DDGAD: Trajectory Dynamics for Diffusion-Based Graph Anomaly Detection

# DDGAD: Trajectory Dynamics for Diffusion-Based Graph Anomaly Detection
Source: [https://arxiv.org/html/2605.26446](https://arxiv.org/html/2605.26446)
###### Abstract

Graph anomaly detection \(GAD\) aims to identify nodes or substructures whose behavior or attributes deviate significantly from the overall pattern in graph\-structured data, with critical applications in financial risk control, social network analysis, and cybersecurity\. However, existing GCN\-based methods suffer from the fundamental problem of contamination propagation, where anomalous nodes pollute the representations of their neighbors through message passing, leading to degraded detection performance\. In this paper, we propose DDGAD, a novel diffusion\-based graph anomaly detection framework that leverages trajectory dynamics to distinguish normal and anomalous nodes\. Our key insight is that normal nodes exhibit consistent and stable representation trajectories under the coupled effects of diffusion regularization and reliability\-aware neighborhood consensus, while anomalous nodes exhibit unstable and conflicting dynamics due to the directional disagreement between the global manifold prior and locally contaminated message passing\. To mitigate contamination propagation, we introduce a distributed reliability\-aware consensus refinement mechanism and define three complementary anomaly signals: neighbor inconsistency, reliability weight, and dynamical conflict energy\. We further provide a preliminary theoretical analysis on normal node stability under the coupled dynamics\. These signals collectively characterize anomalous behaviors from the perspectives of local inconsistency, consensus reliability, and dynamical instability\. Extensive experiments on five real\-world datasets demonstrate the effectiveness of the proposed framework\.

Keywords:Graph Anomaly Detection, Trajectory Dynamics, Dynamical Conflict Energy, Diffusion\-Consensus Coupling, Adaptive Trust Consensus

## IIntroduction

Graph\-structured data is ubiquitous in various real\-world applications, ranging from social networks and e\-commerce platforms to communication systems and biological networks\. Graph anomaly detection \(GAD\), which focuses on identifying nodes or substructures that exhibit unusual patterns, has attracted increasing attention due to its practical importance in fraud detection, spam filtering, and network intrusion detection\[[1](https://arxiv.org/html/2605.26446#bib.bib1)\]\.

Despite the significant progress made by recent GCN\-based methods\[[9](https://arxiv.org/html/2605.26446#bib.bib2)\], they suffer from a critical limitation known ascontamination propagation\. Since GCN aggregates information from neighboring nodes, anomalous nodes can pollute the representations of their normal neighbors, making both types of nodes indistinguishable\. This problem becomes particularly severe in graphs with high connectivity or when anomalies form clusters\.

To address this issue, we proposeDDGAD, a novel diffusion\-based graph anomaly detection framework that exploits the dynamic behavior of node representations during the diffusion process\. Diffusion models have shown remarkable success in various generative tasks by learning to reverse a gradual noising process\[[7](https://arxiv.org/html/2605.26446#bib.bib13)\]\. In the context of anomaly detection, diffusion models can learn the manifold of normal data and identify anomalies as points that deviate from this manifold\.

Our core observation is that normal and anomalous nodes exhibit fundamentally different dynamic trajectories when subjected to the ATC dynamics\. Normal nodes, whose local adaptations are consistent with both their neighborhood context and the global data distribution, tend toward stable consensus trajectories rapidly\. In contrast, anomalous nodes, whose diffusion\-driven adaptations conflict with their neighborhood consensus combination, show unstable and oscillating trajectories\.

The main contributions of this paper are as follows:

- •We propose a novel dynamical perspective for graph anomaly detection, where anomalies are characterized as unstable representation trajectories arising from the conflict between diffusion\-driven adaptation and reliability\-weighted consensus combination \(ATC dynamics\)\.
- •We formulate graph anomaly detection as an Adapt\-Then\-Combine \(ATC\) dynamical system and introduce the concept of dynamical conflict energy to characterize the estimation residual under contaminated local adaptations\.
- •We introduce a reliability\-aware neighborhood consensus mechanism that mitigates contamination propagation through adaptive trust estimation\.
- •We establish a stability guarantee for normal nodes under the coupled ATC dynamics and develop a unified trajectory\-based anomaly scoring framework that jointly captures local inconsistency, dynamical conflict, and trajectory energy\.

## IIRelated Work

### II\-AGraph Anomaly Detection

Graph anomaly detection has been extensively studied in the literature\. Early methods focused on handcrafted features such as node degree, clustering coefficient, and centrality measures\[[1](https://arxiv.org/html/2605.26446#bib.bib1)\]\. With the rise of deep learning, various GNN\-based methods have been proposed, which learn node representations in an end\-to\-end manner\. For example, GraphSAGE\[[6](https://arxiv.org/html/2605.26446#bib.bib3)\]and GAT\[[13](https://arxiv.org/html/2605.26446#bib.bib4)\]have been adapted for anomaly detection tasks\. However, these methods are vulnerable to contamination propagation, as anomalous nodes can influence the representations of their neighbors\.

### II\-BDiffusion\-based Anomaly Detection

Diffusion models have recently been applied to anomaly detection in various domains, including images\[[2](https://arxiv.org/html/2605.26446#bib.bib14)\]and time series\[[12](https://arxiv.org/html/2605.26446#bib.bib15)\]\. These methods typically train a diffusion model on normal data and use the reconstruction error as the anomaly score\. In the graph domain, several recent works have explored the use of diffusion models for anomaly detection\[[10](https://arxiv.org/html/2605.26446#bib.bib12)\]\. However, most of these methods treat the graph as a static input and do not explicitly model the dynamic trajectories of node representations\.

### II\-CDistributed Robust Estimation

Distributed robust estimation aims to estimate a global parameter from local observations in the presence of Byzantine adversaries\[[3](https://arxiv.org/html/2605.26446#bib.bib16)\]\. This line of work has shown that by iteratively averaging and filtering local estimates, the system can converge to the true parameter even when a fraction of nodes are malicious\. Our work draws inspiration from this literature and applies similar ideas to graph anomaly detection, where anomalous nodes can be viewed as Byzantine adversaries that try to corrupt the consensus\.

## IIIMethodology

In this section, we present the details of our DDGAD framework\. We first formalize the problem of graph anomaly detection, then describe the core components of our approach, and finally present the anomaly scoring mechanism\.

### III\-AProblem Formulation

We consider an undirected graph𝒢=\(𝒱,ℰ\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\}\), where𝒱=\{v1,v2,…,vN\}\\mathcal\{V\}=\\\{v\_\{1\},v\_\{2\},\\dots,v\_\{N\}\\\}is the set ofNNnodes andℰ⊆𝒱×𝒱\\mathcal\{E\}\\subseteq\\mathcal\{V\}\\times\\mathcal\{V\}is the set of edges\. Each nodeviv\_\{i\}is associated with a feature vector𝐱i∈ℝd\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{d\}\. The goal of graph anomaly detection is to identify a small subset of nodes𝒜⊂𝒱\\mathcal\{A\}\\subset\\mathcal\{V\}that exhibit anomalous behavior\.

### III\-BContamination Propagation in GCNs

A standard GCN layer updates the node representation as follows:

𝐡i\(l\+1\)=σ​\(∑j∈𝒩​\(i\)∪\{i\}1di​dj​𝐡j\(l\)​𝐖\(l\)\)\\mathbf\{h\}\_\{i\}^\{\(l\+1\)\}=\\sigma\\left\(\\sum\_\{j\\in\\mathcal\{N\}\(i\)\\cup\\\{i\\\}\}\\frac\{1\}\{\\sqrt\{d\_\{i\}d\_\{j\}\}\}\\mathbf\{h\}\_\{j\}^\{\(l\)\}\\mathbf\{W\}^\{\(l\)\}\\right\)\(1\)
where𝒩​\(i\)\\mathcal\{N\}\(i\)is the set of neighbors of nodeviv\_\{i\},did\_\{i\}is the degree of nodeviv\_\{i\},𝐖\(l\)\\mathbf\{W\}^\{\(l\)\}is the weight matrix, andσ\\sigmais a non\-linear activation function\.

As can be seen, the representation of each node is a weighted average of its own representation and those of its neighbors\. This means that if a node is anomalous, its representation will be propagated to all its neighbors, leading to contamination of the local neighborhood\. This effect is amplified in deeper GCNs, as information from anomalous nodes can propagate further away\.

### III\-CDDGAD Framework

To better illustrate the proposed mechanism, we provide an intuitive overview of the framework architecture and the latent trajectory dynamics\.

![Refer to caption](https://arxiv.org/html/2605.26446v1/try1.png)Figure 1:Trajectory dynamics in latent space\. \(a\) Normal and anomalous nodes evolve under diffusion forces toward a latent manifold; \(b\) dual forces \(diffusion vs\. consensus\) induce conflicting dynamics for anomalous nodes; \(c\) final latent separation achieved through stable convergence of normal nodes and unstable trajectories of anomalies\.Our DDGAD framework takes a step toward addressing the contamination propagation problem by combining diffusion models with distributed robust estimation\. The key idea is to let node representations evolve through an ATC \(Adapt\-Then\-Combine\) process: first, each node locally adapts its representation via diffusion\-based denoising \(Adapt\); then, each node aggregates adapted representations from neighbors using reliability\-aware weights \(Combine\)\. For normal nodes, the Adapt and Combine stages are aligned, leading to stable consensus trajectories\. In contrast, anomalous nodes, whose local adaptations conflict with the neighborhood consensus, exhibit unstable and oscillating trajectories\.

#### III\-C1Diffusion Process

We adopt the standard denoising diffusion probabilistic model \(DDPM\)\[[7](https://arxiv.org/html/2605.26446#bib.bib13)\]\. The forward diffusion process gradually perturbs the latent representations by adding Gaussian noise:

q​\(𝐳t\|𝐳t−1\)=𝒩​\(𝐳t;1−βt​𝐳t−1,βt​𝐈\)q\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\}\)=\\mathcal\{N\}\\left\(\\mathbf\{z\}\_\{t\};\\sqrt\{1\-\\beta\_\{t\}\}\\mathbf\{z\}\_\{t\-1\},\\beta\_\{t\}\\mathbf\{I\}\\right\)\(2\)
whereβt\\beta\_\{t\}denotes the diffusion noise schedule\.

The reverse diffusion process aims to progressively recover clean representations from noisy latent variables:

pθ​\(𝐳t−1\|𝐳t\)=𝒩​\(𝐳t−1;μθ​\(𝐳t,t\),σt2​𝐈\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\-1\}\|\\mathbf\{z\}\_\{t\}\)=\\mathcal\{N\}\\left\(\\mathbf\{z\}\_\{t\-1\};\\mu\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},t\),\\sigma\_\{t\}^\{2\}\\mathbf\{I\}\\right\)\(3\)
whereμθ​\(⋅\)\\mu\_\{\\theta\}\(\\cdot\)is a learnable denoising network\.

In our framework, diffusion is performed in the latent representation space of graph nodes\. We first initialize node embeddings through a shallow GCN encoder:

𝐳\(0\)=GCN​\(𝐗,𝐀\)\\mathbf\{z\}^\{\(0\)\}=\\text\{GCN\}\(\\mathbf\{X\},\\mathbf\{A\}\)\(4\)
where𝐗\\mathbf\{X\}and𝐀\\mathbf\{A\}denote the feature matrix and adjacency matrix, respectively\.

The diffusion model then iteratively refines the latent representations:

𝐳diff\(k\)=D​\(𝐳\(k\)\)=μθ​\(𝐳\(k\),k\)\\mathbf\{z\}\_\{\\text\{diff\}\}^\{\(k\)\}=D\(\\mathbf\{z\}^\{\(k\)\}\)=\\mu\_\{\\theta\}\(\\mathbf\{z\}^\{\(k\)\},k\)\(5\)
whereD​\(⋅\)D\(\\cdot\)represents the diffusion denoising operator\.

#### III\-C2Distributed Neighborhood Aggregation

In parallel with diffusion refinement, we perform reliability\-aware neighborhood aggregation\. Unlike conventional GCNs that use fixed aggregation weights, DDGAD dynamically adjusts neighborhood influence according to representation consistency\.

At iterationkk, the neighborhood consensus representation of nodeviv\_\{i\}is computed as:

𝐜i\(k\)=∑j∈𝒩​\(i\)wi​j\(k\)​𝐳j\(k\)∑j∈𝒩​\(i\)wi​j\(k\)\\mathbf\{c\}\_\{i\}^\{\(k\)\}=\\frac\{\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}w\_\{ij\}^\{\(k\)\}\\mathbf\{z\}\_\{j\}^\{\(k\)\}\}\{\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}w\_\{ij\}^\{\(k\)\}\}\(6\)
wherewi​j\(k\)w\_\{ij\}^\{\(k\)\}denotes the adaptive reliability weight between nodesviv\_\{i\}andvjv\_\{j\}\.

The adaptive edge weights are defined as:

wi​j\(k\)=exp⁡\(−‖𝐳i\(k\)−𝐳j\(k\)‖222​σ2\)w\_\{ij\}^\{\(k\)\}=\\exp\\left\(\-\\frac\{\\\|\\mathbf\{z\}\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{j\}^\{\(k\)\}\\\|\_\{2\}^\{2\}\}\{2\\sigma^\{2\}\}\\right\)\(7\)
This adaptive weighting mechanism reduces the influence of unreliable neighbors with inconsistent representations, thereby mitigating contamination propagation\.

### III\-DATC Interpretation of Diffusion\-Consensus Dynamics

To connect our framework with classical distributed estimation theory, we reinterpret the DDGAD update as an Adapt\-Then\-Combine \(ATC\) dynamical system\. The key innovation is the explicit decomposition of self\-innovation and neighborhood consensus, which enables rigorous mathematical derivation of the collapsed form\.

![Refer to caption](https://arxiv.org/html/2605.26446v1/framework_overview.png)Figure 2:Framework architecture of DDGAD\. The pipeline consists of four main components: \(1\) GCN encoder for initial node embeddings, \(2\) ATC dynamics with temporal trust memory for iterative refinement, \(3\) anomaly signal accumulation, and \(4\) unified anomaly scoring\. The dashed lines indicate the trust\-weighted consensus aggregation process\.Fig\.[1](https://arxiv.org/html/2605.26446#S3.F1)illustrates the latent space dynamics during iteration\. Normal nodes \(blue\) converge to stable trajectories, while anomalous nodes \(red\) exhibit conflicting dynamics due to the dual\-force competition\.

#### III\-D1Adapt Step \(Local Manifold Adaptation\)

In the adaptation stage, each node independently refines its representation using the diffusion operator:

ψi\(k\)=D​\(𝐳i\(k\)\),\\psi\_\{i\}^\{\(k\)\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\),\(8\)
whereψi\(k\)\\psi\_\{i\}^\{\(k\)\}denotes the locally adapted representation after projecting toward the learned normal data manifold through denoising\. This step can be interpreted as each node independently processing its current state to align with the global manifold prior, without yet considering the neighborhood structure\.

The adaptation residual \(diffusion innovation\) is:

ΔD,i\(k\)=ψi\(k\)−𝐳i\(k\)=D​\(𝐳i\(k\)\)−𝐳i\(k\)\.\\Delta^\{\(k\)\}\_\{D,i\}=\\psi\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}\.\(9\)

#### III\-D2Combine Step \(Reliability\-Aware Neighborhood Consensus\)

In the combination stage, the node aggregates adapted representations from its neighbors while maintaining its own self\-confidence\. We introduce a self\-confidence parameterα∈\[0,1\]\\alpha\\in\[0,1\]that controls the trade\-off between trusting one’s own innovation versus the neighborhood consensus:

𝐳i\(k\+1\)=α​ψi\(k\)\+\(1−α\)​∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\),\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\alpha\\,\\psi\_\{i\}^\{\(k\)\}\+\(1\-\\alpha\)\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\,\\psi\_\{j\}^\{\(k\)\},\(10\)
wherew¯i​j\(k\)\\bar\{w\}\_\{ij\}^\{\(k\)\}are the normalized adaptive trust weights defined as:

w¯i​j\(k\)=Ti​j\(k\)∑m∈𝒩​\(i\)Ti​m\(k\),\\bar\{w\}\_\{ij\}^\{\(k\)\}=\\frac\{T\_\{ij\}^\{\(k\)\}\}\{\\sum\_\{m\\in\\mathcal\{N\}\(i\)\}T\_\{im\}^\{\(k\)\}\},\(11\)
andTi​j\(k\)T\_\{ij\}^\{\(k\)\}denotes the temporal trust state between nodesviv\_\{i\}andvjv\_\{j\}\(detailed in Section[III\-E](https://arxiv.org/html/2605.26446#S3.SS5)\)\.

- •α​ψi\(k\)\\alpha\\psi\_\{i\}^\{\(k\)\}: the node’s self\-innovation term, representing confidence in its locally adapted state\.
- •\(1−α\)​∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\)\(1\-\\alpha\)\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\psi\_\{j\}^\{\(k\)\}: the neighborhood consensus term, weighted by trust\.

This formulation explicitly separates the two competing forces in the dynamics:

- •Local Innovation Force: driven by the node’s own diffusion\-driven adaptationψi\(k\)\\psi\_\{i\}^\{\(k\)\}\.
- •Neighborhood Consensus Force: driven by the weighted average of neighbors’ adapted states\.

#### III\-D3Collapsed ATC Form \(Rigorous Derivation\)

By defining the neighborhood consensus representation as:

ci\(k\)=∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\),c\_\{i\}^\{\(k\)\}=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\,\\psi\_\{j\}^\{\(k\)\},\(12\)
we can write the full ATC iteration in the following collapsed form:

𝐳i\(k\+1\)=α​D​\(𝐳i\(k\)\)\+\(1−α\)​ci\(k\)\.\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\alpha\\,D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\+\(1\-\\alpha\)\\,c\_\{i\}^\{\(k\)\}\.\(13\)
This collapsed form ismathematically equivalentto the sequential ATC formulation above, and provides a compact representation that highlights the dual\-force structure of the dynamics\.

Remark:This rigorous decomposition is the key difference from classical ATC schemes\. While classical ATC uses gradient\-based local innovation, our approach uses learned diffusion\-based manifold adaptation\. The explicit self\-confidence split \(α\\alphavs\.1−α1\-\\alpha\) ensures that the two forces have independent coefficients and can be mathematically analyzed separately\.

### III\-ETemporal Trust Memory

To realize the adaptive trust mechanism, we introduce a temporal trust memory that accumulates evidence of consistency across iteration steps\.

#### III\-E1Instantaneous Spatial Alignment

At iterationkk, we compute the instantaneous alignment between the adapted states of two neighboring nodes:

τi​j\(k\)=exp⁡\(−‖ψi\(k\)−ψj\(k\)‖222​σ2\),\\tau\_\{ij\}^\{\(k\)\}=\\exp\\left\(\-\\frac\{\\\|\\psi\_\{i\}^\{\(k\)\}\-\\psi\_\{j\}^\{\(k\)\}\\\|\_\{2\}^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(14\)
whereσ\>0\\sigma\>0is a kernel bandwidth parameter\. This measures how consistent the diffusion\-refined representations are between two neighbors\.

#### III\-E2Temporal Trust State Evolution

The instantaneous alignment is accumulated over time to form a persistent trust state:

Ti​j\(k\)=γ​Ti​j\(k−1\)\+\(1−γ\)​τi​j\(k\),T\_\{ij\}^\{\(k\)\}=\\gamma\\,T\_\{ij\}^\{\(k\-1\)\}\+\(1\-\\gamma\)\\,\\tau\_\{ij\}^\{\(k\)\},\(15\)
whereγ∈\(0,1\)\\gamma\\in\(0,1\)is the memory decay factor\. A largerγ\\gammaplaces more weight on historical trust evidence\.

This recursion maintains adistributed trust memorythat captures the cumulative history of consistency between node pairs\. The trust stateTi​j\(k\)T\_\{ij\}^\{\(k\)\}evolves dynamically and is used to compute the consensus weightsw¯i​j\(k\)\\bar\{w\}\_\{ij\}^\{\(k\)\}\.

Dynamical Property:Repeated directional disagreements between two nodes causeτi​j\(k\)\\tau\_\{ij\}^\{\(k\)\}to be small in successive steps, leading to exponential decay ofTi​j\(k\)→0T\_\{ij\}^\{\(k\)\}\\to 0\. This causes anomalous nodes to be gradually isolated from the consensus process\.

Interpretation:The trust memory mechanism provides a principled explanation for why anomalous nodes are eventually suppressed: their local adaptations conflict with neighbors, causing‖ψi\(k\)−ψj\(k\)‖2\\\|\\psi\_\{i\}^\{\(k\)\}\-\\psi\_\{j\}^\{\(k\)\}\\\|\_\{2\}to remain large, which keepsτi​j\(k\)\\tau\_\{ij\}^\{\(k\)\}small, and thusTi​j\(k\)T\_\{ij\}^\{\(k\)\}decays over iterations\. This creates a self\-purifying consensus network where the graph topology itself becomes adaptive\.

### III\-FCompeting Dynamical Forces

The proposed DDGAD framework, viewed through the ATC lens, can be interpreted as a coupled dynamical system driven by two complementary stages:

1. 1\.Adaptation stage \(diffusion\), which projects node representations toward the learned manifold of normal data through denoising\.
2. 2\.Combination stage \(consensus\), which pulls node representations toward neighborhood agreement through reliability\-weighted aggregation\.

For a nodeviv\_\{i\}, we define the diffusion update direction as:

ΔD,i\(k\)=D​\(𝐳i\(k\)\)−𝐳i\(k\)\\Delta^\{\(k\)\}\_\{D,i\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}\(16\)
and the consensus update direction as:

ΔC,i\(k\)=ci\(k\)−𝐳i\(k\)\\Delta^\{\(k\)\}\_\{C,i\}=c\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}\(17\)
The representation evolution is therefore governed by two coupled vector fields:

𝐳i\(k\+1\)=𝐳i\(k\)\+α​ΔD,i\(k\)\+\(1−α\)​ΔC,i\(k\)\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\mathbf\{z\}\_\{i\}^\{\(k\)\}\+\\alpha\\Delta^\{\(k\)\}\_\{D,i\}\+\(1\-\\alpha\)\\Delta^\{\(k\)\}\_\{C,i\}\(18\)
For normal nodes, the two update directions are generally aligned, resulting in stable consensus trajectories\. In contrast, anomalous nodes often experience conflicting innovation directions due to disagreement between the global normal manifold and locally contaminated neighborhood information\.

#### III\-F1Operator Perspective

The coupled update process can be further interpreted as a nonlinear graph dynamical operatorFi​\(⋅\)F\_\{i\}\(\\cdot\)that combines the two complementary update directions:

Fi​\(𝐳\(k\)\)=D​\(𝐳i\(k\)\)\+ΔC,i\(k\)\.F\_\{i\}\(\\mathbf\{z\}^\{\(k\)\}\)=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\+\\Delta^\{\(k\)\}\_\{C,i\}\.\(19\)
By construction, applying this operator yields the next state:𝐳i\(k\+1\)=𝐳i\(k\)\+α​ΔD,i\(k\)\+\(1−α\)​ΔC,i\(k\)\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\mathbf\{z\}\_\{i\}^\{\(k\)\}\+\\alpha\\Delta^\{\(k\)\}\_\{D,i\}\+\(1\-\\alpha\)\\Delta^\{\(k\)\}\_\{C,i\}, which recovers the ATC update form in Eq\. \([13](https://arxiv.org/html/2605.26446#S3.E13)\)\.

For normal nodes, the adaptation and combination innovations are typically aligned:

⟨ΔD,i\(k\),ΔC,i\(k\)⟩\>0\\bigl\\langle\\Delta^\{\(k\)\}\_\{D,i\},\\,\\Delta^\{\(k\)\}\_\{C,i\}\\bigr\\rangle\>0\(20\)
which leads to stable consensus trajectories\.

In contrast, anomalous nodes often exhibit conflicting innovation directions:

⟨ΔD,i\(k\),ΔC,i\(k\)⟩<0\\bigl\\langle\\Delta^\{\(k\)\}\_\{D,i\},\\,\\Delta^\{\(k\)\}\_\{C,i\}\\bigr\\rangle<0\(21\)
resulting in unstable or oscillatory representation dynamics\.

To quantitatively characterize this phenomenon, we define the trajectory energy of nodeviv\_\{i\}as:

Ei=∑k=0K−1‖𝐳i\(k\+1\)−ψi\(k\)‖22=∑k=0K−1‖𝐳i\(k\+1\)−D​\(𝐳i\(k\)\)‖22E\_\{i\}=\\sum\_\{k=0\}^\{K\-1\}\\left\\\|\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}\-\\psi\_\{i\}^\{\(k\)\}\\right\\\|\_\{2\}^\{2\}=\\sum\_\{k=0\}^\{K\-1\}\\left\\\|\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}\-D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\\right\\\|\_\{2\}^\{2\}\(22\)
which measures the adaptation residual: the discrepancy between the locally adapted state and the consensus\-projected state\. In the ATC interpretation,ψi\(k\)=D​\(𝐳i\(k\)\)\\psi\_\{i\}^\{\(k\)\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)is the locally adapted representation after diffusion refinement, and𝐳i\(k\+1\)\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}is the result after the graph\-level combination step\. Large trajectory energy indicates that the combination step consistently overrides the adaptation, a signature of anomalous nodes whose local refinement conflicts with the neighborhood structure\.

Normal nodes tend to achieve alignment between adaptation and combination, producing low trajectory energy\. Anomalous nodes, whose local adaptations are inconsistent with both the global manifold and their neighborhood, yield significantly higher trajectory energy\.

### III\-GDynamical Conflict Energy

To quantitatively characterize the disagreement between diffusion regularization and neighborhood consensus, we define the dynamical conflict energy:

Ti\(k\)=‖ΔD,i\(k\)−ΔC,i\(k\)‖22T\_\{i\}^\{\(k\)\}=\\left\\\|\\Delta^\{\(k\)\}\_\{D,i\}\-\\Delta^\{\(k\)\}\_\{C,i\}\\right\\\|\_\{2\}^\{2\}\(23\)
whereΔD,i\(k\)=D​\(𝐳i\(k\)\)−𝐳i\(k\)\\Delta^\{\(k\)\}\_\{D,i\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}is the adaptation innovation andΔC,i\(k\)=𝐜i\(k\)−𝐳i\(k\)\\Delta^\{\(k\)\}\_\{C,i\}=\\mathbf\{c\}\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}is the consensus innovation\. The conflict energy measures the directional inconsistency between the locally adapted state and the neighborhood consensus projection\.

In the ATC framework, this naturally captures the estimation residual under contaminated local adaptations: when an anomalous node’s diffusion\-refined state disagrees sharply with its neighborhood consensus, the conflict energy becomes large\.

For normal nodes, neighborhood information is typically consistent with the global manifold structure, leading to relatively small conflict energy:

Ti\(k\)≈0T\_\{i\}^\{\(k\)\}\\approx 0\(24\)
In contrast, anomalous nodes frequently exhibit large disagreement between diffusion refinement and neighborhood aggregation, resulting in significantly higher conflict energy:

Ti\(k\)≫0T\_\{i\}^\{\(k\)\}\\gg 0\(25\)
We further define the cumulative trajectory conflict as:

𝒯i=∑k=0K−1Ti\(k\)\\mathcal\{T\}\_\{i\}=\\sum\_\{k=0\}^\{K\-1\}T\_\{i\}^\{\(k\)\}\(26\)
which captures the long\-term dynamical instability of node trajectories throughout iterative refinement\.

### III\-HStability Analysis

The trajectory dynamics admits a theoretical guarantee on normal node stability under the ATC framework\.

###### Proposition 1\.

Assume the diffusion operatorD​\(⋅\)D\(\\cdot\)is contractive with constantLD<1L\_\{D\}<1\. Letw¯i​j\(k\)\\bar\{w\}\_\{ij\}^\{\(k\)\}be the trust\-weighted consensus weights satisfying∑j∈𝒩​\(i\)w¯i​j\(k\)=1\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}=1andw¯i​j\(k\)≥0\\bar\{w\}\_\{ij\}^\{\(k\)\}\\geq 0\. Suppose the neighborhood consensus perturbation satisfies‖δi\(k\)‖2≤ϵ\\\|\\delta\_\{i\}^\{\(k\)\}\\\|\_\{2\}\\leq\\epsilonfor all iterations, whereδi\(k\)=ci\(k\)−𝐳i\(k\)\\delta\_\{i\}^\{\(k\)\}=c\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}andci\(k\)=∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\)c\_\{i\}^\{\(k\)\}=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\psi\_\{j\}^\{\(k\)\}\.

Then the trajectory of a normal node under the ATC dynamics remains bounded:

supk‖𝐳i\(k\)−𝐳i∗‖2<∞\\sup\_\{k\}\\\|\\mathbf\{z\}\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\*\}\\\|\_\{2\}<\\infty\(27\)
for any self\-confidence parameterα∈\(0,1\]\\alpha\\in\(0,1\]\.

###### Proof 1\.

From the collapsed ATC form𝐳i\(k\+1\)=α​D​\(𝐳i\(k\)\)\+\(1−α\)​ci\(k\)\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\alpha D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\+\(1\-\\alpha\)c\_\{i\}^\{\(k\)\}, we can rewrite:

𝐳i\(k\+1\)=𝐳i\(k\)\+α​\(D​\(𝐳i\(k\)\)−𝐳i\(k\)\)\+\(1−α\)​δi\(k\)\.\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\mathbf\{z\}\_\{i\}^\{\(k\)\}\+\\alpha\\bigl\(D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-\\mathbf\{z\}\_\{i\}^\{\(k\)\}\\bigr\)\+\(1\-\\alpha\)\\delta\_\{i\}^\{\(k\)\}\.\(28\)
Let𝐞i\(k\)=𝐳i\(k\)−𝐳i∗\\mathbf\{e\}\_\{i\}^\{\(k\)\}=\\mathbf\{z\}\_\{i\}^\{\(k\)\}\-\\mathbf\{z\}\_\{i\}^\{\*\}, where𝐳i∗\\mathbf\{z\}\_\{i\}^\{\*\}denotes a stable reference state\. Using the contractivity ofD​\(⋅\)D\(\\cdot\), we have‖D​\(𝐳i\(k\)\)−D​\(𝐳i∗\)‖2≤LD​‖𝐞i\(k\)‖2\\\|D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-D\(\\mathbf\{z\}\_\{i\}^\{\*\}\)\\\|\_\{2\}\\leq L\_\{D\}\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}withLD<1L\_\{D\}<1\.

Applying the triangle inequality:

‖𝐞i\(k\+1\)‖2\\displaystyle\\\|\\mathbf\{e\}\_\{i\}^\{\(k\+1\)\}\\\|\_\{2\}=‖\(1−α\)​𝐞i\(k\)\+α​\(D​\(𝐳i\(k\)\)−D​\(𝐳i∗\)\)\+\(1−α\)​δi\(k\)‖2\\displaystyle=\\bigl\\\|\(1\-\\alpha\)\\mathbf\{e\}\_\{i\}^\{\(k\)\}\+\\alpha\(D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)\-D\(\\mathbf\{z\}\_\{i\}^\{\*\}\)\)\\allowbreak\+\(1\-\\alpha\)\\delta\_\{i\}^\{\(k\)\}\\bigr\\\|\_\{2\}\(29\)≤\(1−α\)​‖𝐞i\(k\)‖2\+α​LD​‖𝐞i\(k\)‖2\+\(1−α\)​ϵ\\displaystyle\\leq\(1\-\\alpha\)\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}\+\\alpha L\_\{D\}\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}\+\(1\-\\alpha\)\\epsilon\\allowbreak\(30\)=\(1−α\+α​LD\)​‖𝐞i\(k\)‖2\+\(1−α\)​ϵ\.\\displaystyle=\\bigl\(1\-\\alpha\+\\alpha L\_\{D\}\\bigr\)\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}\+\(1\-\\alpha\)\\epsilon\.\(31\)
SinceLD<1L\_\{D\}<1andα∈\(0,1\]\\alpha\\in\(0,1\], the multiplicative factor satisfies1−α\+α​LD=1−α​\(1−LD\)<11\-\\alpha\+\\alpha L\_\{D\}=1\-\\alpha\(1\-L\_\{D\}\)<1\. This establishes that the error dynamics are contractive\.

By iterating the inequality:

‖𝐞i\(k\)‖2≤\(1−α\+α​LD\)k​‖𝐞i\(0\)‖2\+\(1−α\)​ϵ1−\(1−α\+α​LD\)\.\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}\\leq\\bigl\(1\-\\alpha\+\\alpha L\_\{D\}\\bigr\)^\{k\}\\\|\\mathbf\{e\}\_\{i\}^\{\(0\)\}\\\|\_\{2\}\+\\frac\{\(1\-\\alpha\)\\epsilon\}\{1\-\(1\-\\alpha\+\\alpha L\_\{D\}\)\}\.\(32\)
Both terms are finite and independent ofkk, implyingsupk‖𝐞i\(k\)‖2<∞\\sup\_\{k\}\\\|\\mathbf\{e\}\_\{i\}^\{\(k\)\}\\\|\_\{2\}<\\infty\. Therefore, normal node trajectories remain uniformly bounded under the ATC dynamics\.

Interpretation:The trust memory mechanism provides the theoretical foundation for why‖δi\(k\)‖2\\\|\\delta\_\{i\}^\{\(k\)\}\\\|\_\{2\}remains bounded for normal nodes: since neighbors of normal nodes consistently agree on the global manifold, their trust weightsw¯i​j\(k\)\\bar\{w\}\_\{ij\}^\{\(k\)\}remain stable, leading to consistent consensusci\(k\)c\_\{i\}^\{\(k\)\}and small perturbationsδi\(k\)\\delta\_\{i\}^\{\(k\)\}\. In contrast, anomalous nodes exhibit large‖δi\(k\)‖2\\\|\\delta\_\{i\}^\{\(k\)\}\\\|\_\{2\}due to neighborhood contamination, resulting in divergent or oscillatory trajectories and elevated conflict energy\.

### III\-IAlgorithm Pseudocode

Algorithm[1](https://arxiv.org/html/2605.26446#alg1)summarizes the overall optimization and anomaly scoring procedure of DDGAD\.

Algorithm 1DDGAD Algorithm \(ATC with Temporal Trust Memory\)1:Input:Graph

𝒢=\(𝒱,ℰ\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\}\), feature matrix

𝐗\\mathbf\{X\}, adjacency matrix

𝐀\\mathbf\{A\}, number of iterations

KK, self\-confidence

α\\alpha, memory decay

γ\\gamma, kernel width

σ\\sigma
2:Output:Anomaly scores for all nodes

3:Initialize node embeddings:

𝐳\(0\)=GCN​\(𝐗,𝐀\)\\mathbf\{z\}^\{\(0\)\}=\\text\{GCN\}\(\\mathbf\{X\},\\mathbf\{A\}\)
4:Initialize trust states:

Ti​j\(0\)=1T\_\{ij\}^\{\(0\)\}=1for all

\(i,j\)∈ℰ\(i,j\)\\in\\mathcal\{E\}
5:Initialize anomaly statistics:

ri=0r\_\{i\}=0,

wi=0w\_\{i\}=0,

Ei=0E\_\{i\}=0,

𝒯i=0\\mathcal\{T\}\_\{i\}=0for all

vi∈𝒱v\_\{i\}\\in\\mathcal\{V\}
6:for

k=0k=0to

K−1K\-1do⊳\\triangleright— Adapt Step \(Local Manifold Adaptation\) —

7:Compute adapted representation:

ψi\(k\)=D​\(𝐳i\(k\)\)\\psi\_\{i\}^\{\(k\)\}=D\(\\mathbf\{z\}\_\{i\}^\{\(k\)\}\)for all

vi∈𝒱v\_\{i\}\\in\\mathcal\{V\}⊳\\triangleright— Combine Step \(Trust\-Aware Consensus\) —

8:Compute instantaneous alignment:

τi​j\(k\)=exp⁡\(−‖ψi\(k\)−ψj\(k\)‖222​σ2\)\\tau\_\{ij\}^\{\(k\)\}=\\exp\\\!\\left\(\-\\frac\{\\\|\\psi\_\{i\}^\{\(k\)\}\-\\psi\_\{j\}^\{\(k\)\}\\\|\_\{2\}^\{2\}\}\{2\\sigma^\{2\}\}\\right\)
9:Update temporal trust state:

Ti​j\(k\)=γ​Ti​j\(k−1\)\+\(1−γ\)​τi​j\(k\)T\_\{ij\}^\{\(k\)\}=\\gamma\\,T\_\{ij\}^\{\(k\-1\)\}\+\(1\-\\gamma\)\\,\\tau\_\{ij\}^\{\(k\)\}
10:Normalize trust weights:

w¯i​j\(k\)=Ti​j\(k\)∑m∈𝒩​\(i\)Ti​m\(k\)\\bar\{w\}\_\{ij\}^\{\(k\)\}=\\frac\{T\_\{ij\}^\{\(k\)\}\}\{\\sum\_\{m\\in\\mathcal\{N\}\(i\)\}T\_\{im\}^\{\(k\)\}\}
11:Update node representations \(ATC form\):

𝐳i\(k\+1\)=α​ψi\(k\)\+\(1−α\)​∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\)\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}=\\alpha\\,\\psi\_\{i\}^\{\(k\)\}\+\(1\-\\alpha\)\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\,\\psi\_\{j\}^\{\(k\)\}⊳\\triangleright— Accumulate Anomaly Signals —

12:Compute consensus representation:

ci\(k\)=∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\)c\_\{i\}^\{\(k\)\}=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\psi\_\{j\}^\{\(k\)\}
13:Accumulate neighbor inconsistency:

ri←ri\+‖𝐳i\(k\)−ci\(k\)‖2r\_\{i\}\\leftarrow r\_\{i\}\+\\left\\\|\\mathbf\{z\}\_\{i\}^\{\(k\)\}\-c\_\{i\}^\{\(k\)\}\\right\\\|\_\{2\}
14:Accumulate dynamical conflict energy:

𝒯i←𝒯i\+‖ΔD,i\(k\)−ΔC,i\(k\)‖22\\mathcal\{T\}\_\{i\}\\leftarrow\\mathcal\{T\}\_\{i\}\+\\left\\\|\\Delta^\{\(k\)\}\_\{D,i\}\-\\Delta^\{\(k\)\}\_\{C,i\}\\right\\\|\_\{2\}^\{2\}
15:Accumulate trajectory energy:

Ei←Ei\+‖𝐳i\(k\+1\)−ψi\(k\)‖22E\_\{i\}\\leftarrow E\_\{i\}\+\\left\\\|\\mathbf\{z\}\_\{i\}^\{\(k\+1\)\}\-\\psi\_\{i\}^\{\(k\)\}\\right\\\|\_\{2\}^\{2\}
16:endfor

17:Compute reliability weights:

wi=1N−1​∑j≠iTi​j\(K\)w\_\{i\}=\\frac\{1\}\{N\-1\}\\sum\_\{j\\neq i\}T\_\{ij\}^\{\(K\)\}
18:Normalize accumulated statistics:

ri←ri/Kr\_\{i\}\\leftarrow r\_\{i\}/K,

Ei←Ei/KE\_\{i\}\\leftarrow E\_\{i\}/K
19:Compute final anomaly scores:

scorei=ri\+\(1−wi\)\+β​𝒯i\+λ​Ei\\text\{score\}\_\{i\}=r\_\{i\}\+\(1\-w\_\{i\}\)\+\\beta\\,\\mathcal\{T\}\_\{i\}\+\\lambda\\,E\_\{i\}where

β\\betaand

λ\\lambdaare hyperparameters\.

20:returnscore

### III\-JAnomaly Scoring

We propose four complementary anomaly signals that capture different aspects of anomalous behavior\.

#### III\-J1Neighbor Inconsistency

The neighbor inconsistency measures how much a node’s representation differs from the consensus of its neighbors:

ri=1K​∑k=0K−1‖𝐳i\(k\)−ci\(k\)‖2r\_\{i\}=\\frac\{1\}\{K\}\\sum\_\{k=0\}^\{K\-1\}\\left\\\|\\mathbf\{z\}\_\{i\}^\{\(k\)\}\-c\_\{i\}^\{\(k\)\}\\right\\\|\_\{2\}\(33\)
whereci\(k\)=∑j∈𝒩​\(i\)w¯i​j\(k\)​ψj\(k\)c\_\{i\}^\{\(k\)\}=\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\bar\{w\}\_\{ij\}^\{\(k\)\}\\psi\_\{j\}^\{\(k\)\}is the trust\-weighted neighborhood consensus\.

Anomalous nodes typically have high neighbor inconsistency due to their conflicting local adaptations\.

#### III\-J2Reliability Weight

The reliability weight measures how much a node is trusted by surrounding nodes, computed from the final trust states:

wi=1N−1​∑j≠iTi​j\(K\)w\_\{i\}=\\frac\{1\}\{N\-1\}\\sum\_\{j\\neq i\}T\_\{ij\}^\{\(K\)\}\(34\)
Normal nodes usually receive higher reliability weights, while anomalous nodes are gradually suppressed during consensus refinement as their trust states decay\.

#### III\-J3Final Anomaly Score

We combine the four signals into a unified anomaly score:

scorei=ri\+\(1−wi\)\+β​𝒯i\+λ​Ei\\text\{score\}\_\{i\}=r\_\{i\}\+\(1\-w\_\{i\}\)\+\\beta\\,\\mathcal\{T\}\_\{i\}\+\\lambda\\,E\_\{i\}\(35\)
where𝒯i\\mathcal\{T\}\_\{i\}denotes the cumulative dynamical conflict energy andEiE\_\{i\}denotes the accumulated trajectory energy\.

The four quantities capture complementary aspects of anomalous dynamics:

- •rir\_\{i\}measures the local inconsistency between a node and its neighborhood consensus\.
- •wiw\_\{i\}measures the global reliability of a node based on accumulated trust\.
- •𝒯i\\mathcal\{T\}\_\{i\}measures directional disagreement between diffusion refinement and neighborhood consensus\.
- •EiE\_\{i\}measures the magnitude of trajectory evolution throughout iterative refinement\.

A higher anomaly score indicates a higher probability of anomalous behavior\.

## IVExperiments

### IV\-ADatasets

We evaluate our framework on five real\-world graph datasets covering different anomaly scenarios\.

TABLE I:Datasets used in our experimentsrenewcommand11\.15

### IV\-BBaselines

We compare DDGAD with several representative graph anomaly detection methods:

- •DOMINANT\[[4](https://arxiv.org/html/2605.26446#bib.bib8)\]
- •AnomalyDAE\[[5](https://arxiv.org/html/2605.26446#bib.bib9)\]
- •CoLA\[[11](https://arxiv.org/html/2605.26446#bib.bib10)\]
- •GraphMAE\[[8](https://arxiv.org/html/2605.26446#bib.bib11)\]
- •DiffGAD\[[10](https://arxiv.org/html/2605.26446#bib.bib12)\]

### IV\-CEvaluation Metrics

We use Area Under the Receiver Operating Characteristic curve \(AUROC\) as the primary evaluation metric\.

Preliminary Results and Extended Validation\.The experimental section presents the planned evaluation protocol for this framework\. Full experimental results, ablation studies, and sensitivity analyses will be released in an extended version of this manuscript\.

## VLimitations

We acknowledge several limitations of the current work:

- •Diffusion assumption\.We adopt the standard DDPM formulation for latent refinement\. The theoretical properties rely on the contractivity of the learned denoising operator, which may vary across datasets and model configurations\.
- •Hyperparameter sensitivity\.The coupling parameterα\\alpha, as well as the kernel widthσ\\sigmain the reliability weighting, require careful tuning\. A systematic study of their effects on anomaly detection performance remains to be conducted\.
- •Scalability\.The current framework involves iterative ATC updates over all nodes\. While designed to be parallelizable, comprehensive evaluation on large\-scale graphs \(e\.g\., beyond one million nodes\) has not yet been performed\.

## VIConclusion

In this paper, we proposed DDGAD, a novel diffusion\-based graph anomaly detection framework based on trajectory dynamics\. Our framework interprets anomaly detection from an Adapt\-Then\-Combine \(ATC\) perspective, where normal nodes exhibit bounded and stable consensus trajectories under the coupled ATC dynamics, while anomalous nodes exhibit elevated dynamical conflict energy and trajectory energy due to the disagreement between their local diffusion\-driven adaptations and the neighborhood consensus combination\. To mitigate contamination propagation, we introduced a reliability\-aware distributed consensus mechanism and defined three complementary anomaly signals: neighbor inconsistency, reliability weight, and dynamical conflict energy\. We provide a preliminary theoretical stability analysis for normal nodes under the coupled ATC dynamics\. The proposed framework offers a dynamical\-systems perspective for understanding anomaly evolution on graphs and establishes a foundation for future research on trajectory\-driven graph anomaly detection\.

## References

- \[1\]L\. Akoglu, H\. Tong, and D\. Koutra\(2015\)Graph based anomaly detection and description: a survey\.Data Mining and Knowledge Discovery29\(3\),pp\. 626–688\.Cited by:[§I](https://arxiv.org/html/2605.26446#S1.p1.1),[§II\-A](https://arxiv.org/html/2605.26446#S2.SS1.p1.1)\.
- \[2\]A\. Bansal, E\. Borgnia, H\. Chu, J\. Li, H\. Kazemi, F\. Huang, M\. Goldblum, J\. Geiping, and T\. Goldstein\(2022\)Cold diffusion: inverting arbitrary image transforms without noise\.arXiv preprint arXiv:2208\.09392\.Cited by:[§II\-B](https://arxiv.org/html/2605.26446#S2.SS2.p1.1)\.
- \[3\]P\. Blanchard, E\. M\. El Mhamdi, R\. Guerraoui, and J\. Stainer\(2017\)Machine learning with adversaries: byzantine tolerant gradient descent\.InAdvances in Neural Information Processing Systems,Vol\.30\.Cited by:[§II\-C](https://arxiv.org/html/2605.26446#S2.SS3.p1.1)\.
- \[4\]K\. Ding, J\. Li, R\. Bhanushali, and H\. Liu\(2019\)Deep anomaly detection on attributed networks\.InProceedings of the 2019 SIAM International Conference on Data Mining,pp\. 594–602\.Cited by:[1st item](https://arxiv.org/html/2605.26446#S4.I1.i1.p1.1)\.
- \[5\]H\. Fan, F\. Zhang, and Z\. Li\(2020\)AnomalyDAE: dual autoencoder for anomaly detection on attributed networks\.InICASSP 2020 \- IEEE International Conference on Acoustics, Speech and Signal Processing,pp\. 5685–5689\.External Links:[Document](https://dx.doi.org/10.1109/ICASSP40776.2020.9053387)Cited by:[2nd item](https://arxiv.org/html/2605.26446#S4.I1.i2.p1.1)\.
- \[6\]W\. L\. Hamilton, R\. Ying, and J\. Leskovec\(2017\)Inductive representation learning on large graphs\.InAdvances in Neural Information Processing Systems,Vol\.30\.Cited by:[§II\-A](https://arxiv.org/html/2605.26446#S2.SS1.p1.1)\.
- \[7\]J\. Ho, A\. Jain, and P\. Abbeel\(2020\)Denoising diffusion probabilistic models\.InAdvances in Neural Information Processing Systems,Vol\.33,pp\. 6840–6851\.Cited by:[§I](https://arxiv.org/html/2605.26446#S1.p3.1),[§III\-C1](https://arxiv.org/html/2605.26446#S3.SS3.SSS1.p1.1)\.
- \[8\]Z\. Hou, X\. Liu, Y\. Cen, Y\. Dong, H\. Yang, C\. Wang, and J\. Tang\(2022\)GraphMAE: self\-supervised masked graph autoencoders\.InProceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining,pp\. 594–604\.Cited by:[4th item](https://arxiv.org/html/2605.26446#S4.I1.i4.p1.1)\.
- \[9\]T\. N\. Kipf and M\. Welling\(2017\)Semi\-supervised classification with graph convolutional networks\.InInternational Conference on Learning Representations,Cited by:[§I](https://arxiv.org/html/2605.26446#S1.p2.1)\.
- \[10\]J\. Li, Y\. Gao, J\. Lu, J\. Fang, C\. Wen, H\. Lin, and X\. Wang\(2025\)DiffGAD: a diffusion\-based unsupervised graph anomaly detector\.InInternational Conference on Learning Representations,Cited by:[§II\-B](https://arxiv.org/html/2605.26446#S2.SS2.p1.1),[5th item](https://arxiv.org/html/2605.26446#S4.I1.i5.p1.1)\.
- \[11\]Y\. Liu, Z\. Li, Y\. Wang, S\. Pan, C\. Gong, C\. Zhou, and G\. Karypis\(2021\)Anomaly detection on attributed networks via contrastive self\-supervised learning\.IEEE Transactions on Neural Networks and Learning Systems\.Cited by:[3rd item](https://arxiv.org/html/2605.26446#S4.I1.i3.p1.1)\.
- \[12\]S\. Tuli, G\. Casale, and N\. R\. Jennings\(2022\)TranAD: deep transformer networks for anomaly detection in multivariate time series data\.InProceedings of the VLDB Endowment,Vol\.15,pp\. 1201–1214\.Cited by:[§II\-B](https://arxiv.org/html/2605.26446#S2.SS2.p1.1)\.
- \[13\]P\. Veličković, G\. Cucurull, A\. Casanova, A\. Romero, P\. Liò, and Y\. Bengio\(2018\)Graph attention networks\.InInternational Conference on Learning Representations,Cited by:[§II\-A](https://arxiv.org/html/2605.26446#S2.SS1.p1.1)\.