Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum

Fan XuNan WangHao WuXuezhi WenXibin ZhaoHai Wan

article2024AAAI69 citations

Proposes a semi-supervised graph fraud detector that tackles graph heterophily and severe class imbalance by combining mixed-frequency spectral filtering with local environmental constraints to improve anomaly classification.

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Online fraud in digital payments, e-commerce, and social networks causes severe financial and reputational damage. While organizations increasingly apply graph neural networks to uncover suspicious activity by analyzing connection networks, traditional models struggle in real-world fraud environments. Fraud graphs naturally exhibit high heterophily, meaning fraudulent actors deliberately connect with legitimate users to camouflage their behavior. Standard graph models act as low-pass filters that smooth differences across connected entities, washing out critical signals of anomaly. In addition, existing methods underutilize scarce node labels amidst severe class imbalances, where legitimate users vastly outnumber fraudsters.

The article develops and evaluates a semi-supervised fraud detection framework called SEC-GFD. The primary objective is to improve fraud classification accuracy by capturing mixed-frequency network signals and enhancing the utilization of limited label information through local environmental constraints.

The evaluation tests SEC-GFD across four benchmark datasets of varying scales, ranging from roughly 12,000 to over 5.7 million nodes: product reviews (Amazon), business reviews (YelpChi), financial transactions (T-Finance), and social networks (T-Social). The framework introduces two core mechanisms: a hybrid spectral filter that breaks network signals into mixed high-pass and band-pass frequency components, and a contrastive learning module that compares a masked entity’s multi-hop network behavior with its feature-based nearest neighbors to enforce contextual consistency.

The findings confirm clear performance advantages. Across all four datasets, SEC-GFD outperforms standard graph models, established fraud detectors, and specialized heterophily-aware algorithms in both classification accuracy and ranking metrics. For instance, SEC-GFD achieved an F1-macro score of 77.73% and an AUC of 91.39% on YelpChi, and 87.74% F1-macro with 96.11% AUC on the large-scale T-Social dataset. The analysis reveals that indiscriminately pruning heterophilic connections harms detection performance, whereas processing decomposed frequency bands preserves vital fraud patterns. Ablation tests show that both the hybrid filtering and the environmental constraint modules contribute meaningfully, and that larger network datasets benefit from analyzing higher-order neighbor hops.

These results demonstrate that organizations do not need to rely on complex, error-prone edge-pruning heuristics to combat fraudster camouflage. By deploying hybrid frequency analysis and neighborhood-level contrastive constraints, fraud detection systems can significantly reduce false negatives and false positives without extensive manual labeling. This directly translates to lower operational review costs and reduced financial losses from fraudulent transactions.

Organizations operating large-scale transaction or user networks should consider integrating hybrid spectral filtering architectures into their existing fraud intelligence pipelines. For large datasets, practitioners should calibrate the system to evaluate higher-order neighbor connections to capture extended fraudulent interactions. Because the evaluations focus on static graph benchmarks, engineering teams planning practical deployments should conduct pilot testing on dynamic, streaming transaction flows to verify performance stability under continuous real-time drift.

Beijing Jiaotong University IAT KLISS Tsinghua University University of Science and Technology of China binary classification detection algorithm frequency bands graph-based fraud detection heterophily graphs graph spectrum

arXiv: 2312.06441 Sunxkissed/SEC-GFD

Paper: Beyond Homophily in Graph Neural Networks: Current Limitations and Effective Designs, Jiong Zhu et al. (2020). This foundational work establishes why standard graph neural networks fail on heterophilic networks and outlines essential architectural strategies to overcome structural dissimilarity.
Paper: Graph based anomaly detection and description: a survey, Leman Akoglu et al. (2014). This survey provides essential background on graph-based anomaly and fraud detection principles, motivating why relational network modeling is required to uncover adversarial behaviors.
Paper: Discrete Signal Processing on Graphs, Aliaksei Sandryhaila et al. (2012). This paper establishes the foundational discrete signal processing framework on graphs, which underpins the frequency decomposition and spectral filtering techniques used to separate fraud signals.
Paper: Semi-Supervised Classification with Graph Convolutional Networks, Thomas N. Kipf et al. (2017). This seminal text introduces semi-supervised graph convolutional networks, establishing the baseline low-pass filtering mechanics that the source critiques and adapts for heterophilic fraud settings.
Paper: Finding Global Homophily in Graph Neural Networks When Meeting Heterophily, Xiang Li et al. (2022). This paper introduces methods for identifying global homophily under heterophilous graph structures, contextualizing how to aggregate multi-hop signals when direct neighbors are dissimilar.
Paper: Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs, Cristian Bodnar et al. (2022). This work analyzes the mathematical and topological causes of oversmoothing and performance degradation under heterophily, providing key theoretical grounding for frequency-aware graph models.
Paper: Graph Contrastive Learning with Adaptive Augmentation, Yanqiao Zhu et al. (2020). This paper develops contrastive representation learning paradigms on graphs, providing core methodology for local contextual constraints and self-supervised node representation learning.
Paper: Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning, Qimai Li et al. (2018). This paper demonstrates how graph convolution operates as Laplacian smoothing, clarifying the precise mechanism that washes out anomaly signals in fraud detection.
Paper: Ordered GNN: Ordering Message Passing to Deal with Heterophily and Over-smoothing, Yunchong Song et al. (2023). This study addresses the dual challenges of heterophily and oversmoothing in message passing, offering direct context for architectural solutions to disassortative graphs.
Paper: Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering, Michaël Defferrard et al. (2016). This foundational paper develops localized spectral filtering on graphs using Chebyshev polynomials, establishing the mathematical basis for spectral graph convolutions.

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Abstract

Graph-based fraud detection (GFD) can be regarded as a challenging semi-supervised node binary classification task. In recent years, Graph Neural Networks (GNN) have been widely applied to GFD, characterizing the anomalous possibility of a node by aggregating neighbor information. However, fraud graphs are inherently heterophilic, thus most of GNNs perform poorly due to their assumption of homophily. In addition, due to the existence of heterophily and class imbalance problem, the existing models do not fully utilize the precious node label information. To address the above issues, this paper proposes a semi-supervised GNN-based fraud detector SEC-GFD. This detector includes a hybrid filtering module and a local environmental constraint module, the two modules are utilized to solve heterophily and label utilization problem respectively. The first module starts from the perspective of the spectral domain, and solves the heterophily problem to a certain extent. Specifically, it divides the spectrum into various mixed-frequency bands based on the correlation between spectrum energy distribution and heterophily. Then in order to make full use of the node label information, a local environmental constraint module is adaptively designed. The comprehensive experimental results on four real-world fraud detection datasets denote that SEC-GFD outperforms other competitive graph-based fraud detectors. We release our code at https://github.com/Sunxkissed/SEC-GFD.

Introduction
Related Work
Methodology
Problem Definition
Graph Spectrum and Heterophily
Overview of the Proposed Framework
Hybrid-pass Filter Module
Local Environmental Constraint Module
Fraud Detection
Experiments
Experimental Settings
Experimental Results
Heterophily Edges Clipping Analysis
Effectiveness of Each Component of SEC-GFD
Selection of C for high-order neighbors
Conclusion
Acknowledgments
References

Knowls

Knowl 1 — Hybrid-Pass Spectral Filtering in SEC-GFD
model/method
The hybrid-pass filter module in SEC-GFD decomposes the graph spectrum into $2C$ separate frequency bands to capture both band-pass and clean low-order high-frequency information. Given a normalized graph Laplacian $L = I - D^{-1/2} A D^{-1/2}$ and a maximum neighbor order $C$ , the spectrum is decomposed into $(C+1)$ band-pass filters and $(C-1)$ pure high-pass filters.

The $(C+1)$ band-pass filters $W = (W_{0,C}, W_{1,C-1}, \dots, W_{C,0})$ are derived via Beta wavelet transforms:

$W_{p,q} = U \beta^*_{p,q}(\Lambda) U^T = \frac{\left(\frac{L}{2}\right)^p \left(I - \frac{L}{2}\right)^q}{2 B(p+1, q+1)}$

where $p+q = C$ , $\Lambda$ is the diagonal matrix of Laplacian eigenvalues, $U$ contains the eigenvectors, and $B(\cdot, \cdot)$ is the Beta function.

The $(C-1)$ high-pass filters $R = (R_1, R_2, \dots, R_{C-1})$ capture clean high-frequency signals of low-order neighbors without mixing lower-order low frequencies:

$R_k = (\varepsilon I - \tilde{A})^k = ((\varepsilon - 1)I + L)^k$

where $\tilde{A} = D^{-1/2} A D^{-1/2}$ , $k \in \{1, \dots, C-1\}$ , and $\varepsilon \in [0, 1]$ is a scaling hyperparameter.

Input node features $X \in \mathbb{R}^{N \times d}$ are projected to initial representations $H_0 = \text{MLP}(X)$ . Message passing operates across each of the $2C$ frequency channels independently:

$B_i = W_{i, C-i} H_0, \quad H_j = R_j H_0$

and the resulting representations are aggregated into $H = f_{agg}(B_0, \dots, B_C, H_1, \dots, H_{C-1})$ . The module is supervised via a class-weighted cross-entropy loss:

$\mathcal{L}_{hybrid} = \sum_{v \in \mathcal{V}} [\delta y_v \log p_v + (1 - y_v) \log(1 - p_v)]$

where $y_v \in \{0, 1\}$ is the ground-truth node label, $p_v$ is the predicted anomaly probability for node $v$ , and $\delta$ is the ratio of anomaly instances ( $y_v=1$ ) to normal instances ( $y_v=0$ ) in the training set.
Knowl 2 — Local Environmental Constraint Module
model/method
The local environmental constraint module enhances label utilization by enforcing topological and feature-space consistency between target nodes and their surrounding contexts. To prevent the node's own attributes from dominating its contextual representation, the target node's feature is masked ( $h_t^{(0)} = 0$ ). Multi-hop structural neighborhood information is gathered using Simplifying Graph Convolution (SGC) without non-linearities across $L$ layers:

$h_t^{(l+1)} = \text{UPDATE}\left(h_t^{(l)}, \text{AGG}\left(\{h_v^{(l)} : v \in \mathcal{N}_t\}\right)\right)$

$H_t^{neigh} = \text{AGG}(h_t^1, h_t^2, \dots, h_t^L)$

where $\mathcal{N}_t$ is the neighbor set of node $t$ , and $L$ is set to 2 or 3.

In parallel, a feature-space environmental representation $H_t^{knn}$ is computed by identifying the $k$ -nearest neighbors $K_t$ of node $t$ based on cosine similarity in the original feature space and performing average pooling:

$H_t^{knn} = \frac{1}{|K_t|} \sum_{u \in K_t} x_u$

Assuming that normal nodes should exhibit high similarity between structural multi-hop neighbor features and feature-space $k$ -NN context, whereas anomaly nodes should exhibit large divergence, an InfoNCE-style contrastive loss is optimized:

$\mathcal{L}_{env} = -\log \frac{\frac{1}{|\mathcal{V}_n|} \sum_{v_i \in \mathcal{V}_n} e^{\text{sim}(H_{v_i}^{neigh}, H_{v_i}^{knn})}}{\frac{1}{|\mathcal{V}_a|} \sum_{v_j \in \mathcal{V}_a} e^{\text{sim}(H_{v_j}^{neigh}, H_{v_j}^{knn})}}$

where $\mathcal{V}_n$ and $\mathcal{V}_a$ denote the sets of normal and anomalous nodes in the training set, respectively, and $\text{sim}(\cdot, \cdot)$ is the cosine similarity operator.
Knowl 3 — SEC-GFD Overall Objective Function
equation
The overall training objective of the SEC-GFD fraud detector jointly optimizes the spectrum-enhanced hybrid filtering loss $\mathcal{L}_{hybrid}$ and the local environmental contrastive constraint $\mathcal{L}_{env}$ :

$\mathcal{L} = \alpha \mathcal{L}_{hybrid} + (1 - \alpha) \mathcal{L}_{env}$

where $\alpha \in [0, 1]$ is a balancing hyperparameter, $\mathcal{L}_{hybrid}$ is the weighted binary cross-entropy loss over $2C$ decomposed spectral bands:

$\mathcal{L}_{hybrid} = \sum_{v \in \mathcal{V}} [\delta y_v \log p_v + (1 - y_v) \log(1 - p_v)]$

with $\delta = |\mathcal{V}_a| / |\mathcal{V}_n|$ balancing class frequencies, and $\mathcal{L}_{env}$ is the contrastive environmental alignment loss:

$\mathcal{L}_{env} = -\log \frac{\frac{1}{|\mathcal{V}_n|} \sum_{v_i \in \mathcal{V}_n} \exp\left(\text{sim}(H_{v_i}^{neigh}, H_{v_i}^{knn})\right)}{\frac{1}{|\mathcal{V}_a|} \sum_{v_j \in \mathcal{V}_a} \exp\left(\text{sim}(H_{v_j}^{neigh}, H_{v_j}^{knn})\right)}$

Knowl 4 — Performance Comparison on Real-World Fraud Detection Benchmarks

empirical result

SEC-GFD was evaluated against homophily-based GNNs (GCN, GraphSAGE, GAT, GIN), graph fraud detection algorithms (GraphConsis, CARE-GNN, PC-GNN, GAGA), and heterophily-tailored GNNs (ACM, H2-FDetector, BWGNN, GDN) across four real-world benchmarks: YelpChi, Amazon, T-Finance, and T-Social (split into 40% train, 20% validation, 40% test; trained for 100 epochs).

Methods	YelpChi		Amazon		T-Finance		T-Social
Methods	F1-macro	AUC	F1-macro	AUC	F1-macro	AUC	F1-macro	AUC
Homophily GNN
GCN	53.21	56.80	63.52	80.14	71.47	66.31	61.73	87.84
GraphSAGE	64.34	73.61	75.25	86.73	55.27	69.52	60.49	72.65
GAT	55.86	58.92	82.41	89.73	54.27	75.29	73.72	88.61
GIN	63.42	75.28	71.14	81.38	65.72	81.63	63.38	81.33
GFD Algorithm
GraphConsis	57.91	69.55	78.46	87.27	73.58	91.42	58.32	73.89
CARE-GNN	63.58	79.42	83.31	91.57	65.37	91.93	55.92	68.78
PC-GNN	64.75	77.14	91.73	95.63	77.25	92.83	57.21	71.27
GAGA	76.71	89.77	90.31	95.61	85.13	94.72	76.64	87.93
Heterophily GNN
ACM	69.72	88.28	81.83	93.69	85.48	96.02	81.27	90.79
H2-FDetector	69.38	88.63	83.84	96.41	87.39	95.41	78.89	88.56
BWGNN	76.66	90.56	91.68	97.52	84.59	95.29	84.58	95.27
GDN	76.05	90.79	90.68	97.09	86.12	94.28	80.63	89.35
SEC-GFD (ours)	77.73	91.39	92.35	98.23	89.86	96.32	87.74	96.11

SEC-GFD achieves the best performance across all datasets on both F1-macro and AUC metrics. Compared to the best-performing heterophily-specific baselines, SEC-GFD achieves absolute gains of 2.47% F1-macro and 1.71% AUC on T-Finance, and 3.78% F1-macro and 0.84% AUC on the large-scale T-Social dataset.

Knowl 5 — Filter Sensitivity to Heterophily Edge Pruning
empirical result
The effect of removing heterophily edges depends heavily on the filter type (low-pass, band-pass, high-pass) and whether pruning is applied across the entire graph or restricted to edges connected to labeled training nodes:
1. Whole-graph heterophily edge pruning: As the proportion of deleted heterophily edges increases across the whole graph, performance for low-pass and band-pass filters monotonically improves, whereas high-pass filtering performance monotonically decreases. Band-pass filtering achieves the highest performance overall.
2. Training-graph heterophily edge pruning: When only the heterophily edges connected to known labeled nodes in the training set are deleted, low-pass filters remain poorly performing, while both band-pass and high-pass filters exhibit noticeable performance degradation.
These results demonstrate that heterophily edges provide valuable high-frequency information rather than universal harm, indicating that indiscriminate pruning of heterophily connections degrades spectral representation learning.

Knowl 6 — Ablation of Hybrid Filtering and Environmental Constraint Modules

empirical result

An ablation study on SEC-GFD evaluating the individual contributions of the band-pass bands, the low-order pure high-pass bands, and the local environmental constraint module on Amazon and T-Finance:

Variant	Amazon		T-Finance
Variant	F1-macro	AUC	F1-macro	AUC
SEC-GFD	92.27	98.19	89.74	96.34
w/o high-pass	91.48	97.28	88.65	95.05
w/o band-pass	90.29	96.31	86.79	93.77
w/o env	91.88	97.14	88.14	95.21

Removing the $(C+1)$ band-pass frequency bands causes the largest performance degradation (dropping AUC by 1.88% on Amazon and 2.57% on T-Finance; F1-macro by 1.98% on Amazon and 2.95% on T-Finance). Removing the $(C-1)$ high-pass bands degrades AUC by 0.91% to 1.29%. Omitting the local environmental constraint module $\mathcal{L}_{env}$ causes a 0.86% to 1.13% drop in AUC and 0.39% to 1.60% drop in F1-macro, while still outperforming previous state-of-the-art baselines.

Knowl 7 — Impact of Neighbor Spectral Order C Across Graph Scales
empirical result
The neighbor order $C$ determines the polynomial power of the Laplacian filters and the hop reach of multi-order representations in SEC-GFD. Evaluating $C \in \{1, 2, 3, 4, 5, 6\}$ shows distinct behaviors depending on graph size:
1. For small- to medium-sized fraud graphs (Amazon with 11,944 nodes, YelpChi with 45,954 nodes, and T-Finance with 39,357 nodes), performance gains plateau when $C \ge 2$ .
2. For large-scale fraud graphs (T-Social with 5,781,065 nodes and 73,105,508 edges), performance steadily increases with higher neighbor orders up to $C = 5$ before leveling off.
This demonstrates that large-scale fraud networks benefit significantly from higher-order spectral decomposition ( $C=5$ ) due to broader multi-hop propagation of fraudulent patterns across the graph structure.

Coverage note — None was omitted; all contributed methodology components, theoretical motivations, empirical comparisons, edge-pruning findings, ablation experiments, and hyperparameter analyses are fully covered.

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Citation

MLA

Xu, F., et al. “Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum”. arXiv, 2023, http://arxiv.org/abs/2312.06441v3.

APA

Xu, F., Wang, N., Wu, H., Wen, X., Zhao, X., & Wan, H. (2023). Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum. arXiv. http://arxiv.org/abs/2312.06441v3

Chicago

Xu, F., N. Wang, H. Wu, X. Wen, X. Zhao, and H. Wan. 2023. “Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum”. arXiv. http://arxiv.org/abs/2312.06441v3.

Harvard

Xu, F. et al. (2023) “Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2312.06441v3.

Vancouver

1. Xu F, Wang N, Wu H, Wen X, Zhao X, Wan H (2023) Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum. arXiv

BibTeX

@article{xu2023revisiting,
  title = {Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum},
  author = {Xu, Fan and Wang, Nan and Wu, Hao and Wen, Xuezhi and Zhao, Xibin and Wan, Hai},
  year = {2023},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2312.06441v3},
  eprint = {2312.06441}
}

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Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum

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Table of Contents

Knowls

Knowl 1 — Hybrid-Pass Spectral Filtering in SEC-GFD

Knowl 2 — Local Environmental Constraint Module

Knowl 3 — SEC-GFD Overall Objective Function