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Feature Transformation Enhanced Jacobi Polynomial Graph Filtering for Graph Anomaly Detection

Xiang Wang, Zhijun Cheng, Zhenyu Meng

arXiv:2608.27144v1cs.AI

TL;DR

Graph anomaly detection needs filters that adapt to graph spectral distributions, preserve fine-grained feature information, and use node labels more effectively. JPGFN combines FSTNN, adaptive Jacobi polynomial filtering, and a node-label constraint module; experiments on multiple real-world datasets show it significantly outperforms mainstream approaches.

  • Problem

    Existing frequency-domain GAD methods use static graph-filter bases, insufficiently model attribute importance, and inadequately use node labels.

  • Method

    JPGFN combines FSTNN for fine-grained feature learning, adaptive parameterized Jacobi polynomial filtering, and a node-label constraint module.

  • Results

    JPGFN significantly outperforms mainstream approaches across multiple real-world datasets, with five-dataset ablations showing AUC-ROC and AUC-PR gains from FSTNN over the w/o FSTNN variant.

  • Takeaways & Limitations

    Adaptive filtering, feature separation transformation, and label constraints together form the paper’s proposed framework for graph anomaly detection.

  • Takeaways & Limitations

    The study focuses on semi-supervised anomaly detection on attributed graphs using partial node labels.

Abstract

from arXiv · show

In recent years, graph anomaly detection (GAD) based on frequency-domain filtering have achieved promising results. However, existing approaches still face three major challenges: First, they use static basic function to constructed graph filter which cannot effectively adapt to the frequency-domain distribution of graph data. Second, they fail to adequately consider the importance information of each attribute in the node feature vector, leading to the loss of fine-grained information. Third, they insufficiently utilize node labels for GAD. To address these issues, this paper proposes a novel graph anomaly detection method called JPGFN (Feature Transformation Enhanced Jacobi Polynomial Graph Filtering Network). First, a Feature Separation Transformation Network (FSTNN) is developed to better learn fine-grained node features by feature separation and applying nonlinear transformations to node features across different dimensions. Second, an adaptive Jacobi polynomial graph filtering module is constructed based on Jacobi polynomials to adaptively capture complex frequency-domain features of graph signals. Finally, a node label constraint module is developed to facilitate the use of node labels and enhance the performance of GAD. Experimental results on multiple real-world datasets demonstrate that the proposed method significantly outperforms mainstream approaches.

1. Introduction

Graph anomaly detection must model both node attributes and graph structure, while frequency-domain methods face inflexible filters, insufficient feature-importance modeling, and limited label use. JPGFN addresses these issues with FSTNN, adaptive Jacobi filtering, and node-label constraints, outperforming mainstream approaches across real-world datasets.

  • Graph neural networks improve anomaly detection by jointly capturing node features and topological information through feature aggregation.
  • Frequency-domain GAD methods use graph filters but static basis functions cannot adapt to differing spectral distributions across real-world graphs.
  • JPGFN uses FSTNN to learn fine-grained node features through feature separation and nonlinear transformations across feature dimensions.
  • JPGFN constructs an adaptive graph-filtering module using parameterized Jacobi polynomials to capture complex frequency-domain information.
  • JPGFN adds a node-label constraint module and significantly outperforms mainstream approaches on multiple real-world datasets.

2. Related Work

Related GAD methods include spatial-domain approaches based on neighborhood aggregation and frequency-domain approaches based on graph filtering. Their limitations arise from frequency-domain information loss and assumptions about neighborhood similarity that often fail for anomalous nodes.

  • General-purpose GNNs commonly rely on neighboring nodes having similar labels and features, an assumption that often fails in graph anomaly detection.
  • Spatial-domain GAD methods iteratively aggregate neighborhood information, with examples addressing imbalance, long-range features, inconsistency, and association patterns.
  • Spatial-domain methods cannot learn the frequency-domain information of graph data.
  • Spectral-domain methods extract frequency-domain information using graph filtering, including wavelet, Beta-wavelet, and Chebyshev-polynomial filters.

3. Preliminaries

The paper represents attributed graphs with nodes, features, adjacency, and partial binary anomaly labels, and defines spectral graph filtering through the normalized Laplacian and polynomial filter approximations.

  • Notations and Problem Definition: An attributed graph is represented as G = (V, X, A), with node set V, feature matrix X, and adjacency matrix A.
  • Notations and Problem Definition: Semi-supervised GAD assigns labels 0 and 1 to normal and abnormal nodes and trains a classifier from partial node labels.
  • Graph Filtering: The normalized Laplacian is L = I − D^-1/2 A D^-1/2, and its eigendecomposition provides graph frequencies through eigenvalues and eigenvectors.
  • Graph Filtering: Graph filtering applies g(Λ) in the graph Fourier domain, while a K-order polynomial approximation avoids the high cost of eigenvalue decomposition.

4. Methodology

JPGFN combines feature-specific nonlinear transformation, adaptive Jacobi-polynomial graph filtering, and node-label constraints for graph anomaly detection. Its filtering adapts to diverse frequency-domain distributions while preserving fine-grained feature information.

  • JPGFN consists of FSTNN, an adaptive Jacobi polynomial graph filtering module, and a node label constraint module.
  • 4.1. FSTNN: FSTNN separately transforms node attributes with nonlinear functions, then concatenates the transformed features and applies a linear transformation.The paper uses an MLP as the nonlinear transformation function.
  • 4.2.1. Jacobi Polynomial Basis Functions: Jacobi polynomials provide a flexible basis because adjusting parameters a and b can produce other polynomial families.The parameters satisfy a, b > −1.
  • 4.2.2. Graph Filter Construction: The graph filter uses learnable Jacobi coefficients and basis parameters to capture frequency-domain characteristics across different frequency bands.The filter function is defined on [0, 2], using Laplacian eigenvalues as its spectral input.
  • 4.2.2. Graph Filter Construction: Graph filtering is performed separately for each feature dimension, and outputs from K frequency bands are fused through learnable parameters.The resulting node representation is used for subsequent graph anomaly detection.
  • 4.3. Node Label Constraint Module: The node label constraint module compares central-node representations with local-environment representations to support anomaly discrimination.It uses neighborhood aggregation and cosine similarity, with a contrastive constraint loss separating normal and anomalous embeddings.

5. Experiments

The experiments evaluate JPGFN on five real-world graph anomaly detection datasets using AUC-ROC and AUC-PR. The datasets cover fraudulent reviews, financial fraud, and abnormal users in social networks.

  • Experiments are conducted on five datasets to evaluate the effectiveness of JPGFN.
  • The datasets include Amazon, YelpChi, T-Finance, Elliptic, and Weibo.They address fraudulent reviews, financial fraud, and abnormal-user detection in social networks.
  • JPGFN is evaluated using AUC-ROC and AUC-PR.AUC-PR is described as better reflecting performance on rare anomaly samples under class imbalance.

5.3. Baseline Methods

The baseline comparison covers traditional GNNs, spatial-domain GAD methods, and spectral-domain methods. Experiments use common settings, repeated runs, and dataset-specific splits to support fair comparison.

  • Baselines comprise traditional GNNs, spatial-domain approaches, and spectral-domain methods.The listed models include GCN, GAT, GraphSAGE, GIN, GAS, PC-GNN, GDN, GFCN, BWGNN, AMNet, SEC-GFD, AHFAN, EGNN, and DSGAD.
  • The experiments use identical hardware and experimental conditions for all models.They run on a Tesla A40-48G GPU and Intel Xeon Gold 6326 CPU.
  • Each model is run independently 10 times on each dataset, with mean results used for comparison.The procedure also records standard deviation.
  • Optuna performs parameter grid search, while dataset splitting follows commonly used ratios that vary by dataset.All experiments use the same experimental settings within each dataset.

5.5. Performance Evaluation

JPGFN achieves the strongest anomaly-detection performance across five real-world datasets, while ablations support the contributions of feature separation, adaptive Jacobi filtering, and label constraints.

  • Overall comparison: JPGFN achieves the highest anomaly-detection accuracy across all five datasets compared with the baseline method.Reported improvements span both AUC-ROC and AUC-PR on Amazon, YelpChi, T-Finance, Elliptic, and Weibo.
  • Overall comparison: General GNNs perform poorly in these experiments, whereas GNN-based methods designed for anomaly detection perform better.The paper attributes this contrast to general GNNs’ homogeneity assumption and low-pass neighborhood aggregation.
  • Ablation study: JPGFN improves AUC-ROC and AUC-PR over w/o FSTNN across all five datasets.AUC-PR improvements are 5.75%, 22.9%, 13.68%, 15.79%, and 3.67%, respectively.
  • Ablation study: JPGFN consistently outperforms the fixed-polynomial variants, supporting the greater adaptability of Jacobi polynomials across graph datasets.The comparison includes Chebyshev, Legendre, and Bernstein polynomial variants.
  • Ablation study: Removing the node label constraint module produces small but generally negative performance changes across the five datasets.AUC-PR changes relative to w/o Label are 0.57%, 0.78%, 0.14%, 0.59%, and 0.1%.

5.7. Parameter Analysis

Parameter analysis shows that optimal Jacobi parameters, polynomial orders, and hidden dimensions vary across datasets, reflecting differences in their frequency-domain distributions.

  • Parameter roles: Parameters a and b are learnable variables, allowing the graph filter to adapt to complex spectral distributions.The polynomial order controls the frequency range learned by the filter, while hid-dim controls latent representational capacity.
  • Jacobi parameters: The optimal Jacobi parameters a and b differ across datasets, with distinct values reported for Elliptic, YelpChi, Amazon, T-Finance, and Weibo.The reported optima include (-1.0, 2.0) for Elliptic, (-0.75, -0.5) for YelpChi, (-1.0, 0.25) for Amazon, and (-0.5, -0.25) for T-Finance.
  • Polynomial order: Different datasets exhibit different performance trends as the polynomial order K changes.The paper links this variation to distinct graph frequency-domain distributions and patterns.
  • Hidden dimension: The best hidden layer dimension is 64 for Amazon, YelpChi, and T-Finance, 16 for Elliptic, and 32 for Weibo.These dataset-specific optima indicate that hidden dimensionality requirements vary across datasets.

5.8. Visualization

The visualization study uses t-SNE on YelpChi to compare node representations from JPGFN and its variants, with colors marking normal and abnormal nodes.

  • Visualization setup: The YelpChi visualization maps node representations into two dimensions using t-SNE, coloring normal nodes blue and abnormal nodes yellow.The resulting comparison is shown in Figure 5.
  • Variant comparison: JPGFN produces better visualization quality than the w/ MLP, w/o Jacobi, and w/o Label variants.The paper associates these differences with feature-importance modeling, richer frequency-domain information, and label-information use, respectively.

5.9. Analysis of Time Complexity

JPGFN’s theoretical complexity combines feature transformation, polynomial graph filtering, and label-constraint processing, while runtime depends on implementation factors and differs from competing methods.

  • Theoretical complexity: The stated theoretical complexity of JPGFN is Nnd^2 + KEd.FSTNN has complexity Nnd^2, and the adaptive Jacobi graph-filtering module has complexity KEd.
  • Theoretical complexity: The Jacobi filtering module has time complexity comparable to a general Kth-order polynomial graph filter.The node label constraint module has an overall complexity impact comparable to graph convolution.
  • Empirical runtime: On YelpChi, JPGFN requires less computational time than AHFAN but more training and inference time than SEC-GFD and DSGAD.The paper attributes added overhead partly to JPGFN’s feature separation operation.
  • Empirical runtime: Actual runtime depends on coefficients and model parameters in addition to theoretical complexity.The paper therefore evaluates training and inference costs empirically.

6. Conclusion and Future Work

JPGFN is presented as a graph anomaly detection framework combining fine-grained feature learning, adaptive Jacobi-polynomial filtering, and node-label constraints. Results on five real-life datasets support its superiority over mainstream baselines, while future work targets more effective filters and complex graph settings.

  • JPGFN combines FSTNN for fine-grained node-feature learning, adaptive Jacobi-polynomial filters for frequency-domain information, and node-label constraints for training.The framework addresses feature representation, spectral adaptation, and label utilization within one GAD method.
  • Experimental outcomes on five real-life datasets validate JPGFN’s superiority over existing mainstream baseline methods.
  • Future work will design more effective graph filters for learning frequency-domain information in graph signals.
  • The method will also be extended to more complex scenarios, including dynamic and heterogeneous graphs.
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