Spectral Graph Convolution

CoVariance Neural Networks and their extensions have emerged as effective tools for processing multivariate data, deriving graph shift operators directly from second-order statistics. These architectures, however, are designed for independent and identically distributed observations and do not fully capture the joint structure of temporal and cross-variable dependencies in multivariate time series. In this work, we introduce Inverse Cross-Spectral Neural Networks (iCSNNs), a class of graph neural networks for stationary multivariate time series whose shift operators are the inverse cross-spectral density (iCSD) matrices. These operators encode frequency-specific conditional relationships among variables, exploiting the decomposition provided by the spectral representation theorem. Leveraging spectral smoothness, frequencies are grouped into bands sharing a single iCSD operator, yielding a compact parametrisation that retains the frequency-dependent structure of the process. We further propose a joint learning procedure to estimate both the Fourier-domain dependence structure and the iCSNN parameters, adapting the iCSD operators to the downstream task. When tested on synthetic data, iCSNN outperforms baselines from different methodological families.

We study spectral graph neural networks built from Hermite polynomials and propose HermNet, a simple model that combines a nodewise predictor with normalized Hermite propagation. Its sparse recurrence requires neither eigendecomposition nor a learned basis. We distinguish the basic model from optional coordinate calibration, response normalization and Gaussian derivative regularization. Hermite and other complete polynomial bases span the same degree-bounded filter space, but their coordinates can produce different optimization behavior under limited training budgets. We analyze this behavior through spectral signal energy, label sampling, changes in learned features and the bias--variance trade-off of regularization. Controlled synthetic experiments identify a regime in which plain HermNet outperforms matched polynomial-basis alternatives, including with a jointly trained nonlinear predictor. Curvature regularization further improves HermNet when the same functional penalty is available to every comparator. Fixed-predictor controls support the advantage under short training budgets, but longer training removes the plain-model lead. Matched real-data comparisons show accuracy deficits, and architectural and numerical studies identify further limits. Together, the analysis and experiments clarify when Hermite propagation is useful and how calibration and regularization affect its performance.

Finite element stress fields often exhibit strong local non-smoothness, where stress concentrations near holes, notches, and loading regions induce sharp spatial gradients and high-frequency graph components. Although graph neural networks naturally operate on irregular finite element meshes, conventional message passing is inherently smoothing and progressively attenuates such high-frequency information. Unitary propagation alleviates this problem by preserving spectral magnitudes, but typically relies on matrix functions and high-order approximations with $O(Ked)$ propagation complexity. We propose LiftGCN, an efficient spectrally stable graph network based on Joukowski spectral lifting. LiftGCN maps the real spectrum of a normalized graph operator onto the unit circle through the Joukowski relation and realizes the resulting spectral transformation as a simple second-order recurrence, avoiding matrix exponentials, eigendecomposition, and high-order polynomial truncation. We show that the linear Joukowski backbone has unit-modulus characteristic roots and admits an energy-preserving structure under a positive-definite metric, preventing exponential attenuation of graph-frequency components with depth. Each layer requires only one sparse neighborhood aggregation, yielding $O(ed)$ propagation complexity, while lightweight local nonlinear residuals provide expressive feature transformations. Experiments on finite element stress prediction demonstrate that LiftGCN achieves competitive overall accuracy while improving reconstruction of stress concentrations and local high-gradient structures with substantially reduced computational cost. Our code is available at https://github.com/ChenZeng001/LiftGCN.

While signed-directed graphs have been studied using linear Laplacians in the design of graph neural networks, relatively little research has focused on developing non-linear Laplacian operators for such networks. We introduce a non-linear Laplacian operator specific to signed and directed networks (NLSD). This non-linear operator extends the concepts of the signed Laplacian for signed graphs and the Laplacian for directed graphs. The NLSD calculates node-specific potentials based on features More precisely, if the potential discrepancy is not aligned with the edge direction, we ignore it (and vice versa) leveraging message-passing techniques only across edges where potential discrepancies align with the edge's direction. Utilizing this novel operator, we propose an efficient spectral GNN framework (NLSD-GNN). We conducted comprehensive evaluations focusing on node classification and link prediction, examining scenarios involving signed, directional, or both types of information. Our findings reveal that this spectral GNN framework not only integrates signed and directional data effectively but also achieves superior performance across diverse datasets.

Temporal graph learning is commonly organized around the evolution of node states or the encoding of interaction histories. We study an underexplored, operator-centric question: should the graph propagation mechanism itself evolve over time? We introduce Dynamic Spectral Filtering (DSF), which represents propagation at snapshot t by a Chebyshev polynomial filter with vector-valued, time-dependent coefficients. DSF explicitly treats these compact multi-order coefficients as recurrent temporal states. A recurrent branch proposes updates, while multiplicative global and order-specific gates regulate their magnitude. The temporal state is independent of the number of nodes. On MOOC, Wikipedia, and Reddit temporal link-prediction benchmarks, converged DSF runs attain AP scores of 0.7851, 0.9088, and 0.9860, respectively, with 93K to 133K trainable parameters, 68 to 182 MB peak GPU memory, and 1.6 to 2.1 seconds of training per epoch. Against the closely related DEFT baseline, DSF is better on MOOC, within 0.001 AP on Reddit, and modestly lower on Wikipedia, while using 8.3 to 8.6 times fewer parameters, 25 to 33 times less GPU memory, and 5 to 19 times less time per epoch. Relative to all measured alternatives, it uses 3.3 to 38.6 times less GPU memory. These results support direct spectral-response evolution as a useful temporal inductive bias when computational efficiency is a first-class requirement.

Graph-based fraud detection is essential for safeguarding large-scale transaction systems, where undetected anomalies may lead to substantial financial losses and security risks. Real-world fraud graphs pose two coupled challenges: sparse and imbalanced supervision, where verified fraudulent labels are scarce and heavily skewed toward benign accounts, and representation dilution, where spatial message passing may oversmooth camouflaged anomalies while spectral filters may suppress fraud-relevant mid- and high-frequency irregularities. To address these challenges, we propose ADC-GNN, short for Attention-guided Diffusion-Contrastive Graph Neural Network, a unified framework that combines diffusion-guided feature augmentation, contrastive representation learning, and multi-hop spectral attention for few-shot graph fraud detection. The diffusion component is formulated as a feature-space denoising augmentation mechanism rather than a full topology-generative graph diffusion model: it constructs noise-perturbed node-feature views under a cosine schedule and uses contrastive learning to stabilize node representations across perturbations. The spectral attention module further adaptively emphasizes fraud-relevant hop-level and relation-level cues. We evaluate ADC-GNN primarily on three public benchmarks and additionally report a proprietary real-world telecom transaction dataset with approximately 60,000 records as a private case study. Under the 1% training setting, ADC-GNN achieves consistent improvements over original graph fraud baselines and four protocol-consistent recent graph anomaly/fraud baselines on the public benchmarks. Additional analyses on split stability, training ratios, oversampling alternatives, module-level ablations, diffusion schedules, and runtime and memory-consumption comparisons further characterize the effective operating regime of ADC-GNN.

Spectral graph neural networks (GNNs) interpret message passing as frequency-selective filtering. While low-order spectral filters are efficient, their limited selectivity often leads to weak attenuation outside the passband, whereas high-order alternatives introduce optimization challenges. We propose DCQ-GNN, a spectral GNN based on a compact bank of adaptive convex--concave quadratic filters. By restricting the filter order to two while explicitly exploiting complementary curvature, DCQ-GNN improves spectral selectivity as quantified by Dirichlet energy and entropy measures without resorting to high-order polynomial expansions. The model fuses filter outputs through a node-adaptive gating mechanism to enable node-wise structure-aware spectral selection. We provide a formal spectral analysis grounded in Dirichlet energy attenuation, von Neumann entropy, and curvature polarity, and derive explicit characterizations of filter behavior across varying levels of homophily and structural perturbations. Extensive benchmarks on 10 datasets show that DCQ-GNN ties for the top average rank (3.0) on heterophilic graphs and obtains the second-best rank (4.2) on homophilic graphs, remaining competitive with representative high-order polynomial spectral filters. Furthermore, under strong structural perturbations, DCQ-GNN exhibits substantially smaller performance degradation compared to both first-order and high-order baselines. These results demonstrate that curvature-aware quadratic banks provide a robust and efficient alternative to high-order spectral models while preserving optimization stability and computational efficiency.

Graph foundation models aim to learn transferable knowledge from diverse graphs for generalization to unseen graphs and tasks. Unlike text and images, graphs lack a shared vocabulary or regular spatial grid, making cross-graph transfer challenging. This challenge comes from both feature discrepancies and, more critically, diverse graph structures. Existing GFMs mainly improve transferability by unifying feature spaces or incorporating structural tokens and vocabularies. However, existing topology-aware designs still have limitations. Structural tokens are usually discrete, while structural vocabularies often rely on predefined substructures such as trees and cycles, whose limited coverage may miss richer relational patterns across graphs. Moreover, graph signals contain both high-frequency local patterns and smoother low-frequency patterns, which require different propagation behaviors. These components are often entangled in raw graph signals, while this spectral perspective is rarely explored in existing GFMs. To address these challenges, we propose SPG, a graph foundation model with spectral parsing and prototype-guided spatial propagation. SPG applies learnable Chebyshev filters to decompose node features into multiple spectral responses, reducing the mismatch between frequency-specific graph signals and propagation behaviors. It then constructs a Gromov-Wasserstein prototype geometry to distill transferable pairwise relations beyond predefined substructures into a shared structural space. The learned prototype geometry is further projected back as a prototype-guided propagation operator. Experiments demonstrate consistent improvements in cross-domain generalization.

Self-healing smart grids can quickly adjust their network configuration during outages to minimize power disruptions. During an outage, several actions can be taken, such as network reconfiguration through switching operations and emergency load shedding. However, traditional machine learning methods for outage mitigation are not well suited for smart grids due to their slow response time and high computational cost. To address these challenges, recent studies have explored reinforcement learning to automatically perform network reconfiguration. In these approaches, the control policy is typically modeled using a graph neural network (GNN). However, conventional GNNs operate in the spatial domain and may fail to capture important relationships in the frequency domain. Frequency-domain information is particularly useful for modeling global structural patterns and system-wide interactions in power networks. In this paper, we propose a spectral graph reinforcement learning framework for outage management in distribution networks to enhance system resilience. Our model learns the optimal power restoration policy using a spectral graph neural network. We evaluate the proposed method on three modified IEEE test systems: the 13-bus, 34-bus, and 123-bus networks. Experimental results show that our approach achieves near-optimal performance in real time and generalizes well across a wide range of outage scenarios.

Graph Neural Networks (GNNs) perform computations on graphs by routing the signal between graph regions using a graph shift operator or a message passing scheme. Often, the propagation of the signal leads to a loss of information, where the signal tends to diffuse across the graph instead of being deliberately routed between regions of interest. Two notions that depict this phenomenon are oversmoothing and oversquashing. In this paper, we propose an alternative approach for modeling signal propagation, inspired by quantum mechanics, using the notion of observables. Specifically, we model the place in the graph where the signal lies, how much the signal is concentrated there, and how much of the signal is propagated towards a location of interest when applying a GNN. Using these new concepts, we prove that standard spectral GNNs have poor signal propagation capabilities. We then propose a new type of spectral GNN, termed Schrödinger GNN, which we show has a superior capacity to route the signal across the graph.

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.

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.

With the growing volume of CT examinations, there is an increasing demand for automated tools such as organ segmentation, abnormality detection, and report generation to support radiologists in managing their clinical workload. Multi-label classification of 3D Chest CT scans remains a critical yet challenging problem due to the complex spatial relationships inherent in volumetric data and the wide variability of abnormalities. Existing methods based on 3D convolutional neural networks struggle to capture long-range dependencies, while Vision Transformers often require extensive pre-training on large-scale, domain-specific datasets to perform competitively. In this work, we propose a 2.5D alternative by introducing a new graph-based framework that represents 3D CT volumes as structured graphs, where axial slice triplets serve as nodes processed through spectral graph convolution, enabling the model to reason over inter-slice dependencies while maintaining complexity compatible with clinical deployment. Our method, trained and evaluated on 3 datasets from independent institutions, achieves strong cross-dataset generalization, and shows competitive performance compared to state-of-the-art visual encoders. We further conduct comprehensive ablation studies to evaluate the impact of various aggregation strategies, edge-weighting schemes, and graph connectivity patterns. Additionally, we demonstrate the broader applicability of our approach through transfer experiments on automated radiology report generation and abdominal CT data.