Modeling Spectral Energy Shifts in Spatio-Temporal Graph Anomaly Detection

# Modeling Spectral Energy Shifts in Spatio-Temporal Graph Anomaly Detection
Source: [https://arxiv.org/html/2606.00304](https://arxiv.org/html/2606.00304)
###### Abstract

Graph anomaly detection methods aim to distinguish anomalous nodes\. While prior methods characterize anomalies through increased variation in the spectral energy distributions, they overlook those that result in decreased variation, i\.e\., camouflaged anomalies that appear normal\. We show that this type of anomaly persists across multiple datasets and remains undetectable by existing spectral approaches\. To address this limitation, we propose a node\-level spectral energy formulation that is fully compatible with message passing and enables the detection of camouflaged anomalies\. Building on this formulation, we introduce an energy\-aware graph learning framework that models spectral shifts through energy\-driven message passing in both static and time\-series graphs\. Besides, our unified architecture extends to temporal settings without introducing specialized sequence modules, enabling efficient learning under long sliding windows\. Extensive experiments on large\-scale benchmarks demonstrate the effectiveness and scalability of our approach\. Our code is available at[https://github\.com/AICPS\-Lab/Spectral\-Energy\-Shifts\-in\-GAD](https://github.com/AICPS-Lab/Spectral-Energy-Shifts-in-GAD)\.

Machine Learning, ICML

## 1Introduction

Network systems underpins critical application domains including financial transaction networks\(Maet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib3)\), e\-commerce platforms, social graphs\(Hooiet al\.,[2016](https://arxiv.org/html/2606.00304#bib.bib23)\), and cyber\-physical infrastructures\(Deng and Hooi,[2021](https://arxiv.org/html/2606.00304#bib.bib5)\)\. In these applications, anomalies such as fraudulent transactions, fake reviews, and malicious accounts do not occur in isolation, but propagate through connectivity, amplify their impact, and cause substantial economic losses and system failures\(Liuet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib4); Deng and Hooi,[2021](https://arxiv.org/html/2606.00304#bib.bib5)\)\. Detecting such anomalies in graph\-structured data, known as Graph Anomaly Detection \(GAD\), is therefore essential yet remains an open problem\(Liuet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib39)\)\.

In GAD, existing methods can be broadly categorized into two directions: spatial and spectral\(Tanget al\.,[2023](https://arxiv.org/html/2606.00304#bib.bib41)\)\. Spatial methods aim to enhance graph representation learning, including approaches\(Liet al\.,[2019](https://arxiv.org/html/2606.00304#bib.bib42); Douet al\.,[2020](https://arxiv.org/html/2606.00304#bib.bib24)\)\. On the other hand, spectral methods address the problem by designing different graph filters, AMNet\(Chaiet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib8)\), to capture anomaly patterns in the frequency domain\. Among these works, the spectral formulation proposed in\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)characterizes node attribute anomalies with a high variation trend known as the “right\-shift” phenomenon\. This formulation shows potential to learn anomaly behavior and has inspired subsequent studies\(Linet al\.,[2024](https://arxiv.org/html/2606.00304#bib.bib10); Duanet al\.,[2023](https://arxiv.org/html/2606.00304#bib.bib12)\)\.

The spectral formulation, however, suffers from two key limitations, namely, limited scope and scalability\. The formulation focuses on “right\-shift”, where anomalies exhibiting high variance in energy distribution, and overlooks camouflaged anomalies\(Gaoet al\.,[2023](https://arxiv.org/html/2606.00304#bib.bib11)\)\. These anomalies are embedded in the graph by imitating benign node behavior, resulting in high similarity to normal nodes and little deviation from the majority distribution\(Douet al\.,[2020](https://arxiv.org/html/2606.00304#bib.bib24)\)\. Such anomalies exist in public benchmark datasets but cannot be effectively detected by existing energy\-based formulations\. In our observations on the YelpChi\(Rayana and Akoglu,[2015](https://arxiv.org/html/2606.00304#bib.bib13)\)dataset \(Fig\.[1](https://arxiv.org/html/2606.00304#S2.F1)\), anomalies do not consistently follow the right\-shift pattern\. Among the 32 features, anomalies exhibit lower spectral energy than normal nodes in 26 features\. This result indicates that these anomalies do not significantly deviate from normal nodes, thereby being overlooked by spectral methods\. Moreover, graph\-level spectral energy computation is costly, limiting scalability to large graphs\. To address these challenges, we propose the Energy Graph Neural Network \(EGNN\), a node\-level spectral GAD method\. EGNN broadens the scope of spectral formulation to include camouflaged anomalies by i\) quantifying them using the Rayleigh quotient and ii\) leveraging the observed “left\-shift” phenomenon to describe the low\-variance trend caused by the anomalies\. To improve scalability, we propose a local energy surrogate to approximate graph energy\.

While static detection relies on a single snapshot of spectral footprint, temporal detection relies on the trajectory of energy distributions\. The spectral formulation naturally focuses on the changes in energy distribution, making it a promising representation in the temporal setting\. In time\-series GAD, various approaches span unsupervised and semi\-supervised paradigms, and range from linear models to deep learning methods\(Tuliet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib14); Ruffet al\.,[2018](https://arxiv.org/html/2606.00304#bib.bib16); Xuet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib17); Wuet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib18)\)\. Most existing methods decouple spatial and temporal modeling by stacking temporal modules on static graph backbones\(Parejaet al\.,[2020](https://arxiv.org/html/2606.00304#bib.bib21)\)\. This class of frameworks has a large number of learnable parameters and suffers from poor data efficiency under label\-scarce and highly imbalanced settings\. In addition, their reliance on sliding windows makes performance sensitive to window length\(Huanget al\.,[2023](https://arxiv.org/html/2606.00304#bib.bib20); Maet al\.,[2024](https://arxiv.org/html/2606.00304#bib.bib19)\), and optimization instability\(Pascanuet al\.,[2013](https://arxiv.org/html/2606.00304#bib.bib22)\)\(e\.g\., vanishing or exploding gradients\)\. For temporal graphs, EGNN avoids introducing additional temporal modules; instead, it models temporal dependencies via energy transformations and adaptive gating within sliding windows, yielding a data\-efficient design that is less sensitive to window size\. Our contributions are summarized as follows\.

- •We reveal a previously overlooked camouflaged anomaly pattern that appears across multiple public datasets, and characterize it with spectral formulation\.
- •We propose the EGNN, a node\-level spectral\-energy\-based GAD method\. It broadens the scope of spectral formulation to include camouflaged anomalies and enables energy\-driven message passing through GNNs\. We further mitigate the scalability issue by using local energy surrogates to approximate the graph energy\.
- •We generalize EGNN to time\-series graphs by leveraging a local spectral surrogate to characterize changes over time and adaptive gating within slide windows\. EGNN yields a lightweight, data\-efficient model that outperforms state\-of\-the\-art approaches with orders of magnitude fewer parameters, even under scarce and highly imbalanced supervision\.
- •We conduct a comprehensive evaluation on77benchmarks spanning both static and time\-series graphs, comparing against1414baselines, including state\-of\-the\-art methods\. EGNN consistently outperforms, demonstrating its effectiveness across diverse settings\.

## 2Spectral Energy of Abnormal Graphs

This section introduces the spectral\-energy perspective for graph anomaly detection and summarizes our key observations\. Section[2\.1](https://arxiv.org/html/2606.00304#S2.SS1)presents the GAD problem formulation, defines spectral energy, and reviews the associated spectral right\-shift phenomenon\. Section[2\.2](https://arxiv.org/html/2606.00304#S2.SS2)identifies an underexplored anomaly pattern in real\-world graphs, formalizes it through a precise definition, and validates it on synthetic datasets\. Section[2\.3](https://arxiv.org/html/2606.00304#S2.SS3)further analyzes how spectral energy manifests at the subgraph level, motivating localized modeling for scalable detection\.

### 2\.1Problem Setup and Spectral Energy

#### Graph Anomaly Detection\.

Let𝒢=\(𝒱,ℰ\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\}\)denote an unweighted graph, where𝒱=\{vi\}i=1N\\mathcal\{V\}=\\\{v\_\{i\}\\\}\_\{i=1\}^\{N\}is the set ofNNnodes andℰ⊆𝒱×𝒱\\mathcal\{E\}\\subseteq\\mathcal\{V\}\\times\\mathcal\{V\}is the edge set\. Each nodeviv\_\{i\}is associated with addfeature vectorxi∈𝒳⊆ℝdx\_\{i\}\\in\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}, and we collect the set of node features𝒳\\mathcal\{X\}as𝒳=\{xi\}i=1N\\mathcal\{X\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\. Let𝐱:=\[x1,…,xN\]⊤\\mathbf\{x\}:=\[x\_\{1\},\\ldots,x\_\{N\}\]^\{\\top\}, and𝐱f∈ℝN\\mathbf\{x\}\_\{f\}\\in\\mathbb\{R\}^\{N\}denote the column vector of the graph feature indexedff\. In this work, we focus on*node\-level anomalies*and assume the graph structure is reliable, i\.e\., all edges are anomaly\-free\. We denote the set of anomalous nodes as𝒱a⊆𝒱\\mathcal\{V\}\_\{a\}\\subseteq\\mathcal\{V\}and the set of normal nodes as𝒱n⊆𝒱\\mathcal\{V\}\_\{n\}\\subseteq\\mathcal\{V\}, where𝒱a∩𝒱n=∅\\mathcal\{V\}\_\{a\}\\cap\\mathcal\{V\}\_\{n\}=\\emptyset\. Accordingly, we formulate the GAD as a binary classification problem\. Given the graph𝒢\\mathcal\{G\}, node features𝒳\\mathcal\{X\}, and partial node labels𝒱a\\mathcal\{V\}\_\{a\}and𝒱n\\mathcal\{V\}\_\{n\}, the goal is to learn a classifier that assigns labels to each unlabeled nodev∈𝒱v\\in\\mathcal\{V\}\.

![Refer to caption](https://arxiv.org/html/2606.00304v1/sections/picture/moti.png)Figure 1:Spectral energy distributions in the YelpChi datasets\. Each bin shows the average spectral energy difference between normal and anomalous nodes for each feature\. Blue indicates anomalous nodes have lower energy, and yellow indicates anomalous nodes have higher energy\.
#### Spectral Energy Distribution

LetA∈ℝN×NA\\in\\mathbb\{R\}^\{N\\times N\}be the adjacency matrix of the graph𝒢\\mathcal\{G\}, and letD∈ℝN×ND\\in\\mathbb\{R\}^\{N\\times N\}denote the associated degree matrix, with diagonal entriesDi​i=diD\_\{ii\}=d\_\{i\}, wheredi=∑jAi​jd\_\{i\}=\\sum\_\{j\}A\_\{ij\}\. The normalized Laplacian matrix is defined asL=I−D−1/2​A​D−1/2L=I\-D^\{\-1/2\}AD^\{\-1/2\}, whereIIis the identity matrix of appropriate dimension\. When the underlying graph is undirected, i\.e\.,ℰi​j=ℰj​i\\mathcal\{E\}\_\{ij\}=\\mathcal\{E\}\_\{ji\}, the LaplacianLLis symmetric, i\.e\.,L=LTL=L^\{T\}\. This symmetry guarantees thatLLadmits real eigenvalues and an orthonormal eigenbasis\. LetL=U​Λ​U⊤L=U\\Lambda U^\{\\top\}be the eigendecomposition ofLL, where the eigenvalues satisfy0=λ1≤⋯≤λN0=\\lambda\_\{1\}\\leq\\cdots\\leq\\lambda\_\{N\}, andU=\[u1,u2,…,uN\]U=\[u\_\{1\},u\_\{2\},\\ldots,u\_\{N\}\]is the corresponding orthonormal eigenvectors\. Consider the graph attribute𝐱f\\mathbf\{x\}\_\{f\}at feature indexff, its graph Fourier transform is defined as𝐱^f=U⊤​𝐱f\\hat\{\\mathbf\{x\}\}\_\{f\}=U^\{\\top\}\\mathbf\{x\}\_\{f\}\. The spectral energy at frequencyλk​\(1≤k≤N\)\\lambda\_\{k\}\(1\\leq k\\leq N\)is then defined as𝐱^k,f2∑i=1N𝐱^i,f2\\frac\{\\hat\{\\mathbf\{x\}\}\_\{k,f\}^\{2\}\}\{\\sum\_\{i=1\}^\{N\}\\hat\{\\mathbf\{x\}\}^\{2\}\_\{i,f\}\}\.

#### Right\-Shift Phenomenon

Previous study\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)has shown that anomalous patterns tend to concentrate in high\-frequency components in the spectral domain\. A right\-shift phenomenon is observed when the spectral energy distribution shifts toward higher frequencies Specifically, it characterizes the energy distribution of a graph signal at featureffin the Rayleigh quotient format

Ef\(R\)=∑k=1Nλk​x^k,f2∑k=1Nx^k,f2=𝐱f⊤​L​𝐱f𝐱f⊤​𝐱f\.E^\{\(\\mathrm\{R\}\)\}\_\{f\}=\\frac\{\\sum\_\{k=1\}^\{N\}\\lambda\_\{k\}\\hat\{x\}\_\{k,f\}^\{2\}\}\{\\sum\_\{k=1\}^\{N\}\\hat\{x\}\_\{k,f\}^\{2\}\}=\\frac\{\\mathbf\{x\}\_\{f\}^\{\\top\}L\\mathbf\{x\}\_\{f\}\}\{\\mathbf\{x\}\_\{f\}^\{\\top\}\\mathbf\{x\}\_\{f\}\}\.\(1\)

### 2\.2Camouflage Anomalies

Consider a perfectly smooth, anomaly\-free graph signal in which node features are independently and identically distributed as𝐱f∼𝒩​\(μ​𝟏,σ2​I\)\\mathbf\{x\}\_\{f\}\\sim\\mathcal\{N\}\(\\mu\\mathbf\{1\},\\sigma^\{2\}I\), where𝟏\\mathbf\{1\}is an all\-ones vector,μ\\muandσ2\\sigma^\{2\}stand for mean and variance, respectively\. Increasing the proportion of right\-shift anomalies enlarges the ratio ofσ/\|μ\|\\sigma/\|\\mu\|\. However, not all real\-world anomalies exhibit high variance\. Previous studies\(Hooiet al\.,[2016](https://arxiv.org/html/2606.00304#bib.bib23); Douet al\.,[2020](https://arxiv.org/html/2606.00304#bib.bib24)\)identify a distinct type of anomaly known as camouflage anomalies, where malicious nodes corrupt the graph by intentionally mimicking the behavior of normal, benign nodes\. This anomaly type fundamentally differs from the common high\-variance assumption, as such anomalous data are not out\-of\-distribution\. Instead, these nodes closely resemble normal ones, and their insertion into the graph reduces feature variance\. A representative example of this anomaly arises in Amazon fake review detection, where paid reviewers tend to produce uniformly high ratings\. In this case, the original rating distribution \(which naturally contains diverse opinions\) is contaminated by an excessive number of identical high\-score reviews, causing the overall distribution to concentrate toward a single value and exhibit reduced variance and diversity\.

![Refer to caption](https://arxiv.org/html/2606.00304v1/sections/picture/left.png)Figure 2:Spectral energy distribution on a BA graph under varying fractions \(α\\alpha\) of neighbor\-averaged anomalies\. Higherα\\alphashifts energy toward lower frequencies, indicating a left\-shift phenomenon\.From a probabilistic perspective, this effect can be interpreted as a*variance\-suppressing contamination process*: injecting identical or highly concentrated samples reduces the empirical variance\. In the spectral domain, such a distribution collapse suppresses high\-frequency variations and shifts spectral energy toward low\-frequency components, producing a*left\-shift*phenomenon\.

###### Definition 2\.1\(Left\-Shift\)\.

Anomalies induced by distribution collapse shift spectral energy from high to low frequencies, producing a leftward shift in the spectral energy distribution, i\.e\.,σ/\|μ\|\\sigma/\|\\mu\|decreasing\.

![Refer to caption](https://arxiv.org/html/2606.00304v1/sections/picture/vali.png)Figure 3:Local spectral anomalies under varying anomaly varianceσ\\sigma\(a, c\) and anomaly ratioα\\alpha\(b, d\)\. Plots \(a,b\) correspond to the BA graph, and plots \(c,d\) correspond to the Minnesota network\. In all plots, higher prevalence and degree of anomalies shift the curves to the right, showing the effectiveness of local spectral energy\.#### Camouflage anomalies in real\-world datasets\.

We further validate the presence of camouflage anomalies on two widely used graph anomaly detection benchmarks, YelpChi and Amazon\. Dataset statistics are summarized in TableLABEL:tab:static\_dataset\. The results are shown in Fig\.[1](https://arxiv.org/html/2606.00304#S2.F1), with additional results on Amazon provided in the Appendix[C](https://arxiv.org/html/2606.00304#A3)\. For each dataset, we compute the average local spectral energy of normal and anomalous nodes along each feature dimension\. On YelpChi, anomalous nodes exhibit lower spectral energy in half of the features, suggesting that this pattern is not incidental but closely related to distributional collapse\. A plausible real\-world explanation is that camouflage anomalies may arise from identical or highly concentrated fake reviews, which make anomalous nodes locally smoother and shift their energy toward lower frequencies\.

#### Left\-Shift Phenomenon Validation

To illustrate the left\-shift behavior of camouflage anomalies, we conduct a synthetic experiment on a Barabási–Albert \(BA\) graph with10001000nodes\. We generate one\-dimensional node features from𝒩​\(1,1\)\\mathcal\{N\}\(1,1\)for all nodes\. Camouflage anomalies are then injected by replacing the features of selected nodes with the average feature value of their immediate neighbors, thereby forcing anomalous nodes to mimic benign local patterns\. We vary the anomaly ratio asα∈\{0%,10%,30%,50%\}\\alpha\\in\\\{0\\%,10\\%,30\\%,50\\%\\\}to examine its effect on the spectral energy distribution\. As shown in Fig\.[2](https://arxiv.org/html/2606.00304#S2.F2), the colored histograms report the proportion of spectral energy within four equal\-width eigenvalue intervals, while the curves show the cumulative spectral energy as a function of the eigenvalueλ\\lambda\. Asα\\alphaincreases, spectral energy progressively shifts toward lower frequencies\. This confirms that camouflage anomalies increase local smoothness and exhibit the behavior defined in Definition[2\.1](https://arxiv.org/html/2606.00304#S2.Thmtheorem1)\.

### 2\.3Local Energy Ratio Approximation

Following the experimental setting of the previous work\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\), we conduct controlled experiments on synthetic graphs with injected anomalies to validate the spectral\-energy shift phenomenon using local energy\. We consider two graph types: a BA graph with 500 nodes and the Minnesota road network with 2,642 nodes\. Normal node features are drawn from𝒩​\(1,1\)\\mathcal\{N\}\(1,1\), while anomalous nodes are drawn from𝒩​\(1,σ2\)\\mathcal\{N\}\(1,\\sigma^\{2\}\)withσ\>1\\sigma\>1\. We examine two anomaly settings: \(i\) fixing the anomaly ratio to5%5\\%while varying anomaly magnitudeσ∈\{1,2,5,20\}\\sigma\\in\\\{1,2,5,20\\\}, and \(ii\) fixingσ=5\\sigma=5while varying anomaly ratioα∈\{0%,1%,5%,20%\}\\alpha\\in\\\{0\\%,1\\%,5\\%,20\\%\\\}\.

Our visualization results are shown in Fig\.[3](https://arxiv.org/html/2606.00304#S2.F3)\. We analyze node\-level anomalies and the corresponding spectral energy distributions\. When no anomalies are present, more than 80% of the spectral energy is concentrated in the low\-frequency region\. As the proportion of anomalous nodes increases, spectral energy progressively shifts toward higher frequencies\. Specifically, with 1% anomalous nodes, approximately 10% of the low\-frequency energy shifts to higher frequencies; with 5% anomalies, more than 20% shifts; and with 20% anomalies, over half of the low\-frequency energy moves to the high\-frequency region\.

The effect of anomaly magnitude is more pronounced: when increasing the anomaly variance from 1 to 20, more than 60% of low\-frequency energy is transferred to higher frequencies\. As more anomalous nodes with greater magnitudes are introduced, the spectral energy distribution consistently shifts toward higher frequencies, consistent with the global spectral energy distribution\. This observation indicates that local spectral energy effectively captures anomalous behavior\. The resulting energy curves appear more discrete than their global counterparts, as expected from node\-level computation\.

## 3Experiments

In this section, we compare EGNN with state\-of\-the\-art approaches across both temporal and static settings\. We evaluate detection accuracy, assess scalability on the large\-scale T\-Social dataset, and examine the effectiveness of capturing left\-shift anomalies by contrasting models trained only on right\-shift energy with those trained on both\.

#### Datasets

To evaluate, we conduct experiments on seven public benchmarks, including four static graphs and three time\-series graphs\. TableLABEL:tab:static\_datasetsummarizes their statistics\.Amazon\(McAuley and Leskovec,[2013](https://arxiv.org/html/2606.00304#bib.bib25)\)andYelpChi\(Rayana and Akoglu,[2015](https://arxiv.org/html/2606.00304#bib.bib13)\)target fake\-review detection and exhibit mixed left\-shift and right\-shift spectral patterns, suggesting the coexistence of distribution collapse and outliers\.T\-Finance\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)focuses on anomalous account detection in transaction networks, whileT\-Social\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)considers large\-scale social graphs\. Notably, T\-Social contains over 5M nodes and 73M edges, and is included to demonstrate EGNN’s scalability\.

The Secure Water Treatment \(SWaT\) dataset\(Mathur and Tippenhauer,[2016](https://arxiv.org/html/2606.00304#bib.bib27)\)is collected from a real\-world water treatment testbed and serves as a representative cyber\-physical system benchmark\. The Water Distribution \(WADI\) dataset\(Ahmedet al\.,[2017](https://arxiv.org/html/2606.00304#bib.bib28)\)extends SWaT to a larger\-scale water distribution network with more complex pipeline structures\. TheMSLdataset\(Hundmanet al\.,[2018](https://arxiv.org/html/2606.00304#bib.bib29)\)consists of spacecraft telemetry, where anomalies correspond to system faults and unexpected operational behaviors\.

#### Experimental Setting

To ensure a fair comparison, all baseline methods are trained and evaluated under identical experimental settings, including the same number of training epochs, learning rate, and train/validation/test splits\. We apply a 40%/20%/40% split for training, validation, and testing across all four datasets\. Each method is run 10 times on Amazon, Yelp, and T\-Finance, and we report the mean and standard deviation of the results\. For T\-Social, due to its large scale and high computational cost, we run each method 5 times and report the mean and standard deviation\. For time\-series datasets, all methods are evaluated over 10 runs under the standard train\-on\-normal and test\-on\-mixed protocol for the unsupervised baseline\. To enable semi\-supervised training, we additionally sample a fixed 1% labeled subset from the mixed data\. No point adjustment is applied in any evaluation\. Additional details are provided in Appendix[D](https://arxiv.org/html/2606.00304#A4)\.

#### Baseline Methods

In the static graph setting, we compare the proposed EGNN with a diverse set of representative graph learning models, covering both classical and state\-of\-the\-art graph neural network architectures, ranging from spectral\-based to attention\-based methods\. Specifically, we include the following baseline models:GCN\(Kipf,[2016](https://arxiv.org/html/2606.00304#bib.bib1)\),ChebyNet\(Defferrardet al\.,[2016](https://arxiv.org/html/2606.00304#bib.bib2)\),GAT\(Veličkovićet al\.,[2017](https://arxiv.org/html/2606.00304#bib.bib30)\),GIN\(Xuet al\.,[2018](https://arxiv.org/html/2606.00304#bib.bib6)\),GraphSAGE\(Hamiltonet al\.,[2017](https://arxiv.org/html/2606.00304#bib.bib31)\),SGC\(Wuet al\.,[2019](https://arxiv.org/html/2606.00304#bib.bib32)\),GT\(Dwivedi and Bresson,[2020](https://arxiv.org/html/2606.00304#bib.bib33)\),BernNet\(Heet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib34)\),PC\-GNN\(Liuet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib4)\), andBWGNN\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)\. For time\-series graphs, we includeGDN\(Deng and Hooi,[2021](https://arxiv.org/html/2606.00304#bib.bib5)\),TranAD\(Tuliet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib14)\), andUSAD\(Audibertet al\.,[2020](https://arxiv.org/html/2606.00304#bib.bib35)\)as representative baselines\. For detailed descriptions of these baseline models, we refer the reader to Appendix[B\.2](https://arxiv.org/html/2606.00304#A2.SS2)\.

#### Metrics

To comprehensively evaluate model performance, we adopt three widely used metrics:Macro\-F1, Area Under the Receiver Operating Characteristic Curve \(AUROC\), and Area Under the Precision\-Recall Curve \(AUPRC\), where AUPRC is computed using average precision\. Additional details are provided in the appendix[B\.3](https://arxiv.org/html/2606.00304#A2.SS3)\.

Table 1:Performance comparison on time\-series datasets\.ModelMSLSWaTWADIAvg\. F1\#ParamsAUCPRCF1\-mAUCPRCF1\-mAUCPRCF1\-mTranAD73\.5773\.5792\.7192\.7168\.1481\.4970\.6276\.4543\.5343\.535\.245\.2448\.5064\.36261,243USAD80\.6095\.0263\.7163\.7178\.2178\.2169\.4369\.4371\.5271\.5249\.875\.6348\.3548\.3561\.1961\.191\.28MGDN65\.7965\.7988\.8088\.8057\.5657\.5680\.5780\.5771\.4775\.4575\.4546\.994\.9648\.1860\.3960\.395,121EGNN86\.6397\.2870\.5093\.7070\.1170\.1186\.0367\.8820\.2761\.4172\.6553,953
### 3\.1Experimental Results

#### Performance in Static Graph

EGNN achieves over 95% accuracy in the most challenging dataset and outperforms all the baselines\.The performance of the proposed method and the baselines is reported in TableLABEL:tab:static\_results\. On the small\-scale dataset\(Amazon, YelpChi, and T\-Finance\), strong baselines such as BWGNN, BernNet, and the classical ChebyNet already achieve over 90% F1\-macro\. Although the improvement on Amazon is relatively marginal due to the simplicity of the dataset, EGNN significantly outperforms all baselines on more challenging and large\-scale datasets, including YelpChi, T\-Finance, and T\-Social\. In particular, EGNN improves F1\-macro by more than 16% on the T\-Social dataset\. This substantial gain highlights the scalability of EGNN on large graphs\. Moreover, EGNN consistently achieves superior AUPRC, even on the largest and most challenging dataset, achieving over 95%, while none of the baseline methods reach comparable performance in AUPRC\. This further demonstrates EGNN’s strong capability in identifying anomalous nodes, especially on large\-scale datasets\.

#### Performance in Time\-Series Graph

EGNN achieves the highest average F1 score among three datasets and with fewer parameters\.Table[1](https://arxiv.org/html/2606.00304#S3.T1)reports the performance of different methods on three benchmark time\-series anomaly detection datasets\. EGNN consistently outperforms deep all method across all three datasets in terms of F1\-macro\. On the challenging SWaT dataset, although GDN achieves the highest AUPRC, EGNN attains an F1\-score of 86\.03, significantly outperforming TranAD \(76\.45\) and USAD \(71\.52\)\. Similarly, on WADI, EGNN improves the F1\-score to 61\.41, surpassing all neural baselines by a large margin\. Moreover, EGNN uses substantially fewer parameters than TranAD and USAD, while GDN has only a small number of parameters due to the lack of temporal learning, which partly explains its inferior performance\. These results highlight the effectiveness and efficiency of EGNN in capturing temporal anomaly patterns under weakly supervised settings\.

### 3\.2Sensitivity Analysis

#### Effectiveness of Left\-Shift Spectral Energy

We provide ablation results in Table[2](https://arxiv.org/html/2606.00304#S3.T2), comparing the full model with variants that use only a single energy branch, fixed equal\-weight fusion, and a semantically irrelevant simple random\-walk feature \(SRW\)\. The results show that each component contributes to the final performance\. \(1\) Neither spectral branch alone is sufficient, as Amazon and YelpChi exhibit mixed spectral patterns that require both left\-shift and right\-shift information\. \(2\) Learnable fusion is necessary because fixed equal\-weight fusion cannot adapt to feature\-wise spectral differences and leads to degraded performance\. \(3\) The gain from left\-shift energy is not simply due to increased feature dimensionality, since replacing it with SRW also reduces performance\. These results confirm that left\-shift spectral energy provides meaningful information\.

Table 2:Ablation results of theeffectiveness of left\-shift spectral energy, differentbackbones, andgate designson Amazon and YelpChi datasets\.SettingAmazonYelpChiF1\-mAUCPRCF1\-mAUCPRCE\(R\)E^\{\(R\)\}only87\.2094\.9781\.3672\.2183\.6156\.19E\(L\)E^\{\(L\)\}only87\.7395\.0883\.5071\.7783\.4755\.69Fixed72\.8686\.2745\.7249\.4450\.2214\.78SRW74\.2388\.0450\.7550\.3150\.6714\.97With GCN75\.4087\.8449\.6858\.1363\.6827\.44With GAT60\.0362\.7619\.8862\.5573\.3335\.25CA89\.3595\.1684\.5575\.1986\.4962\.34Bilinear88\.7893\.6282\.2472\.8084\.7457\.84Original91\.5296\.3289\.4076\.8988\.1066\.26

#### Component Analysis

We further evaluate the proposed method with different GNN backbones and gating mechanisms\. Table[2](https://arxiv.org/html/2606.00304#S3.T2)also compares different GNN backbones\. Replacing GraphSAGE with GCN or GAT consistently reduces performance\. We attribute this to GraphSAGE having a better ability to preserve self\-information during neighbor aggregation\. In contrast, GCN introduces additional spectral\-style propagation that may blur the intended spectral semantics, while GAT applies attention\-based reweighting that appears unnecessary in this setting\. As shown in Table[2](https://arxiv.org/html/2606.00304#S3.T2), replacing the MLP gate with cross\-attention \(CA\) or bilinear fusion consistently degrades performance on both datasets, suggesting that simple feature\-wise gating is sufficient to balance left\- and right\-shift spectral energies\. CA is likely redundant because local spectral energy already encodes neighborhood information, while bilinear fusion increases model complexity without improving performance\.

#### Parameter Analysis

![Refer to caption](https://arxiv.org/html/2606.00304v1/sections/picture/abl.png)Figure 4:Performance of EGNN on the Amazon dataset under different hop numbers and on the MSL dataset under different window sizes\.We conduct a parameter sensitivity analysis to examine the effect of several key hyperparameters on EGNN\. Figure[4](https://arxiv.org/html/2606.00304#S3.F4)shows the performance of EGNN under different temporal window sizes on time\-series datasets and different hop numbers on static graph datasets\. The results indicate that varying the temporal window size has only a limited impact on performance; even relatively small \(w=10w=10\) and large \(w=160w=160\) values do not cause noticeable degradation, suggesting that EGNN is robust to this parameter\. As the hop size increases, local spectral energy becomes less discriminative at the node level\. In the extreme case where the hop size approaches the graph diameter, each node recovers nearly the same global spectral energy\. Although 3\-hop achieves the best F1 score, its improvement over 1\-hop is marginal while introducing a higher computational cost\. We refer readers to Appendix[B\.7](https://arxiv.org/html/2606.00304#A2.SS7)for detailed results on the effect of hop size\. Therefore, 1\-hop provides a better trade\-off between effectiveness and efficiency\.

#### Complexity Analysis

The energy computation in Eq\.LABEL:eq:localenergycostsO​\(\|ℰ\|​F\)O\(\|\\mathcal\{E\}\|F\)and is performed once as preprocessing\. The gating module addsO​\(N​F​G​Lg\)O\(NFGL\_\{g\}\)complexity\. Each GraphSAGE layer costsO​\(\|ℰ\|​H\+N​H2\)O\(\|\\mathcal\{E\}\|H\+NH^\{2\}\)\. Therefore, the per\-epoch training complexity isO​\(\|ℰ\|​H\+N​H2\+N​F​G​Lg\)O\(\|\\mathcal\{E\}\|H\+NH^\{2\}\+NFGL\_\{g\}\)\. Since real\-world graphs typically satisfy\|ℰ\|≫N\|\\mathcal\{E\}\|\\gg N, the sparse message\-passing term dominates in practice, yielding an overall complexity of approximatelyO​\(\|ℰ\|​H\)O\(\|\\mathcal\{E\}\|H\)\. We further report empirical computational costs in Appendix[B\.8](https://arxiv.org/html/2606.00304#A2.SS8)\. Although our method introduces additional spectral\-energy and gating computations, it remains scalable and consistently improves detection performance across datasets\.

## 4Related Work

#### Static Graph Anomaly Detection\.

Graph Anomaly Detection \(GAD\) on static graphs aims to identify abnormal nodes, edges, or subgraphs based on structural and attribute information\. Early methods primarily relied on graph representation learning with shallow models, such as random walk embeddings\(Perozziet al\.,[2014](https://arxiv.org/html/2606.00304#bib.bib43)\)or matrix factorization\(Grover and Leskovec,[2016](https://arxiv.org/html/2606.00304#bib.bib44)\)\. With the development of Graph Neural Networks \(GNNs\), a large body of work has adopted message passing architectures for anomaly detection\. Representative approaches include GCN\-\(Kipf,[2016](https://arxiv.org/html/2606.00304#bib.bib1)\), GAT\-\(Veličkovićet al\.,[2017](https://arxiv.org/html/2606.00304#bib.bib30)\), and GraphSAGE\-\(Hamiltonet al\.,[2017](https://arxiv.org/html/2606.00304#bib.bib31)\)based methods, which learn node representations by aggregating neighborhood information\. Several recent studies have explored spectral and frequency\-based perspectives to characterize anomalies\. For example, BWGNN\(Tanget al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib9)\)and BernNet\(Heet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib34)\)use graph spectral filters to model high\-frequency components, motivated by the observation that anomalies tend to exhibit large feature variations\. Other methods, such as PC\-GNN\(Liuet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib4)\)and subgraph\-based approaches, focus on capturing local structural irregularities\. However, most existing static GAD methods implicitly assume that anomalies are associated with high\-frequency or high\-variance patterns, which limits their ability to detect diverse anomaly types\.

#### Time\-Series Graph Anomaly Detection

Time\-Series Anomaly Detection \(TSAD\) is challenging due to anomalies often arising from complex temporal dependencies and inter\-variable correlations\. Sequence\-centric models, such as Anomaly Transformer\(Xuet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib17)\)and TranAD\(Tuliet al\.,[2022](https://arxiv.org/html/2606.00304#bib.bib14)\), mainly rely on Transformer architectures to model temporal dynamics\. These methods are effective for capturing long\-range temporal patterns, but they usually treat variables as independent channels or only implicitly model their interactions, thereby overlooking explicit relational dependencies among variables\. Recent studies have introduced graph\-based formulations for TSAD, where each variable is represented as a node and edges encode inter\-variable dependencies\. For instance, GDN\(Deng and Hooi,[2021](https://arxiv.org/html/2606.00304#bib.bib5)\)learns a dynamic dependency graph based on variable similarity and achieves competitive performance on standard benchmarks even without complex temporal architectures\. However, most existing graph\-based anomaly detection methods are still designed for static settings\(Dinget al\.,[2019](https://arxiv.org/html/2606.00304#bib.bib40)\)\. When applied to dynamic graphs, where both topology and node attributes evolve, these methods struggle to jointly capture structural evolution and temporal dependencies\. As a result, anomaly detection on dynamic graphs remains a challenging and relatively under\-explored problem\(Maet al\.,[2021](https://arxiv.org/html/2606.00304#bib.bib3)\)\.

## 5Conclusion and Future Direction

In this work, we identified camouflaged anomalies, a new class of patterns that induce a left\-shift phenomenon in spectral energy distributions\. Based on this observation, we proposed EGNN, an energy\-aware framework for GAD that broadens the scope of detectable node attribute anomalies\. We further extended spectral energy modeling to temporal graphs with a focus on scalability and data efficiency\. Our experiments demonstrated that EGNN consistently outperforms state\-of\-the\-art methods with orders of magnitude fewer parameters across diverse settings\. Directed graphs and other granularity anomalies, e\.g\., edge\-, subgraph\-, and graph\-, will be explored in future work\. Detailed limitations are further discussed in Appendix[D](https://arxiv.org/html/2606.00304#A4)\.

## Acknowledgment

This work was supported in part by the National Science Foundation under grants NSF 2443803 and 2230087\. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation\.

## Impact Statement

This paper presents EGNN, a spectral\-energy framework for spatio\-temporal graph anomaly detection that captures both high\-variance and camouflaged low\-variance anomalies\. Its main benefit is improving the detection of fraud, coordinated manipulation, and abnormal behaviors in networked systems, thereby reducing economic losses and improving operational reliability\.

However, deployment risks remain\. False positives may unfairly affect legitimate entities, while false negatives may miss critical threats\. Since EGNN is semi\-supervised, it may also inherit biases from historical labels\. Its assumptions on node\-attribute anomalies over undirected and unweighted graphs may further limit robustness under distribution shift or in directed and weighted settings\. Responsible deployment should include threshold calibration, subgroup\-level audits, privacy\-preserving practices, and human oversight in high\-stakes applications\.

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## Appendix ANotations and Derivations

In this section, we summarize the notations utilized in this paper and present the derivation of \(LABEL:eq:left\_right\)\.

### A\.1Notations

Table 3:Notation table\.SymbolMeaningSymbolMeaning𝒢\\mathcal\{G\}Graph𝒱\\mathcal\{V\}Set of nodesℰ\\mathcal\{E\}Set of edgesNNThe number of nodesAAAdjacency matrixDDDegree matrixLLLaplacian matrixUUEigenvector matrixΛ\\LambdaEigenvaluesxxNode feature vector𝒳\\mathcal\{X\}The set of node feature vectors𝐱\\mathbf\{x\}Node feature matrix𝐱^f\\hat\{\\mathbf\{x\}\}\_\{f\}Fourier transform of𝐱\\mathbf\{x\}in featureffL𝒩iL\_\{\\mathcal\{N\}\_\{i\}\}Subgraph Laplacian matrixEi,f\(L\)E^\{\(L\)\}\_\{i,f\}Left\-shift spectral energy at nodeviv\_\{i\}featureffEi,f\(R\)E^\{\(R\)\}\_\{i,f\}Right\-shift spectral energy at nodeviv\_\{i\}featureffggFeature\-wise weightZZRepresentation after gate mechanism𝐇\\mathbf\{H\}Node embeddingℒ\\mathcal\{L\}Training lossα\\alphaAnomaly ratioFFFeature dimensionGGHidden dimension of the gating networkLgL\_\{g\}Number of layers in the gating networkHHHidden dimension of the GNN layer\|ℰ\|\|\\mathcal\{E\}\|The number of edges
### A\.2Derivation of equation \(LABEL:eq:localenergy\)\.

We haveL𝒩i=I−D𝒩i,f−1/2​A𝒩i,f​D𝒩i,f−1/2L\_\{\\mathcal\{N\}\_\{i\}\}=I\-D\_\{\\mathcal\{N\}\_\{i\},f\}^\{\-1/2\}A\_\{\\mathcal\{N\}\_\{i\},f\}D\_\{\\mathcal\{N\}\_\{i\},f\}^\{\-1/2\}\. SubstitutingL𝒩iL\_\{\\mathcal\{N\}\_\{i\}\}into \(LABEL:eq:local\_energy\) for neighborhood𝒩i\\mathcal\{N\}\_\{i\}and expanding via matrix multiplication, we obtainE𝒩i,f\(R\)=En​u​m/Ed​e​nE\_\{\\mathcal\{N\}\_\{i\},f\}^\{\(R\)\}=E\_\{num\}/E\_\{den\}, where the numerator and denominator are given by

En​u​m=∑j∈𝒩​\(i\)wi​j​\(xi,fdi−xj,fdj\)2,Ed​e​n=\(xi,f\)2di\+∑j∈𝒩​\(i\)\(xj,f\)2djE\_\{num\}=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}w\_\{ij\}\\left\(\\frac\{x\_\{i,f\}\}\{\\sqrt\{d\_\{i\}\}\}\-\\frac\{x\_\{j,f\}\}\{\\sqrt\{d\_\{j\}\}\}\\right\)^\{2\},\\ \\ E\_\{den\}=\\frac\{\(x\_\{i,f\}\)^\{2\}\}\{d\_\{i\}\}\+\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\frac\{\(x\_\{j,f\}\)^\{2\}\}\{d\_\{j\}\}

## Appendix BAdditional Experiment Details

### B\.1Experimental Setup

All the experiments were run in a single workstation, which is equipped with an AMD Ryzen Threadripper PRO 7975WX processor \(32 cores, 64 threads\), 128 GB of DDR5 RAM, and an NVIDIA RTX 6000 Ada Generation GPU with 48 GB of VRAM\.

### B\.2Baseline Description

- •GCN: GCN aggregates neighbor information via graph convolution to update node representations\.
- •ChebyNet: ChebyNet approximates spectral graph convolution with Chebyshev polynomials to capture multi\-hop structure efficiently\.
- •GAT: GAT uses attention to assign different weights to neighbors, focusing message passing on the most relevant nodes\.
- •GIN: GIN uses an injective aggregation function and is as expressive as the 1\-Weisfeiler–Lehman test in distinguishing graph structures\.
- •GraphSAGE: GraphSAGE learns node embeddings by sampling and aggregating information from local neighborhoods\.
- •SGC: SGC simplifies GCN by removing nonlinearities and collapsing propagation into a single linear model on pre\-smoothed features\.
- •GT: GT adapts Transformers to graphs by masking self\-attention with graph structure to improve efficiency and inductive bias\.
- •BernNet: BernNet learns adaptive graph filters using a Bernstein polynomial approximation over the normalized Laplacian spectrum\.
- •PC\-GNN: PC\-GNN addresses class imbalance with label\-balanced sampling and learns a distance function to select informative neighbors and suppress noisy links\.
- •BWGNN: BWGNN targets “right\-shift” anomalies using Beta wavelet kernels to construct flexible band\-pass graph filters for high\-frequency signals\.
- •UniGAD: UniGAD is a unified graph anomaly detection framework, which models anomalies across multiple levels \(node, edge, and graph\) within a single architecture\.
- •GDN: GDN models multivariate time series as a variable graph and scores anomalies by deviations from learned dependency patterns\.
- •TranAD: TranAD is a Transformer\-based method that detects anomalies via reconstruction error, enhanced by an adversarial\-style training objective\.
- •USAD: USAD is an unsupervised autoencoder method with two decoders trained iteratively to produce robust reconstruction\-based anomaly scores\.

### B\.3Metrics

#### AUPRC \(Area Under the Precision\-Recall Curve\)

AUPRC is used in a classification model to show the relationship between precision and recall at different threshold levels, which computes the area beneath the Precision\-Recall curve\.

#### AUROC \(Area Under the Receiver Operating Characteristic Curve\)

AUROC evaluates a model’s ability to distinguish the positive and negative classes by measuring the area under the ROC curve\. The ROC curve plots the true positive rate against the false positive rate for varying decision thresholds\. The ROC approach to 1 shows it has perfect ability to distinguish the different classes\.

#### F1\-macro

F1\-macro computes the average score over all classes by treating each class with the same weight\. This metric is widely used in highly imbalanced classification tasks, such as graph anomaly detection\.

### B\.4Details of Training in static graph

All models are trained for a maximum of 200 epochs using the Adam optimizer with a learning rate of 0\.01 and a weight decay of5×10−45\\times 10^\{\-4\}\. Early stopping is applied with a patience of 50 epochs based on validation performance\. Each experiment is repeated 10 times with different random seeds\. The dataset is split into training, validation, and test sets with ratios of 0\.4, 0\.2, and 0\.4, respectively\. We use the model provided in GADBench, and Table[4](https://arxiv.org/html/2606.00304#A2.T4)shows the hyperparameters we used for all baselines\.

Table 4:Baseline models and hyperparameters\.ModelKey ParametersGCNh\_feats=32, num\_layers=1, drop\_rate=0\.15SGCh\_feats=32, k=2, drop\_rate=0\.15GINh\_feats=32, num\_layers=2, drop\_rate=0\.15GraphSAGEh\_feats=32, num\_layers=2, drop\_rate=0\.15GATh\_feats=32, num\_layers=2, num\_heads=4, drop\_rate=0\.15GTh\_feats=32, num\_layers=2, num\_heads=4, drop\_rate=0\.15BWGNNh\_feats=64, num\_layers=2, drop\_rate=0\.00BernNeth\_feats=32, orders=3, drop\_rate=0\.15PCGNNh\_feats=64, num\_layers=2, del\_ratio=0\.4, add\_ratio=0\.4ChebNeth\_feats=32, k=2, drop\_rate=0\.15UniGADh\_feats=32, num\_layers=2, encoder=bwgnn, epoch\_pretrain=50,mask\_ratio=0\.5
### B\.5Details of Training in time\-series graph

All three anomaly detection models \(GDN, TranAD, and USAD\) are evaluated on the TGAD benchmark datasets, using a consistent experimental setup\. The input data is preprocessed using MinMaxScaler normalization, and sliding windows of size 15 are employed to construct temporal sequences from the multivariate time series\. During training, GDN uses a stride of 5, while TranAD and USAD use a stride of 1; all models use a stride of 1 during testing\. The models are trained for 50–150 epochs \(50 for GDN, and 150 for TranAD and USAD\) with a learning rate of 0\.001 and batch sizes of 128 for GDN and USAD, and 32 for TranAD\. A validation split \(10–30% of the training data\) is used for early stopping and threshold tuning\. The optimal classification threshold is selected by sweeping over score percentiles to maximize the Macro F1 score on the validation set\. All experiments report seven evaluation metrics: Macro F1, Binary F1, Recall, Precision, AUROC, and AUPRC\. Multiple runs with different random seeds are conducted to compute the mean and standard deviation, and reproducibility is ensured by fixing random seeds for Python, NumPy, and PyTorch\.

### B\.6Additional Experimental Results

In Table[7](https://arxiv.org/html/2606.00304#A4.T7), we report the average value and standard deviation of 10 runs on YelpChi, Amazon, T\-Finance, and T\-social\.

### B\.7Number of hops analysis

Table[5](https://arxiv.org/html/2606.00304#A2.T5)shows that increasing the hop size does not consistently improve performance\. While 3\-hop achieves the best overall results on YelpChi and competitive performance on Amazon, larger hops, such as 7\-hop, degrade performance, suggesting that overly broad neighborhoods may dilute local anomaly signals\.

Table 5:Ablation study on different hop sizes\.SettingAmazonYelpChiF1\-mAUCPRCF1\-mAUCPRC1\-hop91\.8096\.8890\.3376\.9188\.1266\.103\-hop only92\.1896\.1789\.1878\.2188\.7467\.415\-hop only92\.0296\.3389\.1677\.3588\.1166\.827\-hop only89\.9796\.3184\.7374\.3687\.8463\.07
### B\.8Runtime Analysis

Table[6](https://arxiv.org/html/2606.00304#A2.T6)reports the computational cost of different models for one epoch across four graph anomaly detection datasets\. Our method requires higher GFLOPs than lightweight GNN baselines such as GCN and GraphSAGE, but remains more efficient than GAT on Amazon, YelpChi, and T\-Finance while providing stronger anomaly detection performance\.

Table 6:Computation cost measured by GFLOPs\.ModelsAmazonYelpChiT\-FinanceT\-SocialGCN1\.782\.155\.7069\.5GraphSAGE6\.789\.7728\.27646\.0GAT67\.5761\.51298\.01680BWGNN4\.626\.8219\.31497\.9UniGAD\+BWGNN5\.038\.4920\.41659\.9Ours18\.6530\.3857\.921990

## Appendix CLeft\-Shift Spectral Energy in Amazon dataset

This figure shows the spectral energy distribution on the Amazon dataset\. Each bin represents the average spectral energy for one feature across all nodes\. Among the 25 features, 12 exhibit lower spectral energy for anomalous nodes\.

![Refer to caption](https://arxiv.org/html/2606.00304v1/sections/picture/ama.png)Figure 5:Spectral energy distribution in the Amazon dataset\. Each bin represents the average spectral energy of all nodes in one feature\. Blue indicates normal nodes and yellow indicates anomalies\.
## Appendix DLimitations

The proposed method, EGNN, is built upon spectral energy computation\. In this work, we focus on node attribute anomalies and treat all edges as unweighted\. For the Amazon and YelpChi datasets, we ignore differences in edge connections and assume uniform edge weights\. However, in more realistic settings, edges may carry weights, exhibit nonlinear relationships, and be directed\. In such cases, the spectral formulation must be reformulated, since the eigenvalues of directed graphs are not guaranteed to be real or positive\. EGNN is primarily designed in a semi\-supervised setting, as it requires a small amount of anomaly labels to learn the underlying energy shift patterns\. In a fully unsupervised scenario, where no anomaly labels are available, the performance of EGNN may degrade, since the model lacks explicit supervision to associate spectral energy variations with anomalous behaviors\.

Table 7:Performance comparison on static graph datasets \(40% training\)\.ModelAmazonYelpChiT\-FinanceT\-SocialF1\-mAUCPRCF1\-mAUCPRCF1\-mAUCPRCF1\-mAUCPRCGCN63\.22±\\pm1\.3482\.20±\\pm1\.2634\.15±\\pm4\.2855\.70±\\pm1\.4458\.91±\\pm2\.1722\.42±\\pm2\.6972\.63±\\pm4\.2885\.59±\\pm2\.5343\.45±\\pm11\.7565\.34±\\pm5\.7378\.80±\\pm13\.0623\.96±\\pm13\.61ChebyNet91\.03±\\pm1\.0595\.76±\\pm1\.7386\.18±\\pm2\.3271\.59±\\pm1\.0084\.30±\\pm0\.9154\.46±\\pm2\.5485\.70±\\pm9\.8893\.29±\\pm5\.7475\.37±\\pm22\.6958\.31±\\pm0\.8173\.56±\\pm2\.2112\.04±\\pm3\.17GAT85\.70±\\pm4\.5593\.29±\\pm2\.1075\.37±\\pm10\.9165\.92±\\pm0\.9178\.33±\\pm1\.0541\.34±\\pm2\.3581\.56±\\pm5\.0691\.69±\\pm1\.7155\.90±\\pm14\.4967\.08±\\pm0\.9685\.47±\\pm1\.8125\.65±\\pm2\.05GIN88\.93±\\pm1\.8992\.32±\\pm1\.8580\.29±\\pm2\.3665\.20±\\pm3\.3876\.59±\\pm6\.4239\.51±\\pm6\.5379\.65±\\pm1\.0084\.41±\\pm2\.8451\.83±\\pm2\.7652\.61±\\pm2\.4466\.37±\\pm12\.055\.92±\\pm2\.07GraphSAGE75\.28±\\pm9\.2688\.90±\\pm5\.6166\.20±\\pm16\.0768\.59±\\pm1\.8082\.17±\\pm2\.4645\.80±\\pm4\.3057\.04±\\pm4\.6857\.98±\\pm5\.238\.71±\\pm4\.6058\.42±\\pm1\.0274\.04±\\pm7\.109\.44±\\pm0\.56SGC63\.06±\\pm7\.3877\.10±\\pm9\.0223\.68±\\pm6\.0951\.35±\\pm1\.3551\.91±\\pm2\.2315\.76±\\pm1\.2457\.92±\\pm8\.5351\.91±\\pm12\.4315\.76±\\pm13\.0442\.49±\\pm4\.7848\.93±\\pm7\.623\.04±\\pm0\.41GT89\.14±\\pm3\.3690\.75±\\pm2\.2176\.20±\\pm6\.7767\.27±\\pm1\.8979\.86±\\pm1\.8444\.89±\\pm3\.5564\.63±\\pm6\.3678\.50±\\pm6\.9719\.87±\\pm8\.5163\.66±\\pm4\.4283\.43±\\pm3\.6620\.82±\\pm7\.92BernNet91\.31±\\pm0\.6093\.83±\\pm2\.0984\.01±\\pm2\.6169\.32±\\pm0\.6182\.14±\\pm0\.6548\.89±\\pm1\.3081\.48±\\pm2\.8590\.66±\\pm1\.4452\.90±\\pm9\.9551\.94±\\pm0\.9664\.68±\\pm1\.774\.88±\\pm1\.66PC\-GNN67\.04±\\pm0\.8581\.80±\\pm1\.1729\.36±\\pm4\.3456\.30±\\pm0\.2559\.55±\\pm0\.4022\.91±\\pm1\.0485\.62±\\pm1\.1992\.17±\\pm1\.6173\.70±\\pm3\.1751\.19±\\pm6\.2072\.88±\\pm9\.9513\.77±\\pm11\.98BWGNN91\.40±\\pm0\.7896\.17±\\pm1\.0986\.42±\\pm2\.1771\.78±\\pm1\.2084\.33±\\pm1\.1554\.67±\\pm2\.6284\.67±\\pm9\.8893\.12±\\pm5\.7473\.84±\\pm22\.6981\.78±\\pm2\.2994\.32±\\pm0\.8160\.81±\\pm5\.72UniGAD90\.46±\\pm0\.7396\.60±\\pm0\.6586\.65±\\pm2\.2171\.23±\\pm0\.6583\.69±\\pm0\.6453\.97±\\pm1\.6389\.34±\\pm0\.4495\.14±\\pm0\.2784\.71±\\pm0\.4878\.67±\\pm1\.2191\.65±\\pm0\.7158\.33±\\pm4\.68Ours91\.52±\\pm0\.4796\.32±\\pm0\.7889\.40±\\pm1\.1076\.89±\\pm0\.6688\.10±\\pm0\.6866\.26±\\pm1\.5289\.60±\\pm0\.7495\.39±\\pm0\.4884\.39±\\pm1\.5195\.40±\\pm0\.1199\.69±\\pm0\.0195\.89±\\pm0\.13