Graph Neural Networks

Graphs with heterophily, where adjacent nodes carry different labels, are prevalent in real-world applications, from social networks to molecular interactions. However, existing spectral Graph Neural Network (GNN) approaches tailored for heterophilous graph classification suffer from hub-dominated (node with large degree) aggregation and oversmoothing, as their suboptimal polynomial filters introduce approximation errors and blend distant signals. To address the degree-biased aggregation and suboptimal polynomial filtering, we introduce a Hierarchical Multi-view HAAR (HMH), a novel spectral graph-learning framework that scales in near-linear time . HMH first learns feature- and structure-aware signed affinities via a heterophily-aware encoder, then constructs a soft graph hierarchy guided by these embeddings. At each hierarchical level, HMH constructs a sparse, orthonormal, and locality-aware Haar basis to apply learnable spectral filters in the frequency domain. Finally, skip-connection unpooling layers combine outputs from all hierarchical levels back into the original graph, effectively preventing hub domination and long-range signal bottleneck (over-squashing). Experimentation shows that HMH outperforms state-of-the-art spectral baselines, achieving up to a 3% improvement on node classification and 7% points on graph classification datasets, all while maintaining linear scalability.

In recent years, the use of edge information provided by knowledge graphs together with the advantages of higher-order connectivity in graph neural networks for recommendation systems has become an important research direction. However, existing approaches are often limited by sparse labels, insufficient graph structure learning, and noisy entities in the knowledge graph, which reduce recommendation accuracy. To address these limitations, we propose a multi-view graph contrastive learning framework. The proposed method enhances user representations through multi-view knowledge graph distillation, enabling more accurate modeling of user preferences over entities and relations. The network aggregates neighborhood entity information to construct informative item representations. Furthermore, we design a multi-level self-supervised contrastive learning module that performs comparisons across three perspectives: Inter-Level, Intra-Level, and Interaction-Level. This design improves the model's ability to generalize across intra-class samples while increasing discrimination between inter-class samples, thereby enabling more effective multi-dimensional feature modeling. We conduct extensive experiments on three public datasets using both baseline and ablation settings. Experimental results demonstrate that the proposed framework consistently outperforms existing state-of-the-art methods. Ablation studies further verify the effectiveness of each module in the proposed model.

Conformal prediction (CP) provides a distribution-free approach to uncertainty quantification with finite-sample guarantees. However, applying CP to graph neural networks (GNNs) remains challenging as the combinatorial nature of graphs often leads to insufficiently certain predictions and indiscriminative embeddings. Existing methods primarily rely on embedding-space proximity for localization, which can be unreliable for graphs and yield inefficient prediction sets. We propose GRAPHLCP, a proximity-based localized CP framework that explicitly incorporates graph topology and inter-node dependencies into localization and weighting. Our approach introduces a feature-aware densification step to mitigate locality bias in sparse graphs, followed by a Personalized PageRank-based kernel computation to model structural proximity. This enables topology-dependent anchor sampling and calibration weighting that captures both local and long-range dependencies. Extensive experiments on several regression and classification datasets demonstrate that GRAPHLCP guarantees marginal coverage with finite samples while efficiently attaining favorable test conditional coverage across various conditioning scenarios.

Physics-based digital twins aim to predict the dynamics of real-world objects under interaction, enabling real-to-sim-to-real applications in robotics. Current approaches reconstruct such twins as explicit physical models (such as spring-mass systems) to predict the dynamics, but the resulting models often inherit the resolution of the visual reconstruction rather than being reduced to the physical complexity required to reproduce task-relevant dynamics. This mismatch introduces redundant topology, making repeated forward-dynamics rollouts unnecessarily expensive. To address this challenge, we present PhySPRING, an fully differentiable GNN-based method to reduce complexity in spring--mass digital twins. PhySPRING jointly learns a hierarchy of coarsened graph topologies and their mechanical parameters from observations. At each reduction level, PhySPRING merges nodes with similar learned dynamic responses to optimize the topology, while maintaining every reduced layer as an explicit spring--mass system. On the PhysTwin benchmark, PhySPRING improves dense reconstruction and prediction accuracy over PhysTwin, while reduced models retain stable physical and visual fidelity with up to a 2.30 times speed-up. We further demonstrate the effectiveness of PhySPRING in a Real2Sim robot policy-evaluation pipeline, where the reduced models are substituted zero-shot into ACT and $π_0$ evaluations, maintaining comparable manipulation success rates across downsampling levels while improving action-sampling effectiveness. Together, PhySPRING enables efficient and structure-preserving spring--mass reduction without sacrificing fidelity or robotic utility.

Accurate prediction of structural displacements under external loading is fundamental to structural health monitoring and seismic safety assessment. Although the finite element method (FEM) remains the prevailing approach because of its high accuracy, its considerable computational cost restricts its suitability for real-time monitoring applications. To address this limitation, this study proposes a data-driven framework based on Graph Neural Networks (GNNs), in which structural systems are represented as graphs with joints modeled as nodes and structural members as edges. By incorporating both geometric and mechanical properties into the graph representation, the proposed model learns the relationship between applied loads and structural responses directly from simulated data. A synthetic dataset was generated from a two-story frame structure using ANSYS, and both a conventional Neural Network (NN) and a GNN were trained for comparison. The results show that the proposed GNN framework predicts displacements and rotations with high accuracy and outperforms the NN model, demonstrating its potential as a fast and efficient alternative to traditional FEM-based analysis.

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.

Recent work has observed that explanations produced by Self-Interpretable Graph Neural Networks (SI-GNNs) can be self-inconsistent: when the model is reapplied to its own explanatory graph subset, it may produce a different explanation. However, why self-inconsistency arises remains poorly understood. In this work, we first identify re-explanation-induced context perturbation as the direct cause of score variation. We then introduce a latent signal assignment hypothesis to explain why only some edges are sensitive to this perturbation, and analyze how conciseness regularization affects latent signal assignment. Given that self-inconsistent edges do not provide stable evidence for the model's prediction, we propose Self-Denoising (SD), a model-agnostic and training-free post-processing strategy that calibrates explanations with only one additional forward pass. Experiments across representative SI-GNN frameworks, backbone architectures, and benchmark datasets support our hypothesis and show that SD consistently improves explanation quality while adding only about 4--6% computational overhead in practice.

EDA problems are graph-structured, but not all graph-structured problems call for the same GNN computation. We argue that successful GNN-for-EDA methods are those whose propagation, aggregation, and supervision align with the native algebra of the target task. Concretely: static timing analysis is a max-plus/min-plus recurrence on a topologically ordered DAG, structurally aligned with asynchronous DAG-GNNs; placement is governed by hypergraph wirelength and density penalties and is exploited by differentiable placers rather than by message-passing GNNs alone; routing congestion is a sparse demand-supply field over a layout grid; switching-activity propagation is a probabilistic recurrence on a directed netlist; IR drop is a linear system on the power-delivery network; and analog symmetry extraction is a discrete constraint-prediction problem on schematic graphs. Through these task-by-task alignments we (i) review the GNN architectural toolkit relevant to circuits, (ii) formalize how circuit graphs differ from generic graphs (directed, heterogeneous, multi-scale, with sequential and clock structure), (iii) characterize where current methods succeed and where the algebra-architecture mismatch limits them, and (iv) identify failure modes--stage leakage, proxy-to-signoff gap, calibration, and design-distribution shift--that we believe are likely to dominate the next phase of work. We position the paper as a GNN-for-EDA, task-aligned analysis rather than a comprehensive AI-for-chip-design survey. Continuous SE(3)-equivariant geometric GNNs are usually mismatched to Manhattan digital layout, and LLM-for-RTL, HLS, and RL/diffusion-based topology generation are outside our scope.

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.

GNNs have become a standard tool for learning on relational data, yet they remain highly vulnerable to backdoor attacks. Prior defenses often depend on inspecting specific subgraph patterns or node features, and thus can be circumvented by adaptive attackers. We propose PRAETORIAN, a new defense that targets intrinsic requirements of effective GNN backdoors rather than surface-level cues. Our key observation is that flipping a victim node's prediction requires substantial influence on the victim: attackers tend to either inject many trigger nodes or rely on a small set of highly influential ones. Building on this observation, PRAETORIAN (i) analyzes internal correlations within potential trigger subgraphs to detect abnormally large injected structures, and (ii) quantifies external node influence to identify triggers with disproportionate impact. Across our evaluations, PRAETORIAN reduces the average attack success rate (ASR) to 0.55% with only a 0.62% drop in clean accuracy (CA), whereas state-of-the-art defenses still yield an average ASR of >20% and a CA drop of >3% under the same conditions. Moreover, PRAETORIAN remains effective against a range of adaptive attacks, forcing adversaries to either inject many trigger nodes to achieve high ASR (>80%), which incurs a >10% CA drop, or preserve CA at the cost of limiting ASR to 18.1%. Overall, PRAETORIAN constrains attackers to an unfavorable trade-off between efficacy and detectability.

Aging clocks aim to estimate biological age, a measure of physiological state distinct from chronological age, from observable biomarkers, and are widely used for health assessment and disease analysis. DNA methylation is a particularly informative biomarker due to its stability and strong association with aging, and recent learning-based approaches have improved predictive performance. However, most existing methods treat CpG sites as independent features, overlooking the complex and heterogeneous biological relationships among them. We propose RelAge-GNN, a multi-relational graph neural network framework for DNA methylation-based age prediction. Our method constructs three complementary graphs capturing co-methylation patterns, genomic co-localization, and gene-level associations among CpG sites. Each graph is modeled by an independent GNN branch, and a learnable gating mechanism adaptively fuses the resulting representations. Experiments on large-scale datasets show that RelAge-GNN achieves competitive accuracy and stronger correlation with chronological age compared to state-of-the-art methods. Moreover, the model exhibits improved sensitivity in detecting age acceleration across diverse disease cohorts, highlighting its potential utility for disease characterization. Finally, through post hoc interpretability analyses, we quantify the contributions of different relational structures and CpG sites, providing biologically meaningful insights and suggesting potential directions for aging-related research. Our code is available at: https://anonymous.4open.science/r/RelAge-GNN-F1E3/.

Graph Anomaly Detection (GAD) is a critical task in graph machine learning with vital applications in financial fraud detection and social platform governance. However, existing GAD benchmarks are often restricted to small-scale, curated graphs with relatively balanced anomaly ratios, leaving a substantial gap between academic evaluation and real-world deployment. To bridge this gap, we present a multi-dimensional benchmark that systematically evaluates GAD models under three deployment-relevant challenges: million-scale graphs, extreme anomaly scarcity, and missing node attributes. We derive a family of controlled benchmark variants from five diverse graphs, including two native industrial-scale datasets with over 3.7 million nodes. Our extensive evaluation of nine representative GAD models reveals three major limitations: (1) most GNN-based methods fail to scale to million-node graphs due to prohibitive memory requirements; (2) detection performance drops sharply under realistic anomaly ratios (e.g., 0.1%), often resulting in zero recall; and (3) reconstruction-based models are highly sensitive to attribute imputation strategies. Our findings suggest that strong performance in laboratory settings does not guarantee robustness in production environments. We release this benchmark and empirical evaluation as a diagnostic testbed to promote the development of robust and scalable GAD systems for large-scale, imperfect graphs encountered in practice. Code is available at https://anonymous.4open.science/r/Benchmark_GAD-E7A3.

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

Identifying vulnerable transmission lines in power grids before a cascading failure occurs is challenging: existing methods can learn inter-line failure correlations from cascade data, but they are trained and evaluated on a single grid, and transferring the learned knowledge to an unseen grid remains an open problem. We address this by training a single Gated Recurrent Unit (GRU)-gated Graph Attention Network on combined cascading failure data from limited training grids and applying it directly to any unseen grid without retraining. A GRU gate controls what information each node retains or discards at each cascade iteration. Empirical evaluation shows that the model transfers zero-shot to multiple new grids spanning inter-time and inter-domain settings. Using information extracted from the trained model, we consistently identify more vulnerable lines than established structural and electrical baselines.

Learning-based approaches to NP-hard problems have shown increasing promise, but their progress is fundamentally constrained by the high cost of generating labeled training data. In domains such as Boolean satisfiability (SAT), standard pipelines rely on solver-in-the-loop labeling, which scales poorly with problem size and limits the amount of usable supervision. This bottleneck hinders the broader goal of leveraging machine learning to capture structure in hard combinatorial problems. In this work, we propose a target-aware, solver-free data generation framework for SAT that produces correctly labeled SAT and UNSAT instances by construction, eliminating the need for expensive solver calls. Our method aligns generated instances with the structural properties of a target benchmark, making synthetic data effective for downstream learning. We further develop a linear-programming-aware graph neural network (LPGNN) architecture that incorporates constraint-violation residuals into message passing, enabling the model to exploit underlying optimization structure. Together, these contributions support a data-centric paradigm for learning on NP-hard problems, where scalable, task-aligned data generation is as critical as model design. Our approach yields orders-of-magnitude speedups in data generation, demonstrating that benchmark-aligned synthetic data can effectively augment solver-labeled datasets for GNN-based SAT prediction.

Previous methods that predict system-wide travel time, predominantly grounded in graph neural networks, remain limited to typical and recurring demand patterns. While they successfully predict future congestion following daily commute, they inherently approximate a single demand realisation and fail to capture varying route choices. In this work, we propose a Generalised Travel Time Predictor (GenTTP) that successfully differentiates route choices and offers accurate flow and travel time predictions. Our framework learns to uncover complex spatiotemporal traffic patterns and microscopic relationships between route choices and the resulting travel times. This addresses a critical gap: the lack of travel time prediction models that generalise across varying route assignments, where the same demand can produce substantially different network-wide outcomes depending on how travellers are distributed over available paths.

We present an accelerated relax-and-round algorithm for concave coverage problems, which generalize the classic maximum coverage problem. Building on the relax-and-round framework of Barman et al. [STACS 2021], we propose two significant improvements. First, we replace the linear programming (LP) relaxation step with a projected accelerated gradient method applied to a smooth surrogate objective to achieve a $\widetilde{O}(mn \varepsilon^{-1})$ running time. Second, we use a specialized rounding scheme for the hypersimplex that combines the Carathéodory decomposition algorithm in Karalias et al. [NeurIPS 2025] with randomized swap rounding of Chekuri et al. [FOCS 2010]. We prove tight approximation ratios for new reward functions, including a $0.827$-approximation for the logarithmic reward $\varphi(x) = \log(1 + x)$. Finally, we conduct maximum multi-coverage experiments on synthetic and real-world graphs, demonstrating that our algorithm outperforms approaches that use state-of-the-art LP solvers.

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 Edit Distance (GED) is a fundamental, albeit NP-hard, metric for structural graph similarity. Recent neural graph matching architectures approximate GED by first encoding graphs with a Graph Neural Network (GNN) and then applying either a graph-level regression head or a matching-based alignment module. Despite substantial architectural progress, the role of encoder geometry in neural GED estimation remains poorly understood. In this paper, we develop a theoretical framework that connects encoder geometry to GED estimation quality for two broad classes of neural GED estimators: graph similarity predictors and alignment-based methods. On fixed graph collections, where the doubly-stochastic metric $d_{\mathrm{DS}}$ is comparable to GED, we show that graph-level bi-Lipschitz encoders yield controlled GED surrogates and improved ranking stability; for matching-based estimators, node-level bi-Lipschitz geometry propagates to encoder-induced alignment costs and the resulting optimized alignment objective. We instantiate this perspective using FSW-GNN, a bi-Lipschitz WL-equivalent encoder, as a drop-in replacement in representative neural GED architectures. Across representative baselines and benchmark datasets, the resulting geometry-aware variants significantly improve GED prediction and ranking metrics. A faithfulness case study of untrained encoders, together with ablations and transfer experiments, supports the view that these gains arise from improved representation geometry, positioning encoder geometry as a useful design principle for neural graph matching.

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.

Sparse autoencoders (SAEs) have become central to mechanistic interpretability, decomposing transformer activations into monosemantic features. Yet existing analyses characterise features almost exclusively through top-activating token lists or decoder weight vectors, leaving the higher-order co-occurrence structure shared across features largely unexamined. We introduce a graph-structured representation in which each SAE feature is modelled as a token co-occurrence graph: nodes are the tokens most frequent near strong activations, and edges connect pairs that co-occur within local context windows. A custom WL-style, frequency-binned graph kernel then provides a similarity measure over this structural space. Applied as a proof of concept to features from a large SAE trained on GPT-2 Small and probed with a synthetic mixed-domain corpus, our clustering recovers heuristic motif families (punctuation-heavy patterns, language and script clusters, and code-like templates) that are not recovered by clustering on decoder cosine similarity. A token-histogram baseline achieves higher overall purity, so the contribution of the graph view is complementary rather than dominant: it surfaces structural relationships that token-frequency and decoder-weight views alone do not capture. Cluster assignments are stable across graph-construction hyperparameters and random seeds.

Mechanistic interpretability aims to reverse-engineer transformer computations by identifying causal circuits through activation patching. However, scaling these interventions across diverse prompts and task families produces high-dimensional, unstructured datasets that are difficult to compare systematically. We propose a framework that reframes mechanistic analysis as a graph machine-learning problem by representing activation-patching profiles as patch-effect graphs over model components. We introduce three graph-construction methods: direct-influence via causal mediation, partial-correlation, and co-influence and apply graph kernels to analyze the resulting structures. Evaluating this approach on GPT-2 Small using Indirect Object Identification (IOI) and related tasks, we find that patch-effect graphs preserve discriminative structural signals. Specifically, localized edge-slot features provide higher classification accuracy than global graph-shape descriptors. A screened paired-patching validation suggests that CI and PC selected candidate edges correspond to stronger activation-influence effects than random or low-rank candidates. Crucially, by evaluating these representations against rigorous prompt-only and raw patch-effect controls, we make the evidential scope of the benchmark explicit: graph features compress structured patching signal, while raw tensors and surface cues define strong baselines that any circuit-level claim should address. Ultimately, our framework provides a compression and evaluation pipeline for comparing patching-derived structures under controlled baselines, separating robust slice-discriminative evidence from stronger task-general causal-circuit claims.

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.

Progress on graph foundation models is hindered by benchmark practices that conflate the contributions of node features and graph structure, making it hard to tell whether a model actually learns from connectivity, or whether it even needs to. We propose addressing this using graph invariants, i.e., permutation-invariant, task-agnostic structural descriptors that serve as a diagnostic framework for graph benchmarks. We show that (i) invariants are more expressive than standard GNNs, (ii) invariants characterize structural heterogeneity within and across benchmark datasets, (iii) invariants predict multi-task performance, and (iv) simple invariant-based models are competitive with, and sometimes exceed, transformer and message-passing baselines across 26 datasets. Our results suggest that expressivity is not the main driver of predictive performance, and that on tasks where structure matters, a non-trainable structural proxy often matches trained message-passing models. We thus posit that invariant baselines should become a standard for evaluating whether structure is required for a task and whether a model picks up on it, serving as a stepping stone towards graph foundation models.

Subgraph pattern detection aims to uncover complex interaction structures in graphs. However, state-of-the-art graph neural network (GNN)-based solutions assume centralized access to the entire graph. When graphs are instead distributed across multiple parties, client-local GNN computations diverge from those of a centralized model, resulting in a representation-equivalence gap. We formalize this as a structural observability problem, where subgraph patterns crossing partition boundaries become locally unidentifiable. To bridge this gap, we propose a per-step, layer-wise embedding exchange framework in which clients synchronize intermediate node representations at each layer of the forward pass, without exposing raw features or labels. Under an extended-subgraph assumption and shared model parameters across clients, this framework recovers the same node representations as a centralized GNN over the full graph. Experiments on synthetic directed multigraphs with cycles, bicliques, and scatter-gather patterns show that embedding exchange and federated parameter aggregation are complementary rather than interchangeable: their combination recovers most of the representation gap, provided exchanged embeddings are fresh per-step rather than stale per-epoch.

Random directed acyclic graphs (DAGs) based on imposing an order on Erdős-Rényi and scale free random graphs are widely used for evaluating causal discovery algorithms. We show that in such DAGs, the set of nodes reachable via open paths, termed relatives, increases monotonically along the causal order. We assess the prevalence of this pattern numerically, and demonstrate that it can be exploited for causal order recovery via sorting by the estimated number of relatives. We note that many simulations in the literature feature settings where this yields an excellent proxy for the causal order, and show that a strict increase of relatives along the causal order leads to a singular Markov equivalence class. We propose sampling time-series DAGs as a possible alternative and discuss implications for causal discovery algorithms and their evaluation on synthetic data.

Graph Federated Learning (GFL) enables collaborative representation learning across distributed subgraphs while preserving privacy. However, heterogeneity remains a critical challenge, as subgraphs across clients typically differ significantly in both semantics and structures. Existing methods address heterogeneity by enforcing the rigid alignment of model parameters or prototypes between clients and the server. However, these alignments implicitly rely on a restrictive global linearity assumption that summarizes local data distributions using a single and globally consistent representation space. This severely compresses the personalized representation space of clients and fails to preserve diverse local graph distributions. To overcome these limitations, we propose Federated Graph Manifold Calibration (FedGMC), a novel paradigm that tackles semantic heterogeneity and structural heterogeneity from a unified manifold perspective. Instead of enforcing rigid alignment, FedGMC introduces a dual manifold calibration mechanism that preserves global commonalities while maximizing the personalized representation space of local clients. Specifically, for semantic heterogeneity, the server constructs a geometrically optimal semantic manifold via equidistant semantic anchors, so as to guide the calibration of local semantic manifolds. For structural heterogeneity, the server constructs a global structural manifold by building global structural templates, so as to guide the calibration of local structural manifolds. Finally, the server dynamically refines both global semantic manifolds and structural manifolds by aggregating local manifolds. Extensive experiments on eleven homophilic and heterophilic graphs demonstrate that FedGMC effectively balances global commonality and local personalization, thereby significantly outperforming state-of-the-art baseline methods.

Graph pooling is commonly applied in graph classification, yet its empirical gains over standard WL-1 expressive GNNs are often marginal or inconsistent. We study this gap by analysing the interaction between node features and graph topology and their effect on pooling objectives. Our analysis reveals that pooling operators require node features that are well-aligned with the graph's topology -- a condition often overlooked and not guaranteed in empirical networks. We formalise fundamental requirements for node features to enable effective pooling, and introduce a quantitative measure of feature quality. Our empirical evaluation shows that, when these requirements are satisfied, pooling can be beneficial and improve performance on appropriate datasets.

Foundation models excel at language, where sentences become tokens, and vision, where images become pixels, because both reduce to discrete symbols on a shared, fixed grid. Knowledge Graphs share the discreteness, but not the geometry. Their entities and relations are discrete symbols, yet their arrangement is relational and lacks a common, fixed grid. Knowledge Graphs (KGs) share the discreteness, but not the geometry. They form irregular, non-Euclidean topologies whose local neighborhoods differ from graph to graph. Therefore, Knowledge Graph Foundation Models (KGFMs) rely on identifying structural invariances to produce transferable representations. Without a universal token set, KGFMs are limited in their ability to transfer representations across unseen KGs. We close this gap by treating graphlets, small connected graphs, as structural tokens that recur in heterogeneous KGs. In this paper, We introduce a model-agnostic framework based on a vocabulary of graphlets that mines a KG between relations via pattern matching. In particular, we considered closed and open 2- and 3-path, and star graphlets, to obtain robust invariances. The framework is evaluated on 51 KGs from a wide range of domains, for zero-shot inductive and transductive link prediction. Experiments show that adding simple graphlets to the vocabulary yields models that outperform prior KGFMs.

Dynamic text-attributed graphs (DyTAGs) provide a powerful framework for modeling evolving systems in which node semantics and time-dependent interactions are tightly coupled. Recently, multimodal learning has emerged as a promising yet underexplored direction for enhancing DyTAG representation learning. However, existing methods typically rely on rigid modality partitions and one-shot fusion strategies, which limit their ability to capture the intrinsic and evolving dependencies between node semantics and interaction behaviors. To address these limitations, we propose \textbf{PRISM}, an iterative cross-modal posterior refinement framework for DyTAG representation learning. PRISM organizes DyTAG information into semantic and behavioral modalities, providing a more intrinsic alternative to carrier-level modality partitions. Instead of fusing the two modalities in a single step, PRISM learns a refinement trajectory that progressively transforms semantic priors into behavior-conditioned posterior states through cross-modal interaction with behavioral evidence. Extensive experiments on DTGB benchmark datasets show that PRISM achieves strong performance on temporal link prediction and destination node retrieval tasks. Further ablation studies validate the effectiveness of semantic--behavioral modeling and iterative posterior refinement.

The Weisfeiler--Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry, i.e., meshes with identical connectivity but different embeddings are indistinguishable. We introduce the Geometric Simplicial Weisfeiler--Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. In addition, we show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), a complete invariant for geometric simplicial complexes, this yields a geometric expressivity characterization together with an approximation framework. Experiments on synthetic and mesh datasets serve to validate our theory, showing a clear hierarchy from combinatorial to geometry-aware models.

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.

It is well established that spectral graph neural networks (GNNs) can universally approximate node signals; however, their expressive power remains bounded by the 1-dimensional Weisfeiler-Lehman test, which is mirrored in their lack of universality for higher-order signals. To go beyond this bound, we propose the Full-Spectrum GNNs (FSpecGNNs), a second-order generalization of classical spectral GNNs. FSpecGNN advances spectral filtering from two perspectives: (1) it lifts signals from the node domain to the node-pair domain; and (2) it extends the univariate spectral filter over eigenvalues to a bivariate filter over eigenvalue pairs. We show that classical spectral GNNs arise as a diagonal special case of FSpecGNNs, and prove that FSpecGNNs can be at most as expressive as Local 2-GNN while universally approximating node-pair signals, the latter being particularly beneficial for heterophilic graph learning. Moreover, FSpecGNN admits scalable implementations that avoid explicit node-pair-level computations; combined with a low-rank approximation that reduces full-spectrum convolution to a combination of polynomial spectral filters, it enables learning on large graphs. Empirically, FSpecGNN validates the predicted expressivity and delivers strong performance on heterophilic benchmarks.

Adversarial learning and the robustness of Graph Neural Networks (GNNs) are topics of widespread interest in the machine learning community, as documented by the number of adversarial attacks and defenses designed for these purposes. While a rigorous evaluation of these adversarial methods is necessary to understand the robustness of GNNs in real-world applications, we posit that many works in the literature do not share the same experimental settings, leading to ambiguous and potentially contradictory scientific conclusions. In this benchmark, we demonstrate the importance of adopting fair, robust, and standardized evaluation protocols in adversarial GNN research. We perform a comprehensive re-evaluation of seven widely used attacks and eight recent defenses under both poisoning and evasion scenarios, across six popular graph datasets. Our study spans over 453,000 experiments conducted within a unified framework. We observe substantial differences in adversarial attack performance when evaluated under a fair and robust procedure. Our findings reveal that previously overlooked factors, such as target node selection and the training process of the attacked model, have a profound impact on attack effectiveness, to the extent of completely distorting performance insights. These results underscore the urgent need for standardized evaluations in adversarial graph machine learning.

Knowledge graphs automatically constructed from text are increasingly used in real-world applications. However, their inherent noise, fragmentation, and semantic inconsistencies significantly affect the performance of Graph Neural Networks (GNNs) on downstream tasks. Assessing their performance and robustness remains difficult, as it is often unclear whether observed results stem from the learning model or from the quality of the constructed graph itself. In this work, we introduce a dual-purpose benchmark designed to jointly evaluate (i) the performance of GNNs on noisy, text-derived graphs and (ii) the effectiveness of graph construction methods on a downstream task. The benchmark is built in the biomedical domain from a single textual corpus and includes two automatically constructed graphs generated using different extraction methods, alongside a high-quality reference graph curated by experts that serves as an upper performance bound. This design enables controlled comparison of construction methods and systematic evaluation of GNN robustness through semi-supervised node classification. We further provide a standardized, reproducible, and extensible evaluation framework, facilitating the integration of new graph extraction methods and learning models.

Given two GNNs that output node embeddings, how can we determine if they were trained independently? An adversary could have trained one GNN specifically to mimic the other GNN's embeddings. To obscure this relationship between the GNNs, the adversarial GNN might then transform its output embeddings. The two GNNs could have different architectures, weights, and embedding dimensions, and the adversary can transform the embeddings. Despite these stringent conditions, our algorithm (named CopyCop) can identify such copycat GNNs, unlike existing watermarking and fingerprinting methods. We also provide theoretical guarantees for CopyCop. Finally, experiments on 14 datasets and 5 GNN architectures demonstrate that CopyCop is accurate and robust against a broad class of adversarial attacks and transformations. Code is available at: https://anonymous.4open.science/r/CopyCop-Graph-Ownership-Verification-8143/README.md

We introduce Graph Normalization (GN), a principled dynamical system on graphs that serves as a differentiable approximation engine for the NP-hard Maximum Weight Independent Set (MWIS) problem. MWIS encompasses many combinatorial challenges, including optimal assignment, scheduling, set packing, and MAP inference in discrete Markov Random Fields. Unlike Belief Propagation, we prove GN always converges to a binary indicator of a Maximum Independent Set. GN realizes a fast quasi-Newton descent through an exact Majorization-Minimization step, systematically improving the MWIS relaxed primal objective. We establish an equivalence between GN and the Replicator Dynamics of a nonlinear evolutionary game, where vertices compete for inclusion in an independent set. While a non-potential game, the GN game follows Fisher's Fundamental Theorem of Natural Selection, where the average fitness equals the MWIS primal objective and strictly increases. This connection leads to a weighted extension of the Motzkin-Straus theorem, showing MISes are in bijection with the local minima of a quadratic form over a tilted simplex. For the Assignment Problem, GN acts as a variant of the Sinkhorn algorithm that naturally converges to a hard assignment while generalizing to arbitrary constraint graphs. We demonstrate GN's performance as a fast binarization engine for the state-of-the-art Bregman-Sinkhorn relaxed MWIS solver. On real-world benchmarks with up to 1M edges, GN identifies solutions within 1% of the best known results in seconds on a CPU. GN opens new avenues for deep learning architectures requiring differentiable, "hard" decisions under constraints, with applications in structured sparse attention, dynamic network pruning, and Mixture-of-Experts. Beyond core AI, the GN framework enables end-to-end learning of constrained optimization in computer vision, computational biology, and resource allocation.

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.

System Neural Diversity (SND) measures behavioral heterogeneity in multi-agent reinforcement learning by averaging pairwise distances over all $\binom{n}{2}$ agent pairs, making each call quadratic in team size. We introduce Graph-SND, which replaces this complete-graph average with a weighted average over the edges of an arbitrary graph $G$. Three regimes follow: $G=K_n$ recovers SND exactly; a fixed sparse $G$ defines a localized diversity measure at $O(|E|)$ cost; and random edge samples yield an unbiased Horvitz-Thompson estimator and a normalized sample mean with $O(1/\sqrt{m})$ concentration in the sampled edge count $m$. For fixed sparse graphs we prove forwarding-index distortion bounds for expanders and a spectral refinement under low-rank distance structure; for random $d$-regular graphs we prove an unconditional probabilistic $\widetilde{\mathcal{O}}(D_{\max}/\sqrt{n})$ bound. On VMAS we verify recovery, unbiasedness, concentration, and wall-clock scaling, with a PettingZoo TVD panel checking non-Gaussian transfer. In a 500-iteration $n=100$ PPO run, Bernoulli-$0.1$ Graph-SND tracks full SND while reducing per-call metric time by about $10\times$, and frozen-policy GPU timing up to $n=500$ follows the predicted $\binom{n}{2}/|E|$ speedup. Random $d$-regular expanders empirically achieve $\mathrm{SND}_{G}^{\mathrm{u}}/\mathrm{SND} \in [0.9987, 1.0013]$ at $Θ(n \log n)$ edges. In DiCo diversity control at $n=50$, Bernoulli-$0.1$ Graph-SND preserves set-point tracking with paired reward differences indistinguishable from zero across nine matched cells while cutting per-call metric cost by ${\sim}9.5\times$. Together, these results show that the SND aggregation bottleneck can be removed without changing the metric's semantics, yielding a drop-in sparse alternative that scales beyond complete-graph SND and supports both passive measurement and closed-loop diversity control.

LLM-empowered multi-agent systems offer new potential to accelerate scientific discovery by generating novel research ideas. However, existing methods typically coordinate agents through temporary texts, such as drafts or chat logs; it is difficult to pinpoint the weaknesses in the generated ideas and how the agents refine them. To this end, we introduce \textbf{Evolving Idea Graphs} (EIG), a graph-based multi-agent scientific ideation framework that can generate high-performance research ideas across various benchmark-native metrics, such as novelty, feasibility, and clarity. Instead of coordinating solely through texts, EIG represents a partially formed proposal as an evolving idea graph, where nodes capture scientific claims and edges encode relations (e.g., support and conflict), enabling unresolved weaknesses to remain identifiable throughout the idea evolving process. Specifically, a learned two-head controller operates over the evolving graph to guide the ideation: one head selects graph edits for agents to execute, while the other decides when the graph is ready for commit as final proposal synthesis. On AI Idea Bench 2025 and LiveIdeaBench, EIG outperforms all compared systems on both automatic benchmark scores and blind expert ratings. Ablations further show that explicit graph state provides the main performance gains, and learned edit-and-commit control adds consistent improvements.

Uncertainty quantification (UQ) in graph neural networks (GNNs) is crucial in high-stakes domains but remains a significant challenge. In graph settings, message passing often relies on strong assumptions such as exchangeability, which are rarely satisfied in practice, and achieving reliable UQ typically requires costly resampling or post-hoc calibration. To address these issues, we introduce Quantile-free Prediction Interval GNN (QpiGNN), a framework that builds on quantile regression (QR) to enable GNN-based UQ by directly optimizing coverage and interval width without requiring quantile inputs or post-processing. QpiGNN employs a dual-head architecture that decouples prediction and uncertainty, and is trained with label-only supervision through a quantile-free joint loss. This design allows efficient training and yields robust prediction intervals, with theoretical guarantees of asymptotic coverage and near-optimal width under mild assumptions. Experiments on 19 synthetic and real-world benchmarks show QpiGNN achieves average 22% higher coverage and 50% narrower intervals than baselines, while ensuring efficiency and robustness to noise and structural shifts.

Unlike vision and language domains, graph learning lacks a shared input space, as input features differ across graph datasets not only in semantics, but also in value ranges and dimensionality. This misalignment prevents graph models from generalizing across datasets, limiting their use as foundation models. In this work, we propose ALL-IN, a simple and theoretically grounded method that enables transferability across datasets with different input features. Our approach projects node features into a shared random space and constructs representations via covariance-based statistics, thus eliminating dependence on the original feature space. We show that the computed node-covariance operators and the resulting node representations are invariant in distribution to permutations of the input features. We further demonstrate that the expected operator exhibits invariance to general orthogonal transformations of the input features. Empirically, ALL-IN achieves strong performance across diverse node- and graph-level tasks on unseen datasets with new input features, without requiring architecture changes or retraining. These results point to a promising direction for input-agnostic, transferable graph models.

Graph neural networks have been widely used in Boolean satisfiability (SAT) tasks to learn structural information from SAT formulas. The goal of these studies is to solve SAT instances or to enhance SAT solvers, including tasks such as unsat-core prediction. However, most existing approaches model a SAT formula as a bipartite graph or a directed acyclic graph, which are less direct in capturing clause-level and higher-order interactions among literals and clauses. Moreover, these approaches are limited in modeling intrinsic polarity-related properties of SAT, such as the complementary relationship between the positive and negative literals of a variable. To address these limitations, we propose a polarity-aware representation learning framework over clause-literal hypergraphs. We model SAT formulas as clause-literal hypergraphs augmented with a clause incidence graph to capture higher-order structural interactions. We then introduce a polarity-aware decomposition mechanism that separates variable representations into polarity invariant and equivariant components, explicitly modeling the relationship between positive and negative literals, with the resulting literal representations propagated along the hypergraph structure. We further incorporate a polarity-inversion consistency regularization to reinforce polarity-consistent representations during training. Experimental results on multiple SAT datasets demonstrate the effectiveness of the proposed approach.

Humans can acquire a highly structured intuitive understanding of musical patterns, yet these patterns often require multiple iterations of reflection and re-listening to internalize fully. To capture such an internalization process, we present a computational model for the learning of jazz harmonic patterns based on library learning. Given a corpus of harmonic progressions, our model searches over a space of programs composed of primitive harmonic relations in order to discover concise generative explanations of the corpus. The model first enumerates possible programs for each piece, and then jointly learns a library of harmonic patterns and refactored programs. To efficiently navigate the vast joint space of programs and libraries, we integrate deductive parsing with library learning on e-graphs. We explore how well our model captures aspects of human musical pattern learning by evaluating the intuitiveness of both programs and libraries, as well as similarities to human-written harmonic derivations.

Many real-world heterogeneous graphs exhibit pronounced heterophily, where connected nodes often have dissimilar labels or play different semantic roles. In such settings, standard heterogeneous graph neural networks that aggregate messages along metapaths or meta-relations primarily based on feature similarity can propagate misleading information, since feature similarity may be misaligned with underlying relational semantics. In this paper, we propose HeterSEED, a semantics-structure decoupling framework for heterogeneous graph learning under heterophily. HeterSEED decouples representation learning into a heterogeneous semantic channel that captures type- and relation-aware local semantics and a structure-aware heterophily channel that separates homophilic and heterophilic neighborhoods via pseudo-label-guided partitioning and aggregates them using metapath-based structural weights. A node-level adaptive fusion mechanism then combines the two channels to produce context-dependent node representations. Theoretically, we establish that, on heterogeneous graphs under heterophily, HeterSEED is strictly more expressive than standard heterogeneous graph neural networks that rely primarily on feature similarity and provably reduces the prediction bias introduced by heterophilic neighbors. Experiments on five real-world heterogeneous graphs, including two large-scale networks at the million-node and hundred-million-edge scale, demonstrate that HeterSEED consistently outperforms representative heterogeneous graph neural networks and recent heterophily-aware baselines, especially in strongly heterophilic regimes.

The integration of deep learning approaches in biomedical research has been transformative, enabling breakthroughs in various applications. Despite these strides, its application in protein inference is impeded by the scarcity of extensively labeled datasets, a challenge compounded by the high costs and complexities of accurate protein annotation. In this study, we introduce GraphPI, a novel framework that treats protein inference as a node classification problem. We treat proteins as interconnected nodes within a protein-peptide-PSM graph, utilizing a Graph Neural Network-based architecture to elucidate their interrelations. To address label scarcity, we train the model on a set of unlabeled public protein datasets with pseudo-labels derived from an existing protein inference algorithm, enhanced by self-training to iteratively refine labels based on confidence scores. Contrary to prevalent methodologies necessitating dataset-specific training, our research illustrates that GraphPI, due to the well normalized nature of Percolator features, exhibits universal applicability without dataset-specific fine-tuning, a feature that not only mitigates the risk of overfitting but also enhances computational efficiency. Our empirical experiments reveal notable performance on various test datasets and deliver significantly reduced computation times compared to common protein inference algorithms.

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem $(H_L, X_R, P)$, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies $C_{\mathcal{K}}^2<μ_Lμ_R$. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.

Nonstabilizerness, commonly referred to as magic, is a fundamental resource underpinning quantum advantage. In this paper, we propose a magic-informed quantum architecture search (QAS) technique that enables control over a quantum resource within the general framework of circuit design. Inspired by the AlphaGo approach, we tackle the problem with a Monte Carlo Tree Search technique equipped with a Graph Neural Network (GNN) that estimates the magic of candidate quantum circuits. The GNN model induces a magic-based bias that steers the search toward either high- or low-magic regimes, depending on the target objective. We benchmark the proposed magic-informed QAS technique on both the structured ground-state energy problem and on the more general quantum state approximation problem, spanning different sizes and target magic levels. Experimental results show that the proposed technique effectively influences the magic across the search tree and notably also on the resulting final circuit, even in regimes where the GNN operates on out-of-distribution instances. Although introducing a problem-agnostic magic bias could, in principle, constrain the search dynamics, we observe consistent improvements in solution quality across all problems tested.

Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both $\mathbb{Z}_2$ and $\mathrm{U}(1)$ lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in $\mathrm{U}(1)$ quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.

We present a method for finding hierarchy-aware embeddings of knowledge graphs (KGs) using graph neural networks (GNNs) enriched with a semantic loss derived from underlying ontologies. This method yields embeddings that better reflect domain knowledge. To demonstrate their utility, we predict and interpret the effects of gene deletions in the yeast Saccharomyces cerevisiae and learn box embeddings for KGs in the absence of a prediction task. We further show how box embeddings can serve as the basis for evaluating KG revisions. Our yeast KG is constructed from community databases and ontology terms. Low-dimensional box embeddings combined with GNNs are used to predict cell growth for double gene knockouts. Over 10-fold cross validation, these predictions have a mean $R^2$scoreof~0.360, significantly higher than baseline comparisons, demonstrating that high-level qualitative knowledge is informative about experimental outcomes. Incorporating semantic loss terms in the training of the models improves their predictive performance ($R^2$=0.377) by aligning embeddings with ontology structure. This shows that class hierarchies from ontologies can be exploited for quantitative prediction. We also test the trained models on triple gene knockouts, showing they generalise to data beyond those seen in training. Additionally, by identifying co-occurring relations in the yeast KG important for the cell-growth predictions, we construct hypotheses about interacting traits in yeast. A biological experiment validates one such finding, revealing an association between inositol utilisation and osmotic stress resistance, highlighting the model's potential to guide biological discovery.

Understanding how biological and artificial neural networks implement computation from connectivity is a central problem in neuroscience and machine learning. In neural systems, structural and functional connectivity are known to diverge, motivating approaches that move beyond direct connections alone. Here, we show that the spatial and temporal function of recurrent neural networks (RNNs) trained on hierarchically modular tasks can be recovered by modelling the network as a graph and analysing the multi-hop pathways between input and output units. In particular, decomposing these pathways by hop length reveals how the network temporally routes information. This perspective reframes regularisation: if function is implemented through multi-hop communication, then standard penalties such as L1 regularisation, which act only on individual weights, constrain single-hop structure rather than the multi-hop pathways that support computation. Motivated by this view, we introduce resolvent-RNNs (R-RNNs), which constrain multi-hop pathways and thereby induce temporal sparsity beyond that achieved by standard L1 regularisation. Compared with L1 regularisation, R-RNNs achieve improved performance by inducing temporal sparsity that matches the task structure, even when the task signal is sparse. Moreover, R-RNNs exhibit stronger sparsity-function alignment, reflected in their increased robustness under strong regularisation. Together, our results identify multi-hop communication as a key principle linking structure to function in recurrent networks, and suggest that sparsity should be defined over functional pathways rather than individual parameters.

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.

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

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.

Regulatory frameworks such as GDPR increasingly require that ML predictions be accompanied by post-hoc explanations, even when raw data and trained models cannot be released. Differential privacy (DP) is the standard mitigation for the residual privacy risk of releasing these explanations. We show that DP is not sufficient: an adversary observing only DP-perturbed GNN explanations can reconstruct hidden graph structure with high accuracy. Our attack, PRIVX, exploits the fact that the Gaussian DP mechanism is a single DDPM forward step at known noise level σ(ε), recasting reconstruction as reverse diffusion conditioned on the corrupted signal, a principled Bayesian denoiser under known DP corruption. We formalise a stratified adversary model parameterised by (M, \hatε, \hatδ, S, ρ) that interpolates between oblivious and oracle attackers, and derive endpoint-matched two-sided bounds on reconstruction AUC. For practitioners, we provide regime-stratified guidance on explainer choice: on homophilic graphs, neighbourhood-aggregating explainers (GraphLIME, GNNExplainer) leak more structure than per-node gradient explainers under the same DP budget; on strongly heterophilic graphs the ordering reverses. We introduce PRIVF as an auxiliary diagnostic sharing the same diffusion backbone to decompose leakage into explainer-induced and intrinsic graph-distribution components. Experiments across seven benchmarks, three DP mechanisms, and three GNN backbones show PRIVX achieves AUC above 0.7 at ε = 5 on five of seven datasets, with the attack succeeding well within typically deployed privacy budgets.

Graph Neural Networks (GNNs) achieve strong performance on node classification tasks but remain difficult to interpret, particularly with respect to which input features drive their predictions. Existing global GNN explainers operate at the structural level identifying recurring subgraph motifs, but none explain model behaviour globally at the level of input node attributes. We propose GRAFT, a posthoc global explanation framework that identifies class-level feature importance profiles for GNNs. The method combines diversity-guided exemplar selection, Integrated Gradients-based attribution, and aggregation to construct a global view of feature influence for each class, which can be further expressed as concise natural language rules using a large language model with self-refinement. We evaluate GRAFT across multiple datasets, architectures, and experimental settings, demonstrating its effectiveness in capturing model-relevant features, supporting bias analysis, and enabling feature-efficient transfer learning. In addition, we introduce a structured human evaluation protocol to assess the interpretability of generated rules along dimensions such as accuracy and usefulness. Our results suggest that GRAFT provides a practical and interpretable approach for analysing feature-level behaviour in GNNs, bridging quantitative attribution with human-understandable explanations.

Graph unlearning remains a critical technique for supporting privacy-preserving and sustainable multimodal graph learning. However, we observe that existing unlearning strategies tend to apply uniform parameter selection and editing across all graph neural network (GNN) layers, which is especially harmful for multimodal graphs where high-dimensional input projections encode dominant cross-modal knowledge. As a result, over-editing these sensitive layers often leads to catastrophic utility degradation after forgetting, undermining both stable learning and effective privacy protection. To address this gap, we propose FDQ, a Feature-Dimension Aware Quantile framework for multimodal graph unlearning. FDQ adaptively identifies high-dimensional input projection layers and applies more conservative, FDQ-guided quantile thresholds when constructing suppression sets, while keeping the underlying importance estimation mechanism unchanged. FDQ is seamlessly integrated with diagonal sensitivity-based parameter importance analysis to enable efficient node and edge unlearning under general forget requests. Through extensive experiments on Ele-Fashion and Goodreads-NC, we demonstrate that FDQ consistently achieves strong utility preservation while maintaining effective forgetting against membership inference attacks. Overall, FDQ offers a principled and robust solution for privacy-aware unlearning in high-dimensional multimodal graph systems.

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.

Traditional Simultaneous Localization and Mapping (SLAM) algorithms rely heavily on the static environment assumption, which severely limits their applicability in real-world spaces populated by moving entities, such as pedestrians. In this work, we propose DynoSLAM, a tightly-coupled Dynamic GraphSLAM architecture that integrates socially-aware Graph Neural Networks (GNNs) directly into the factor graph optimization. Unlike conventional approaches that use rigid constant-velocity heuristics or deterministic single-agent neural priors, our framework formulates pedestrian motion forecasting as a stochastic World Model. By utilizing Monte Carlo rollouts from a trained GNN, we capture the multimodal epistemic uncertainty of human interactions and embed it into the SLAM graph via a dynamic Mahalanobis distance factor. We demonstrate through extensive simulated experiments that this stochastic formulation not only maintains highly accurate retrospective tracking but also prevents the optimization failures caused by the deterministic "argmax problem". Ultimately, extracting the empirical mean and covariance matrices of future pedestrian states provides a mathematically rigorous, probabilistic safety envelope for downstream local planners, enabling anticipatory and collision-free robot navigation in densely crowded environments.

Learning data-efficient object dynamics models for robotic manipulation remains challenging, especially for deformable objects. A popular approach is to model objects as sets of 3D particles and learn their motion using graph neural networks. In practice, this is not enough to maintain physical feasibility over long horizons and may require large amounts of interaction data to learn. We introduce PIEGraph, a novel approach to combining analytical physics and data-driven models to capture object dynamics for both rigid and deformable bodies using limited real-world interaction data. PIEGraph consists of two components: (1) a \textbf{P}hysically \textbf{I}nformed particle-based analytical model (implemented as a spring--mass system) to enforce physically feasible motion, and (2) an \textbf{E}quivariant \textbf{Graph} Neural Network with a novel action representation that exploits symmetries in particle interactions to guide the analytical model. We evaluate PIEGraph in simulation and on robot hardware for reorientation and repositioning tasks with ropes, cloth, stuffed animals and rigid objects. We show that our method enables accurate dynamics prediction and reliable downstream robotic manipulation planning, which outperforms state of the art baselines.