational dependencies missing from the original schema. Based on these findings, we develop an end-to-end structural optimizer that applies both operations to adapt relational graphs automatically. Across 26 tasks spanning classification, regression, and recommendation, the optimized graphs consistently improve accuracy while often reducing inference cost.

Modern image-analysis pipelines often convert images into structured semantic variables, such as facial attributes, object concepts, and scene descriptors. Learning directed dependencies among these variables can produce interpretable visual semantic graphs, but continuous directed acyclic graph learning is limited by the cost of enforcing acyclicity. We present polyDAG, a polynomial acyclicity framework for efficient continuous causal discovery in visual semantic graphs. polyDAG replaces the matrix-exponential acyclicity constraint with a finite polynomial trace constraint and proves that the new constraint is zero exactly for acyclic graphs. We further derive a geometric-series implementation that avoids the explicit summation loop while preserving the same acyclicity condition. Experiments on synthetic Erdos-Renyi graphs and CelebA facial visual attributes show that polyDAG improves efficiency and structure recovery. Averaged over the revised synthetic protocol with d in {100, 200, 500}, polyDAG reduces mean structural Hamming distance from 318.4 to 285.4 and improves mean F1 score from 0.725 to 0.756. At 100 nodes, the geometric variant runs in 3.44 seconds compared with 5.16 seconds for the exponential baseline, corresponding to a 33.4 percent speedup. Code and data are publicly available at https://github.com/wenhaoz-fengcai/polyDAG.

Huntington's disease (HD) is a progressive brain disorder that gradually affects movement, cognitive function, and behavior. Identifying the stage of the disease accurately and consistently is important for understanding its course, grouping patients, personalized care, and discovering treatment. Existing clinical staging frameworks rely primarily on predefined clinical measurement thresholds and clinical expert decisions, yet these discrete cut-offs may obscure meaningful intra-stage variability and remain vulnerable to inter-rater differences, especially in motor and functional assessments. To address these limitations, we developed an unsupervised machine learning framework based on dynamic graph representation learning to capture temporal relationships within and across patients from longitudinal clinical measurements. Using the learned representations, we applied K-means++ clustering to identify well-separated groups. We then iteratively increased the number of clusters (k), using stability analysis to assess robustness and reveal additional meaningful clusters beyond the initial optimal solution. We applied the framework to 302 individuals from the Enroll-HD cohort (1,477 visits, 44 clinical variables per visit; 80% manifest participants), enabling data-driven discovery of HD stages reflecting natural clinical progression. Despite the limited cohort size, the proposed framework achieved robust clustering performance using a four-dimensional latent space, identifying four meaningful and statistically distinct disease stages through clustering stability analysis. Each stage corresponded to well-defined clinical measurement boundaries, with minimal overlap compared to previously established clinical staging methods.

We establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings, where the goal is to determine with vanishing error which of two structured planted mechanisms generated the observed data. We prove matching low-degree upper and lower bounds for counting communities in the planted submatrix and planted dense subgraph models. The resulting testing threshold coincides, down to the sharp constant, with the known low-degree recovery threshold. In contrast, the task of weak testing, where the goal is to outperform random guessing, does not have a sharp threshold but rather a smooth transition, which we identify. To prove our results, we develop a framework for planted-vs-planted testing that builds on a latent-variable expansion originating in low-degree recovery and employs new methods to identify and prune non-signal contributions.

Early and accurate detection of Alzheimer's disease (AD) is important for timely intervention and disease management. Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM) and its Universum-based variants have shown promising results for AD classification. However, existing methods treat Universum samples as independent points and do not consider the geometric relationships among them. This paper proposes two graph-guided Universum learning models, namely UG-GEPSVM and IUG-GEPSVM, for AD versus cognitively normal (CN) classification using structural MRI data. In the proposed framework, mild cognitive impairment (MCI) subjects are used as Universum data to provide intermediate information between AD and CN classes. A graph is constructed over the Universum samples using Gaussian similarity, Minimum Spanning Tree connectivity, and multi-hop propagation. From this graph, a Laplacian matrix is derived that captures the geometric structure of the MCI samples. This Laplacian-based regularization is incorporated into the learning process in place of the conventional independent Universum penalty term. UG-GEPSVM integrates this regularization into the generalized eigenvalue formulation, while IUG-GEPSVM extends the numerically stable improved GEPSVM framework using a standard eigenvalue formulation. Experiments on ADNI MRI dataset variants using ICA- and PCA-based features at five different noise levels show that both proposed models consistently outperform existing GEPSVM and Universum-based methods. UG-GEPSVM achieves the highest average AUC of 88.07% and maintains stable performance under increasing noise levels. Statistical tests further confirm the significance of the observed improvements.

Active learning (AL) for node classification typically focuses on selecting the most informative nodes for annotation within one or a few large graphs (e.g., in social network analysis). However, in other domains, such as molecular chemistry or electronic design automation, datasets consist of thousands of independent graphs. In many of these inductive settings, annotating an individual node requires a full-graph analysis, which effectively yields the remaining node labels on-the-fly. Therefore, these scenarios require AL strategies that select entire graphs instead of single nodes, a problem which has not been tackled in the literature so far. Thus, we introduce ALINC, an AL framework for inductive node classification via graph sampling. It bridges the existing methodological gap by elevating node-level utility measures to graph-level selection criteria through various aggregation mechanisms. In an extensive benchmark including ten strategies, three aggregation methods, and four datasets, we identify CoreSet, TypiClust, and BADGE as the top-performing graph sampling strategies. Our detailed analysis further reveals that the choice of the aggregation method is pivotal, as it substantially affects model performance and annotation costs. Finally, we demonstrate the effectiveness of ALINC in two use case studies: site-of-metabolism prediction in molecules and design automation of printed circuit board schematics.

Graph-based deep learning methods have been widely employed in encrypted traffic analysis to exploit latent correlations across different granularities. However, while complex preprocessing pipelines and sophisticated model structures often achieve strong performance, they may obscure inherent protocol semantics during representation learning. Moreover, the hierarchical structure of protocol layers and their corresponding fields, defined by protocol specifications and routinely utilized in manual traffic analysis, remains underexplored in existing learning frameworks. In this paper, we propose Protocol Tree Graph Attention with Mixture of Experts (PTGAMoE), a semantic-preserving hierarchical graph-based expert framework for encrypted traffic analysis. The field-based graph construction and expert committee design enable PTGAMoE to quantify the model's preferences for specific fields and protocols. Extensive experimental results on representative benchmark datasets under strict no-data-leakage settings demonstrate that PTGAMoE significantly outperforms state-of-the-art (SOTA) models. Furthermore, the semantic-preserving design provides interpretable insights into protocol-level feature importance and expert-level contributions, reflecting the model's decision-making logic in encrypted traffic classification tasks.

Graph neural networks have achieved strong performance on graph-structured data, but their effectiveness depends heavily on the quality of the observed graph. In real applications, graph topology is often imperfect: noisy edges may connect unrelated nodes, while missing edges may prevent useful information from being propagated. Existing robust graph learning methods mainly address this problem by removing suspicious edges or by learning a new graph structure during training. However, edge removal alone cannot recover missing connections, and graph structure learning may introduce additional optimization complexity. In this paper, we propose Topology-Aware Gaussian Repair (TAGR), a simple graph repair framework for robust message passing in graph neural networks. Instead of learning a dense adjacency matrix, TAGR constructs a sparse feature-neighborhood graph using an adaptive Gaussian kernel and combines it with a topology-aware residual correction of the observed graph. The Gaussian repair component introduces auxiliary edges between feature-similar nodes, while the residual correction preserves and reweights the original topology according to local feature and structural consistency. The repaired graph can be used directly with standard graph neural networks without changing their architectures. Extensive experiments on benchmark citation networks show that TAGR improves the robustness of GNNs under both noisy-edge and missing-edge settings. The analysis further show that Gaussian feature-neighborhood repair provides the main robustness gain, while topology-aware residual correction improves stability when the observed graph is incomplete. These results suggest that effective graph robustness can be achieved through lightweight sparse graph repair rather than dense graph structure learning.

Graph foundation models (GFMs) emerged as a dominant paradigm in graph representation learning by leveraging large-scale pre-training for cross-domain inference. However, the parameterized knowledge encoded within these models is insufficient to cope with distribution shifts, limiting their generalization ability. To mitigate this issue, retrieval-augmented generation (RAG) has been introduced to incorporate external knowledge at inference time. Nevertheless, existing RAG frameworks operating in Euclidean space suffer from a fundamental geometric limitation: the polynomial volume growth of Euclidean space is inherently mismatched with the tree-structured external knowledge bases. This mismatch leads to the loss of semantic granularity in retrieval and gives rise to the hubness phenomenon.To address this limitation, we propose a Hyperbolic Retrieval-Augmented Generation (HyRAG) framework designed to enhance the generalization capabilities of GFMs. Specifically, the introduced Hyperbolic Knowledge Indexing module retains the tree-like hierarchies of the external knowledge base by modeling them within hyperbolic space. The Multi-granularity Retrieval module then provides GFMs with the global semantic anchors and local semantic nuances through coarse-grained and fine-grained knowledge retrieval, respectively. Finally, the Dual-path Fusion module achieves effective knowledge integration for graph tasks at both the feature and structural levels. Experiments on multiple graph benchmarks demonstrate significant improvements in the zero-shot setting, highlighting the generalization of our method for robust GFMs inference.

Dynamic graph learning (DGL) is essential for modelling evolving graph data, but existing methods suffer from significant computational overhead due to repeated full-snapshot retraining and are not well-suited for collaborative settings with partitioned data. In realistic graph systems, cross-partition edges are unavoidable, but direct sharing of graph structure between clients may violate privacy constraints. We propose DG-CoLearn, a client-oblivious collaborative dynamic graph learning framework built on incremental graph snapshot processing, which focuses computation on graph regions affected by temporal updates while preserving historical information through temporal modelling. This incremental design is consistently applied across the entire graph processing pipeline, including a server-mediated embedding exchange mechanism to enable accurate multi-hop message passing without exposing raw cross-client structural information. Extensive experiments demonstrate that DG-CoLearn achieves up to 33.8$\times$ speedup in training time and 27.4$\times$ reduction in communication overhead, while consistently improving predictive performance on both node classification (up to 13.36% F1 improvement) and link prediction (up to 8.27% MAP improvement) tasks. These results highlight the effectiveness of DG-CoLearn in bridging efficiency, scalability, and client-to-client structural privacy in collaborative dynamic graph learning.

Graph neural networks (GNNs) are limited to modeling pairwise interactions, while higher-order models based on cell complexes achieve greater expressivity but often suffer from poor scalability. We introduce simplified and factored cellular Weisfeiler Leman tests (sCWL and fCWL), which preserve the expressivity of the CWL test while improving computational efficiency. We further introduce the maximal clique complex, enabling scalable CWNs with reduced time and memory complexity while retaining strong empirical performance. To avoid explicit clique enumeration, we propose CliqueWalk, a biased random walk that samples maximal cliques and scales linearly with graph size. These contributions yield a scalable topological learning framework for higher-order graph representation.

Spatio-temporal graph-based models have typically been used to forecast new cases of infectious diseases such as COVID-19 and chickenpox outbreaks. However, the use of stochastic modelling into their learning process has been surprisingly under-investigated and rarely considered entire data sets of large countries. As a result, it is unknown whether these models would provide accurate forecasts in real-world disease spread scenarios. In this work, we propose a spatio-temporal stochastic graph-based architecture that integrates a stochastic formulation and uncertainty approximation process to forecast new infectious disease cases. We find that our approach can adapt to encode large and small population geographical networks within a single model architecture. Using two real-world data sets, COVID-19 in the US and chickenpox in Hungary, we report an enhanced effect of the proposed architecture across predictions of the 2022 first wave for COVID-19 in the US and comparative results of chickenpox waves during 2012-2014 in Hungary. By benchmarking with four spatio-temporal graph-based models, quantitative results show competitive overall weekly performance of the proposed approach on forecasting new cases for all 3,218 US counties and all 20 Hungary counties. The proposed approach can represent overall epidemic progression relative to baselines, though with a one-step delay; while exhibiting a reduced sensitivity to high-frequency and low-amplitude variability.

Graph Machine Learning as a Service (GMLaaS) platforms increasingly implement explainability interfaces to meet regulatory transparency requirements. However, this transparency creates exploitable vulnerabilities for model extraction attacks. We present the first model extraction attack specifically designed for graph classification under strict black-box constraints where the attacker observes only discrete class labels and binary explanation masks (no probability scores, gradients, or confidence values). Our method (1) uses model explanation outputs to guide Monte Carlo edge sensitivity estimation toward decision boundaries, with Hoeffding concentration guarantees on estimation accuracy and (2) exploits explanation subgraphs to efficiently narrow the boundary search space. Extensive experiments on benchmark graph datasets across multiple domains demonstrate our method's superiority over comparable baselines. These findings demonstrate that such explainability interfaces create exploitable attack surfaces, informing both defensive mechanisms and policy frameworks for explainable AI mandates. The implementation code is provided in https://github.com/LabRAI/XSTEAL/.

Graph clustering is essential in graph analysis for revealing structural patterns and node communities. Despite recent advances in self-supervised contrastive learning that have improved clustering via structural and attribute signals, existing methods still struggle to flexibly capture high-order local structures and often overlook global semantics in complex graphs. These limitations lead to suboptimal node representations, especially in real-world graphs with fragmented structures and ambiguous cluster boundaries. To address these limitations, a contrastive graph clustering framework is proposed to jointly integrate multi-scale local structures with global semantics via attention mechanisms. At the local level, GNN-based topological signals extracted from multiple propagation depths are adaptively fused through attention-based weighting to capture multi-scale neighborhood features. At the global level, semantic prototypes derived from dynamically evolving cluster centers are adaptively aggregated through attention to guide node representations and enhance inter-cluster separability. The model is trained under a dual-view contrastive learning paradigm with a hybrid objective that combines instance-level and structure-aware losses to improve representation robustness and discrimination. Experiments on eight real-world graph datasets demonstrate that our method achieves competitive clustering performance. Code is available at https://github.com/vege12138/w2.

International Classification of Diseases (ICD) is a globally recognized coding system that records diagnostic events during each patient encounter, providing a standardized data foundation for various clinical tasks. However, the irregular and hierarchical nature of ICD code sequences poses challenges for N-D lattice-based sequential modeling methods, leading to overly complex model designs. In this paper, we propose GraD-IBD, a graph diagnosis model that reformulates longitudinal ICD trajectories as visit-bucketized, temporally directed graphs to detect the risk of inflammatory bowel disease (IBD). A novel context-aware, time-decay message passing mechanism was developed to capture temporal dependencies while reducing model complexity. The experimental results using a real-world clinical dataset demonstrated consistent and robust improvements in IBD detection over state-of-the-art methods, with significant reductions in computational complexity compared to sequential models. These findings highlight the potential of graph representation learning to enable efficient, scalable, and accurate disease risk prediction from longitudinal ICD diagnosis codes.

The rich information underlying graphs has inspired further investigation of unsupervised graph representation. Existing studies mainly depend on node features and topological properties within static graphs to create self-supervised signals, neglecting the temporal components carried by real-world graph data, such as timestamps of edges. To overcome this limitation, this paper explores how to model temporal evolution on dynamic graphs elegantly. Specifically, we introduce a new inductive bias, namely temporal translation invariance, which illustrates the tendency of the identical node to keep similar labels across different timespans. Based on this assumption, we develop a dynamic graph representation framework CLDG that encourages the node to maintain locally consistent temporal translation invariance through contrastive learning on different timespans. Except for standard CLDG which only considers explicit topological links, our further proposed CLDG++ additionally employs graph diffusion to uncover global contextual correlations between nodes, and designs a multi-scale contrastive learning objective composed of local-local, local-global, and global-global contrasts to enhance representation capabilities. Interestingly, by measuring the consistency between different timespans to shape anomaly indicators, CLDG and CLDG++ are seamlessly integrated with the task of spotting anomalies on dynamic graphs, which has broad applications in many high-impact domains, such as finance, cybersecurity, and healthcare. Experiments demonstrate that CLDG and CLDG++ both exhibit desirable performance in downstream tasks including node classification and dynamic graph anomaly detection. Moreover, CLDG significantly reduces time and space complexity by implicitly exploiting temporal cues instead of complicated sequence models.

Graph anomaly detection (GAD) commonly relies on message passing to jointly encode node attributes and neighborhood context. However, once the two are mixed, an abnormal post-encoding state may reflect either an intrinsic node deviation or incompatible contextual influence, making its source ambiguous. We propose Disagreement-Driven Graph Anomaly Detection (DDGAD), which treats persistent incompatibility between node-wise and contextual estimates as anomaly evidence. Inspired by Adapt-Then-Combine (ATC), DDGAD reverses its consensus objective: Adapt produces a node-wise estimate without new same-step neighbor aggregation, while Combine forms a neighborhood-dependent contextual estimate, and their pre-consensus disagreement is accumulated across ATC steps for detection. We further characterize this signal from graph-spectral and source-response perspectives and derive sufficient conditions for anomaly--normal separation under contextual mixing. Experiments on six benchmarks show the highest average AUROC among the evaluated methods, while controlled interventions and Adapt-operator controls further support persistent disagreement as an effective detection signal.

Graph fraud detection has long depended on Graph Neural Networks (GNNs) to propagate and aggregate information across relational data. A critical obstacle in practice, however, is that fraudsters frequently disguise themselves by forging numerous connections with benign users, causing fraud signals to be progressively diluted during neighborhood aggregation and undermining detection reliability. While recent efforts have used Large Language Models (LLMs) to provide rich semantic cues for fraud detection, the underlying intent behind suspicious connections remains insufficiently explored. Compounding this issue, the scarcity of annotated fraud samples makes it difficult to train detectors that remain robust under heavy camouflage. To address these gaps, we propose L2IR, an LLM-driven Latent Intent Revealing framework for graph fraud detection. By uncovering latent intent from both user behaviors and suspicious connections, L2IR extracts intent-aware representations from raw behavioral traces and reasons about the true purpose behind individual connections, effectively distinguishing supportive links from misleading ones. It further incorporates adaptive self-training to enhance robustness under limited supervision. Evaluations on two real-world datasets characterized by pervasive camouflage demonstrate that L2IR surpasses strong baselines and can function as a plug-in enhancement for a range of GNN-based detectors, improving AUPRC by up to 8.27%.

In biomedical and neurodegenerative disorders, accurate and early disease identification remains challenging due to the scarcity of labeled data and the complexity of imaging patterns. To address these challenges, we introduce ARMA-C3, a unified unsupervised and semi-supervised graph learning framework for node classification based on contrastive learning and graph-cut regularization to learn structurally meaningful and discriminative representations. By modeling samples or images as graph nodes and exploiting inter-sample relationships, the proposed framework captures subject-level dependencies that conventional machine learning methods typically overlook. We conduct extensive binary classification experiments across five clinically relevant datasets: the Alzheimer's Disease Neuroimaging Initiative (ADNI), the Neuroimaging in Frontotemporal Dementia (NIFD) dataset, and three medical imaging benchmarks (BreastMNIST, PneumoniaMNIST, and a liver ultrasound dataset). Experimental results demonstrate that ARMA-C3 achieves competitive and frequently superior performance compared to classical clustering techniques, state-of-the-art machine learning models, and existing graph-based deep learning approaches across multiple evaluation settings, particularly under limited supervision and severe class imbalance. The proposed framework further demonstrates robust representation learning and strong cross-modal generalization across diverse biomedical imaging modalities.

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{tΔ}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly defined, and we recover the classical graph-Laplacian pointwise rate $O(N^{-2/(d+6)})$ up to logarithmic factors, for diffusion times $t$ up to $O(1)$ and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of $Q_t f$ at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density $p$ is $C^3$ and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.

Chain-of-Thought (CoT) prompting has shown promise in enhancing the reasoning capabilities of large language models (LLMs) on text-attributed graphs (TAGs). This work reframes CoT-based graph learning through the principle of clustering as reasoning, offering a $k$-means interpretation of how iterative reasoning operates over graph-structured data. We observe that existing graph CoT methods rely on disjoint architectures and fixed graph representations, limiting step-by-step semantic-topological interaction and interpretability. To overcome this limitation, we propose a unified framework named KCoT that integrates CoT reasoning with graph representation learning. Our key theoretical result reveals a formal mathematical correspondence between a Transformer block and the $k$-means algorithm, allowing reasoning to be interpreted as iterative assignment and update steps. Based on this insight, we introduce a Semantic Discriminating Prompt that explicitly formulates these steps as structured CoT reasoning, together with a structure-grounded alignment strategy to fuse topological priors with evolving thought-conditioned representations. Experiments on standard benchmarks demonstrate consistent improvements over state-of-the-art methods, validating clustering as a principled mechanism for CoT-based graph learning.

Biological systems are governed by structured molecular interactions, where pathways, regulatory circuits, and functional gene relationships shape cellular behavior and disease progression. Much of this knowledge is naturally represented as graphs. However, most biomedical AI models cannot directly use graph-encoded biological knowledge and instead require compressed low-dimensional representations, which can lose important structure and reduce performance, especially in limited-sample clinical studies. Here, we introduce Graph-in-Graph (GiG), a knowledge graph-modulated deep learning framework for data-efficient clinical prediction. GiG represents each patient as a standalone modular graph, in which curated biological knowledge graphs define edges and patient-specific measurements, such as gene expression, define node features. This design allows multiple biological knowledge graphs to be integrated while preserving gene-gene interactions and pathway topology during patient-level representation learning. Across cohorts comprising nearly 9,700 patients and five clinical tasks, including liquid biopsy cancer detection, prostate cancer diagnosis, and 32-class pan-cancer classification, GiG consistently outperforms traditional and state-of-the-art methods, with the largest gains in limited-sample settings. On the challenging prostate cancer diagnosis task, GiG improves macro-F1 by up to 49 percentage points relative to competing methods. Control experiments replacing real pathway graphs with random topologies confirm that these gains arise from biologically grounded knowledge graph structure rather than graph modeling alone. These findings show that knowledge graph-modulated deep learning can improve robustness, interpretability, and sample efficiency in clinical data analysis, and provide a principled framework for integrating biological knowledge graphs into predictive modeling.

Generating graph ensembles with precisely controlled structural properties is central to investigating how network structure shapes function. Canonical ensembles impose constraints only in expectation (soft constraints), letting individual realizations fluctuate around the target, whereas enforcing hard constraints with prescribed precision in every realization remains challenging beyond fixing the degree sequence. Here we show that a reinforcement learning framework can drive a graph through degree-preserving rewirings to satisfy a prescribed assortativity, which characterizes the degree--degree correlation of adjacent nodes. By replacing the entropically dominated Metropolis--Hastings random walk with directed transport, the learned policy reduces generation cost by at least an order of magnitude while retaining over 98% of configurational diversity. Trained on small graphs, the framework generalizes across sizes and topologies without retraining, enabling quantitative isolation of secondary observables such as the clustering coefficient. These results establish reinforcement learning as a practical paradigm for hard-constrained graph generation.

Spectral clustering largely depends on the affinity graph, yet constructing a graph that preserves reliable local connectivity while adapting to heterogeneous data structures remains challenging. Existing granular-ball-based spectral clustering methods usually reduce graph complexity by using coarse-grained representatives. However, the learned local regions are often treated as graph nodes or anchors, and their structural information is not sufficiently used to regularize the original sample-level graph. To address this issue, this paper proposes a Minimum Description Length based Granular-Ball Tree-Regularized Spectral Clustering method, termed MDL-GBTRSC. The proposed method constructs a granular-ball tree through local MDL model selection, with reciprocal neighborhood continuity used to discourage splits that break reliable local connections. The stable leaf balls obtained from the tree provide coding-scale information for regularizing the sample-level affinity graph. In addition, a shared-neighbor bridge code is introduced to adjust weak local bridge relations without requiring an additional user-specified threshold. In this way, MDL-GBTRSC connects interpretable local representation learning with affinity graph construction in a unified spectral clustering framework. Experiments on real and synthetic datasets show that MDL-GBTRSC achieves the best average ARI and NMI under the adopted fixed-configuration protocol compared with classical spectral clustering baselines and representative granular-ball, micro-cluster, and anchor-based methods.

Blind image quality assessment (BIQA) for ultrahighdefinition (UHD) images remains challenging because native-resolution inference is computationally expensive, whereas aggressive resizing or isolated cropping may suppress scale-sensitive distortions and weaken the relationship between local artifacts and global scene context. This paper aims to improve UHD-BIQA by explicitly modeling the structural dependencies among sampled image regions rather than treating them as independent views, and a graph representation learning framework UHD-GCN-BIQA is proposed. The framework samples aspect-ratio-aligned patches from each UHD image, encodes them as graph nodes, and constructs a hybrid k-nearest-neighbor graph using spatial proximity and feature similarity. Residual graph convolution is used to propagate contextual information across regions, and gated attention pooling aggregates patchlevel evidence into an imagelevel quality prediction. An exponential moving average normalized multiobjective loss function is adopted to stabilize the joint optimization of regression, correlation, and ranking objectives. Experiments on the UHD-IQA benchmark show that UHD-GCN-BIQA achieves PLCC = 0.7784, SRCC = 0.8019, and RMSE = 0.0519, obtaining competitive correlation performance and the lowest RMSE among the compared methods. These results indicate that graph-based region relation modeling is effective for UHD image quality assessment, particularly for improving absolute quality score estimation under high-resolution visual content.

Relational prediction tasks are fundamental in many real-world applications, where data are naturally stored in relational databases (RDBs). Relational Deep Learning (RDL) addresses this problem by modeling RDBs as graphs and applying graph neural networks (GNNs) for end-to-end learning. However, the full-resolution property is commonly adopted as a design principle in graph construction for RDBs to preserve relational semantics, which leads most existing methods to rely on fixed graph structures. In this paper, we propose FROG, a Full-Resolution and Optimizable Graph Structure Learning} framework for RDL that formulates relational structure learning as a learnable table role modeling problem, allowing tables to contribute as nodes and edges in message passing. We further design role-driven message passing mechanisms to capture relational semantics, enabling joint optimization of graph structure and GNN representations. To ensure semantic consistency, we introduce functional dependency constraints that regularize representations across table and entity levels. Extensive experiments demonstrate that our method outperforms existing approaches and reveal how table roles impact downstream tasks, offering new insights into graph construction for RDL

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

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 recommended item 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 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 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 nodes evaluations.

This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations generated by a linear structural equation model. DAG learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property. Continuous acyclicity constraints have led to important algorithmic advances by replacing the discrete DAG constraint with smooth equality constraints. However, existing formulations still involve difficult non-convex optimization landscapes and may suffer from degenerate first-order optimality conditions. Here, we restrict attention to DAGs with non-negative edge weights and exploit this additional structure to obtain a simpler characterization of acyclicity. Building on this characterization, we formulate a regularized non-negative DAG learning problem and develop an algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by non-negativity. In the population regime, we show that the true DAG is the unique global minimizer of the proposed augmented-Lagrangian formulation; moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic KKT point. Numerical experiments on synthetic and real-world data show that the proposed method improves over state-of-the-art continuous DAG-learning alternatives.

We introduce Contrastive FUSE, a fast and unified framework for scalable node representation learning in graphs with partially available pairwise node labels and no available node features. Unlike existing methods, we directly optimize a spectral contrastive objective that integrates community-aware structural signals with signed pairwise constraints. To support large-scale training, we replace the expensive modularity gradient with a lightweight approximation, which preserves the structure-seeking behavior of modularity while reducing the computational cost significantly. This yields an efficient optimization scheme with a natural gradient decomposition and adaptive learning-rate scaling, enabling fast iterative updates even on million-edge graphs. Extensive experiments on benchmark citation networks, large co-purchase graphs, and OGB datasets show that Contrastive FUSE achieves competitive or superior contrastive classification performance without relying on node features, while offering substantial runtime gains over existing baselines. These results highlight the effectiveness of coupling modularity-inspired structural learning with contrastive supervision for efficient and scalable contrastive node representation learning.

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vector representation of the data. The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric, induced by the differentials of the embedding functions. The resulting Euler-Lagrange equations are derived in a coordinate-free form, enabling direct evaluation of all operators from the distance graph alone. Though non-linear and missing an explicit expression for their non-linearity, these equations are shown to be resolved as an iteratively updated sparse linear problem. The main contributions of the proposed approach are (a) the derivation of the functional equations governing the optimal Euclidean embedding in the continuum, (b) a representation-free formulation that requires only a neighborhood distance graph and no feature vectors and (c) an estimation procedure based exclusively on local graph operations. We experimentally evaluate the resulting non-parametric algorithm on synthetic manifolds and real datasets, demonstrating consistent preservation of local metric structure and neighboring relations, while approximating the global isometric embedding.

Learning causal relations from observational data is challenging because the graph search space grows super-exponentially with the number of variables. Ordering-based methods reduce this space by first identifying the topological ordering, whereas continuous optimization methods explore most likely regions of the space by casting DAG learning as a differentiable objective with an acyclicity constraint. Despite their conceptual appeal, both paradigms face significant scalability limitations in high-dimensional settings, restricting their practical applicability. In this work, we introduce a new formulation for linear causal discovery that tightly integrates these two paradigms to achieve substantial gains in scalability without sacrificing accuracy. Our approach, TriOpt, decomposes the problem into two efficient stages. First, it recovers the topological ordering by exploiting the Sherman-Morrison rank-1 downdate together with the additive structure of linear kernels, enabling fast and scalable ordering estimation. Second, given this ordering, we reformulate structure learning as a convex continuous optimization problem that entirely avoids the need for enforcing costly acyclicity constraints. We theoretically show that, under the true ordering, TriOpt exactly recovers the underlying linear DAG. Empirically, across synthetic, semi-synthetic, and real-world datasets, TriOpt achieves orders-of-magnitude speedups over state-of-the-art linear causal discovery methods in high-dimensional regimes, while maintaining comparable or superior accuracy.

Coordination graphs are a central abstraction in cooperative multi-agent reinforcement learning (MARL), yet existing sparse-graph learners lack a theoretically grounded mechanism to decide which edges should exist and how much information each edge should carry. Current methods rely on heuristic criteria that offer no formal guarantee on the learned topology, and no principled way to allocate different communication capacities to structurally different agent relationships. To address this, we propose Heterogeneous Information-Bottleneck Coordination Graphs (HIBCG), which learns a group-aware sparse graph in which both edge existence and message capacity are theoretically justified. With the graph information bottleneck (GIB) serving as the underlying tool, HIBCG first constructs a group-aligned block-diagonal prior that provides a closed-form criterion for edge retention -- determining which edges should exist and at what density per group block -- and then controls per-agent feature bandwidth on the resulting topology, compressing messages to retain only task-relevant content. We prove that the group-aligned prior strictly tightens the variational bound on topology learning, that the objective decomposes per group block, enabling differential edge control, and that capacity allocation follows a water-filling principle.

Node representation learning, such as Graph Neural Networks (GNNs), has emerged as a pivotal method in machine learning. The demand for reliable explanation generation surges, yet unsupervised models remain underexplored. To bridge this gap, we introduce a method for generating counterfactual (CF) explanations in unsupervised node representation learning. We identify the most important subgraphs that cause a significant change in the k-nearest neighbors of a node of interest in the learned embedding space upon perturbation. The k-nearest neighbor-based CF explanation method provides simple, yet pivotal, information for understanding unsupervised downstream tasks, such as top-k link prediction and clustering. Consequently, we introduce UNR-Explainer for generating expressive CF explanations for Unsupervised Node Representation learning methods based on a Monte Carlo Tree Search (MCTS). The proposed method demonstrates superior performance on diverse datasets for unsupervised GraphSAGE and DGI.