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RagGAD: Rationale-Aware Conditional Gaussian Mixture Normalizing Flow for Unsupervised Graph Anomaly Detection
Junxin Lu, Jing Zhao, Shiliang Sun
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
The paper introduces RagGAD, an unsupervised graph anomaly detection framework that addresses the limitations of existing methods relying on the homophily assumption. RagGAD utilizes an adaptive rationale disentangler to separate stable rationales from spurious correlations in node interrelationships, further decomposing stable rationales into robust and fragile components. It employs a rationale-aware conditional Gaussian mixture normalizing flow to model the intricate distributions of normal and abnormal nodes, identifying anomalies as low-density regions in a structure-aware distribution space.
Entities (8)
Relation Signals (7)
RagGAD → solves → Graph Anomaly Detection
confidence 100% · we propose RagGAD, an unsupervised graph anomaly detection framework
Adaptive Rationale Disentangler → disentangles → Spurious Correlations
confidence 95% · disentangle stable rationales from spurious correlations within node interrelationships
RagGAD → identifies → Anomalies
confidence 95% · RagGAD identifies anomalies as low-density regions within a structure-aware distribution space.
RagGAD → uses → Adaptive Rationale Disentangler
confidence 95% · RagGAD introduces an adaptive rationale disentangler to disentangle stable rationales from spurious correlations
Adaptive Rationale Disentangler → decomposes → Robust Rationales
confidence 90% · decomposes stable rationales into robust and fragile components
Adaptive Rationale Disentangler → decomposes → Fragile Rationales
confidence 90% · decomposes stable rationales into robust and fragile components
RagGAD → mitigates → Homophily Assumption
confidence 90% · By mitigating spurious homophilic correlations and embracing the heterogeneity of normal patterns
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Abstract
Abstract:Graph anomaly detection aims to identify nodes that deviate from normal behavioral patterns within graphs. However, existing methods largely rely on the homophily assumption, which makes it difficult to distinguish spurious affinities and to capture the diverse behaviors of normal nodes,limiting their robustness in complex real-world scenarios. To address this problem, we propose RagGAD, an unsupervised graph anomaly detection framework based on rationale-aware conditional Gaussian mixture normalizing flow. RagGAD introduces an adaptive rationale disentangler to disentangle stable rationales from spurious correlations within node interrelationships, and further decomposes stable rationales into robust and fragile components. The learned rationales capture underlying interaction patterns that characterize normal behaviors under varying conditions, while anomalies emerge as deviations associated with unstable or spurious correlations. To model the intricate distributions of normal and abnormal nodes, RagGAD integrates rationale-non-rationale Gaussian mixture modeling with a robust-fragile rationale mixture learning strategy. By mitigating spurious homophilic correlations and embracing the heterogeneity of normal patterns, RagGAD identifies anomalies as low-density regions within a structure-aware distribution space. Extensive experiments on multiple benchmark datasets demonstrate that RagGAD outperforms state-of-the-art methods.
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- Source: https://arxiv.org/abs/2608.16018v1
- Canonical: https://arxiv.org/abs/2608.16018v1
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RagGAD: Rationale-Aware Conditional Gaussian Mixture Normalizing Flow for Unsupervised Graph Anomaly Detection Junxin Lu Jing Zhao Shiliang Sun Thanks: Junxin Lu and Jing Zhao are with the School of Computer Science and Technology, East China Normal University, Shanghai 200062, China. (E-mail: junxinlu.ecnu@gmail.com, jzhao2011@gmail.com) Thanks: Shiliang Sun is with the School of Computer Science and Technology, East China Normal University, Shanghai 200062, China, and also with the State Key Laboratory of Submarine Geoscience, School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China. (E-mail: shiliangsun@gmail.com). Thanks: Corresponding author: Shiliang Sun. Abstract Graph anomaly detection aims to identify nodes that deviate from normal behavioral patterns within graphs. However, existing methods largely rely on the homophily assumption, which makes it difficult to distinguish spurious affinities and to capture the diverse behaviors of normal nodes, limiting their robustness in complex real-world scenarios. To address this problem, we propose RagGAD, an unsupervised graph anomaly detection framework based on rationale-aware conditional Gaussian mixture normalizing flow. RagGAD introduces an adaptive rationale disentangler to disentangle stable rationales from spurious correlations within node interrelationships, and further decomposes stable rationales into robust and fragile components. The learned rationales capture underlying interaction patterns that characterize normal behaviors under varying conditions, while anomalies emerge as deviations associated with unstable or spurious correlations. To model the intricate distributions of normal and abnormal nodes, RagGAD integrates rationale-non-rationale Gaussian mixture modeling with a robust-fragile rationale mixture learning strategy. By mitigating spurious homophilic correlations and embracing the heterogeneity of normal patterns, RagGAD identifies anomalies as low-density regions within a structure-aware distribution space. Extensive experiments on multiple benchmark datasets demonstrate that RagGAD outperforms state-of-the-art methods. Index Terms: Graph anomaly detection, node homophily, stable rationales, rationale-aware conditional Gaussian mixture normalizing flow I Introduction Graph anomaly detection (GAD) aims to identify nodes in a graph that exhibit behaviors or characteristics significantly deviating from the majority [27, 41, 26]. GAD is crucial in various real-world applications, such as detecting irregularities in financial [25, 30] and social [52] networks. Existing GAD methods are predominantly based on supervised or semi-supervised learning paradigms to model node-level abnormalities, relying on labeled data to guide model training [10, 48]. However, in real-world scenarios, data is often vast and complex, with anomalies that cannot be pre-identified or labeled. Fig. 1: In UGAD, (a) existing methods fall into the homophily trap when abnormal nodes mimic normal connection patterns (e.g., abnormal node v2v_2 imitating the affinity of normal node v1v_1 with its neighbors). Additionally, they incorrectly identify v4v_4 as abnormal due to overlooking the diversity of normal patterns. (b) RagGAD disentangles stable rationales, enabling the model to capture true mutual influence while eliminate spurious connections/affinities. Furthermore, it decomposes rationales into robust and fragile components, allowing for fine-grained modeling of both stable and context-sensitive normal patterns. In this context, unsupervised graph anomaly detection (UGAD) [35, 54, 40] emerges as a more practical yet challenging paradigm, as it eliminates the need for labeled supervision, thereby expanding the practical applicability of GAD. Existing UGAD works are comparatively few, which can be roughly categorized into two groups: (i) data reconstruction-based methods, which identify anomalies by quantifying discrepancies between reconstructed node attributes/structures and the original inputs [17, 43, 34, 13]; (i) self-supervised learning-based methods, which utilize pre-text tasks such as proxy classification, surrogate contrastive learning, or pre-trained models to provide additional supervisions [29, 32, 65]. Despite achieving promising detection performance, when dealing with complex and diverse graphs, above-mentioned works inherently face the following limitations: (i) Over-reliance on the homophily assumption, which posits that normal nodes have stronger affinity with each other than anomalies, may lead to the homophily trap [23, 40, 17, 8]. When normal nodes exhibit weak affinity or abnormal nodes deliberately conceal their suspicious activity patterns, the homophily assumption is violated, resulting in misidentification. As illustrated in Figure 1(a), the abnormal node v2v_2 disguises its abnormal behavior by mimicking the normal node v1v_1, sharing same neighbors and inter-nodes affinities with v1v_1. Consequently, v2v_2 is incorrectly identified as a normal node. This is attributed to their neglect of deeper interaction patterns among nodes, focusing solely on statistical co-occurrence or superficial similarity. However, there exist intricate yet stable interdependencies among nodes, referred to as robust rationales. These rationales capture the intricate yet stable interdependencies among nodes, explaining the robust interaction patterns underlying specific nodes. By modeling stable rationales, the model can uncover the genuine mutual influences between nodes, eliminating spurious connections or affinities even when abnormal nodes obscure their behavior, as shown in Figure 1(b). (i) In real-world graphs, nodes often exhibit diverse normal patterns, such as varying preferences and personalities within a community. As shown in Figure 1(a), nodes v3v_3 and v4v_4 are normal and connected to abnormal node v2v_2. If the model fails to distinguish the varying directions and strengths of interactions with v2v_2, it may misclassify v4v_4 as abnormal. Therefore, the model must be sufficiently sensitive to these variations, ensuring that diverse and fine-grained normal behaviors are not misidentified as anomalies. To address these challenges, we propose RagGAD, a novel framework based on Rationale-aware conditional gaussian mixture normalizing flow for UGAD, as shown in Figure 2. RagGAD employs an adaptive rationale disentangler (ARD) to disentangles stable rationale from spurious correlations in node interrelationships. The rationales encapsulate the underlying influence mechanisms governing normal node behaviors, while anomalies are revealed via unstable spurious correlations. To address the diversity of normal patterns, RagGAD employs a learnable rationale soft mask to decompose stable rationales into robust and fragile components. This enables fine-grained modeling of both stable and context-sensitive normal patterns. Based on the disentangled stable rationales and unstable correlations, we propose a node-level rationale-aware conditional Gaussian mixture normalizing flow (RGMN) model to capture complex normal-abnormal distributions, with anomalies identified as low-density regions within the mixture distribution. A key component of RGMN is rationale-non-rationale Gaussian mixture modeling (RRGM), which leverages conditional Gaussian mixture modeling to effectively separate and model the distributions of rationale and non-rationale correlations. Additionally, to capture nuanced variations within rationale representations, we introduce a robust-fragile rationale mixture learning (RFRM) strategy. This strategy models diverse combinations of robust and fragile components, enabling RagGAD to learn fine-grained divergent traits across normal patterns and provide a comprehensive understanding of the underlying mixture distribution. The main contributions of this paper are summarized as follows: • We disentangle stable rationales from spurious correlations within complex node interrelationships, further refining rationales into robust and fragile components using an adaptive rationale disentangler. This enables RagGAD to capture the underlying influence mechanisms governing normal node behaviors, while anomalies are revealed through unstable non-rationale correlations. • We propose a node-level rationale-aware conditional Gaussian mixture normalizing flow model, which learns the diversity of node normal behaviors and models the intricate distributions of both normal and abnormal patterns, enabling efficient anomaly detection in low-density regions. • Experimental results on multiple datasets demonstrate that RagGAD significantly outperforms state-of-the-art baselines for unsupervised graph anomaly detection. Fig. 2: Overview of RagGAD. RagGAD introduces an adaptive rationale disentangler (ARD) to disentangle rationales cG_c and non-rationale correlations oG_o from complex node interrelationships ℐI. The robust rationales r|cG_r|c and fragile rationales f|cG_f|c are further decomposed from cG_c using a learnable soft mask ℳ M. Based on r|cG_r|c, f|cG_f|c, and oG_o, RagGAD extracts the robust rationale representation Hr|cH_r|c, fragile rationale representation Hf|cH_f|c, and non-rationale representation HoH_o, respectively, which are then transformed into latent conditional embeddings zr|cz_r|c, zf|cz_f|c, and zoz_o through NCNF. Subsequently, RRGM instantiates rationales and non-rationale correlations as distinct classes for Gaussian mixture modeling to fit the distribution of node attribute x~ x conditioned on zr|cz_r|c, zf|cz_f|c and zoz_o. Meanwhile, RFRM captures the latent diversification of normal distributions among nodes through fine-grained robust-fragile rationale Gaussian mixture learning. I Relative Works I-A Unsupervised Graph Anomaly Detection Compared to supervised [47, 48] and semi-supervised [3, 19, 42, 15] paradigms, unsupervised graph anomaly detection (UGAD) aims to identify abnormal nodes among predominantly normal nodes without relying on labeled data [10, 34]. A practical and intuitive strategy is data reconstruction-based, where nodes are reconstructed based on their similarity or affinity with neighbors, and those with high reconstruction errors are identified as anomalies [17, 43, 34, 10]. Another line of work leverages self-supervised learning, incorporating contrastive learning [32, 6, 65, 7], proxy classification [56], and auxiliary objectives [20, 36, 37, 63, 13]. Additionally, general anomaly detection methods, such as ARC [31], UNPrompt [35] and FreeGAD [64], aim to detect anomalies across domains without requiring fine-tuning and retraining, but suffer from low accuracy [53, 31, 35]. FreeGAD [64] leverages an affinity-gated encoder and anchor-guided statistical deviations to compute anomaly scores, however, its reliance on heuristic anchors and shallow statistical measures limits its ability to capture complex dependencies. ADA-GAD [17] and HUGE [36] alleviate the homophily trap through learning-free augmentation strategy and label-free heterogeneity measurement to reduce false detection rates. CoCo [54] leverages Transformers to jointly model multi-hop local and global contextual features, and detects anomalies by measuring discrepancies in their correlations. GCTAM [61] extends TAM [40], which identifies anomalies by maximizing normal-node similarity while truncating anomalous ones, by incorporating contextual information and global similarities to avoid rigid thresholds thereby improving the modeling of node-specific characteristics and high-order affinities. However, they are still limited to the statistical node associations provided by the dataset, without further exploring more robust underlying rationale relationships among variables. The aforementioned works typically rely on statistical co-occurrence/similarity or homophily-aware anomaly metrics to model intrinsic normal patterns for UGAD. However, they struggle to eliminate spurious connections and affinities, making them susceptible to the homophily trap when anomalies conceal their suspicious behavior. In contrast, RagGAD disentangles more robust dependencies, i.e., rationales, from unstable non-rationale correlations, thereby revealing the underlying mechanisms of interactions and generative patterns among normal nodes, while exposing anomalies through non-rationale correlations. I-B Normalizing Flows in Anomaly Detection Normalizing flows (NFs) leverage invertible transformations to map complex data distributions to simpler ones (e.g., standard normal distribution), enabling precise probability density estimation. NFs maximize the log-likelihood of normal samples, assuming that normal samples map to high-density regions, while anomalies fall into low-density regions [38, 5]. NFs have been widely explored for anomaly detection in non-graph data [44, 9, 16, 45, 59]. DifferNet [44] uses NFs to assign meaningful likelihoods to images and develops a scoring function to detect defects. BGAD [60] enhances model distinguishability through a boundary-guided semi-pull contrastive learning mechanism based on NFs, while HGAD [59] introduces a hierarchical Gaussian mixture normalizing flow for unified anomaly detection. GANF [28] combine normalizing flow with a graph auto-encoder to create a generative model of graph structures. For graph data, FANFOLD [5] proposes a graph normalizing flows-driven asymmetric network for unsupervised graph-level anomaly detection. However, how to utilize normalizing flows for the more challenging task of node-level UGAD, while disentangling inter-node robust rationale relationships, remains unexplored in previous works. These works inspire us to leverage NFs for learning data distributions and identifying anomalies in low-density regions. However, existing methods based on normalizing flows rely solely on normal data during training, which is impractical and leads to ambiguous decision boundaries between normal and abnormal data, thereby reducing distinguishability. In contrast, RagGAD facilitates coexistence of normal and abnormal data during training, leveraging disentangled robust rationales and non-rationale correlations for discriminative rationale Gaussian mixture modeling in UGAD. Furthermore, existing methods overlook the diversity of normal patterns, misclassifying diverse fine-grained normal behaviors as anomalies. I Preliminaries I-A Problem Statement An attributed graph is represented as G=,ℰ,G=\V,E,X\, where =v1,v2,⋯,vNV=\v_1,v_2,·s,v_N\ and ℰ=…,eij,…i,j=1NE=\…,e_ij,…\_i,j=1^N are the sets of nodes and edges, respectively. =xii=1N∈ℝN×dxX=\x_i\_i=1^N ^N× d_x is the node-level attribute matrix, where xix_i indicates the attribute vector of node viv_i with dxd_x dimensions. The inter-node connectivity of G is represented by an adjacent matrix A=aiji,j=1N∈ℝN×NA=\a_ij\_i,j=1^N ^N× N, where aij=1a_ij=1 indicates an edge between nodes viv_i and vjv_j, and aij=0a_ij=0 indicates no edge. Given an attributed graph G=,ℰ,G=\V,E,X\, UGAD aims to identify abnormal nodes aV_a (the minority) from normal nodes nV_n (the majority), where a∪n=V_a _n=V, a∩n=∅V_a _n= and |a|≪|n| _a _n , without access to any class labels during training. The objective of UGAD model is to learn an anomaly scoring function :→ℝS:V , such that (vn)<(va)S(v_n)<S(v_a) for all ∀vn∈n∀ v_n _n and ∀va∈a∀ v_a _a. I-B Normalizing Flow The normalizing flow [11, 22, 12] is a probabilistic density estimation model that maps an unknown data distribution p()p(X) to a tractable latent distribution p(Z)p(Z) by a sequence of invertible transformations. Specifically, a normalizing flow model is defined as a bijective transformation ℱ:x∈ℝdx↔z∈ℝdzF:x ^d_x z ^d_z, establishing a mapping from the data space x∈x to the latent space z∈Zz∈ Z. According to the change of variables formula in calculus [51], the log-likelihood of any x∈x can be computed as: logp(x)=logpZ(ℱθ(x))+log|detJ|,logp_X(x)=logp_Z (F_θ(x) )+log|detJ|, (1) where J=∇xℱθ(x)=δℱθ(x)δxJ= _xF_θ(x)= _θ(x)δ x is the Jacobian matrix of the transformation. θ represents the learnable model parameters of ℱF. In most flow-based models, pZ(z)p_Z(z) is typically assumed to follow a standard multivariate normal distribution (0,)N(0,I) for simplicity. The normalizing flow ℱθF_θ can be optimized by maximizing the log-likelihood over the training distribution p()p(X). Consequently, the loss function is defined as follows: ℒnf=x∼p()[−logp(x)].L_nf=E_x p(X)[-logp_X(x)]. (2) Conditional normalizing flow [4, 67, 1] extends normalizing flow by introducing an external condition C, allowing the model to estimate the conditional distribution p(x|)=pZ(ℱθ(x,))|det∇xℱθ(x,)|p_X(x|C)=p_Z (F_θ(x;C) ) _xF_θ(x;C) , thereby enabling conditional density estimation. The log-density of x conditioned on C is defined as follows: logp(x|)=logpZ(ℱθ(x,))+log|det∇xℱθ(x,)|.logp_X(x|C)=logp_Z (F_θ(x;C) )+log _xF_θ(x;C) . (3) IV Methodology We propose RagGAD, a node-level unsupervised graph anomaly detection framework based on rationale-aware conditional Gaussian mixture normalizing flow. IV-A Adaptive Rationale Disentanglement In UGAD, only the node attribute matrix X and the adjacent matrix A are accessible during training, while the stable rationales cG_c and unstable non-rationale correlations oG_o are unknown. To tackle this, we propose an adaptive rationale disentanglement module, which effectively disentangles the cG_c and oG_o from node interdependencies ℐI. RagGAD first employs an attribute projector (⋅)P(·) to reduce the attribute dimension of nodes [62, 31]. Specifically, given the attribute matrix ∈ℝN×dxX ^N× d_x, the attribute projection is defined as follows: ~∈ℝN×dx~=()=W, X ^N× d_ x=P(X)=XW_P, (4) where ~ X is the projected attribute matrix. W∈ℝN×dx~W_P ^N× d_ x is a dataset-specify linear projection weight matrix, and dx~d_ x is a pre-defined dimension of the projected attribute for nodes. The attribute projector (⋅)P(·) can employ commonly used dimensionality reduction methods, such as singular value decomposition [46] or principal component analysis [2]. a) Adaptive Rationale Disentangler. We propose an adaptive rationale disentangler (ARD) to capture the stable rationales cG_c among nodes, while simultaneously disentangling robust rationales r|cG_r|c and fragile rationale relationships f|cG_f|c. Specifically, ARD integrates a node interrelation attention module to learn the structure interrelationships between nodes, forming the node interrelationships ℐI. This involves calculating the cross attention coefficient εij _ij, which reflects the interrelationship of node vjv_j to viv_i, defined as: εij=[aij=1]⋅exp(QiKjT/d~)∑k[aik=1]⋅exp(QiKkT/d~), _ij= I [a_ij=1 ]·exp (Q_iK_j^T/ d_ X ) _kI [a_ik=1 ]·exp (Q_iK_k^T/ d_ X ), (5) where Q=x~iWQ∈ℝdx~Q= x_iW_Q ^d_ x is the query embedding of viv_i, and K=x~jWK∈ℝdx~K= x_jW_K ^d_ x is the key embedding of vjv_j. WQ∈ℝdx~×dx~W_Q ^d_ x× d_ x and WK∈ℝdx~×dx~W_K ^d_ x× d_ x are the query and key weights, respectively. According to the adjacent matrix A, node vjv_j is constrained to be a neighbor of node viv_i, i.e., vj∈(vi)=vk|aik=1v_j (v_i)=\v_k|a_ik=1\. RagGAD treats the disentanglement of cG_c and oG_o as an edge selection problem. Some edges reflect stable underlying dependencies between nodes, i.e., rationales, while others represent unstable relationships determined by proximity or statistical correlations. Therefore, we use a binary mask as disentangler, denoted as ℳ=miji,j=1N∈0,1N×NM=\m_ij\_i,j=1^N∈\0,1\^N× N, as shown in Figure 2. This allows us to disentangle c∈ℝN×NG_c ^N× N and o∈ℝN×NG_o ^N× N from ℐ=εiji,j=1NI=\ _ij\_i,j=1^N as follows: c=ℐ⊙ℳ,o=ℐ⊙(1−ℳ),s.t.ℳ=m~ij>τσ(ℳ~), casesG_c=I ,\\ G_o=I (1-M ), cases\;s.t.\;M=I_ m_ij>τσ( M), (6) where ℳ~=m~iji,j=1N∈ℝN×N M=\ m_ij\_i,j=1^N ^N× N is the trainable mask of ℳM. σ(⋅)σ(·) is the sigmoid function, τ is the hyperparameter for selecting rationale edges, and ⊙ represents element-wise multiplication. In UGAD, relying solely on the disentanglement of rationales and non-rationale correlations remains insufficient for fully capturing the complex and diverse patterns underlying normal nodes. This limitation may cause certain nodes, which exhibit diverse behaviors but are inherently normal, to be mistakenly identified as anomalies. To address this, we further decompose cG_c into stable rationales r|cG_r|c and fragile rationales f|cG_f|c. r|cG_r|c captures persistent, robust relationships, while f|cG_f|c reflects context-sensitive relationships that are more susceptible to data perturbations or neighborhood fluctuations. To achieve this decomposition, RagGAD introduce a learnable soft mask ℳ^∈ℝN×N M ^N× N, which partitions cG_c into r|cG_r|c and f|cG_f|c as follows: r|c=c⊙σ(ℳ^),f|c=c⊙(1−σ(ℳ^)),G_r|c=G_c σ( M), _f|c=G_c (1-σ( M)), (7) where ℳ^=m^iji,j=1N M=\ m_ij\_i,j=1^N, with m^ij∈[0,1] m_ij∈[0,1] representing the learnable weight for the stability of the rationales from node vjv_j to node viv_i. This decomposition enables a fine-grained re-weighting partitioning of cG_c, where each rationale edge is assigned a stability score via σ(ℳ^)σ( M). By combining diverse robust and fragile rationales, RagGAD effectively delineates the boundaries between stable and fragile rationale relationships, offering a nuanced understanding of the underlying anomalies within the graph while comprehensively capturing the behavioral diversity of normal nodes. Upon extracted r|cG_r|c and f|cG_f|c, we introduce two rationale augmented graph convolution layers, GCNr|c(⋅)GCN_r|c(·) and GCNf|c(⋅)GCN_f|c(·), followed by a reconstructor ℛ(⋅)R(·), to generate the node attribute reconstruction. The reconstruction process is defined as follows: ^=ℛ([Hr|c∥Hf|c];θℛ),Hr|c=GCNr|c(r|c,~)=Wr|c2(ReLU(r|c~Wr|c1))+br|c2,Hf|c=GCNf|c(f|c,~)=Wf|c2(ReLU(f|c~Wf|c1))+bf|c2, split -14.22636pt& X=R ( [H_r|c\|H_f|c ]; _R ),\\ -14.22636ptH_r|c -2.84544pt=&GCN_r|c(G_r|c, X) -1.42271pt=W^2_r|c -1.42271pt ( -1.42271ptReLU -1.42271pt (G_r|c XW^1_r|c ) -1.42271pt ) -2.84544pt+ -1.42271ptb^2_r|c,\\ -14.22636ptH_f|c -2.84544pt=&GCN_f|c(G_f|c, X) -1.42271pt=W^2_f|c -1.42271pt ( -1.42271ptReLU -1.42271pt (G_f|c XW^1_f|c ) -1.42271pt ) -2.84544pt+ -1.42271ptb^2_f|c, split (8) where W∗|c1∈ℝdx~×dx~W^1_*|c ^d_ x× d_ x, W∗|c2∈ℝdx~×dhW^2_*|c ^d_ x× d_h, and b∗|c2∈ℝdhb^2_*|c ^d_h are the learnable parameters of GCNr|c(⋅)GCN_r|c(·) and GCNf|c(⋅)GCN_f|c(·), respectively. Inspired by the success of GCNs in node representation learning through the aggregation of neighborhood effects [33, 57], GCNr|c(⋅)GCN_r|c(·) and GCNf|c(⋅)GCN_f|c(·) aggregate the attributes of parent nodes to encode dependency relationships, yielding node-level rationale representations Hr|c∈ℝN×dhH_r|c ^N× d_h and Hf|c∈ℝN×dhH_f|c ^N× d_h. The [⋅∥⋅] [·\|· ] denotes the concatenation of representations. ℛ(⋅)R(·), parameterized by θℛ _R, is a MLP that transforms the fused rationale representation H=[Hr|c∥Hf|c]H= [H_r|c\|H_f|c ] into the reconstructed node attributes X. Similarity, another graph convolution layer, GCNo(⋅)GCN_o(·), is employed to extract the non-rationale representation Ho∈ℝN×dhH_o ^N× d_h, i.e., Ho=GCNo(o,~)H_o=GCN_o(G_o, X). IV-B Rationale-Aware Node-Level Gaussian Mixture Normalizing Flow Model Inspired by the success of anomaly detection strategies based on normalizing flow, we leverage normalizing flow to learn a robust anomaly detection criterion for UGAD. Normalizing flow maps data into learned distributions, with the assumption that abnormal instances tend to reside in low-density regions, while normal instances are concentrated in high-density regions [66, 55, 5]. However, applying normalizing flow to UGAD presents several challenges: (i) how to accurately estimating the density of a mixture distribution containing both normal and abnormal data, thereby distinguishing high-density (normal) and low-density (abnormal) regions; (i) how to capture subtle variations in rationale representations to enable the model to learn the diverse patterns inherent in normal nodes, providing a more nuanced understanding of the underlying mixture distribution. To address these challenges, we propose a node-level rationale-aware conditional Gaussian mixture normalizing flow model (RGMN), as illustrated in Figure 2. RGMN comprises three core modules: (i) a Node-Level Conditional Normalizing Flow (NCNF) block, which conditionally transforms both rationale and non-rationale node representations into latent conditional embeddings, thereby facilitating conditional density estimation; (i) a Rationale-Non-Rationale Gaussian Mixture Modeling (RRGM) module, which separates and models rationale and non-rationale distributions using conditional Gaussian mixture modeling, enabling a more precise distinction between normal and abnormal nodes; and (i) a Robust-Fragile Rationale Mixture Learning (RFRM) strategy, which learns the Gaussian mixture distribution of rationale representations, capturing fine-grained divergent patterns among normal nodes and characterizing their diverse traits. a) Node-Level Conditional Normalizing Flows. The normalizing flow ℱF in RagGAD is built upon the Real-NVP [12], comprising η affine coupling layers with an identical structure, as shown in Figure 2. Specifically, given the robust rationale representation hr|ci∈ℝdhh_r|c^i ^d_h, the affine transformation that maps the projected attribute x~i∈ℝdx~ x_i ^d_ x of node viv_i, conditioned on hr|cih_r|c^i (i.e., condition =hr|ciC=h_r|c^i), to the latent conditional embedding zr|ci∈ℝdzz_r|c^i ^d_z is defined as: zr|ci=ℱθ(x~i;hr|ci)⇒zr|ci=[zr|ci,1∥zr|ci,2], z_r|c^i=F_θ( x_i;h_r|c^i) z_r|c^i= [z_r|c^i,1\|z_r|c^i,2 ], (9) x~i1,x~i2=Θ(x~i),zr|ci,1=x~i1, x_i^1, x_i^2= ( x_i),z_r|c^i,1= x_i^1, (10) zr|ci,2=x~i2 -8.5359ptz_r|c^i,2= x_i^2 ⊙exp(s([x~i1∥hr|ciWh]))+t([x~i1∥hr|ciWh]), -2.84544pt ( -2.84544pts -2.84544pt ( -1.42271pt [ x_i^1\|h_r|c^iW_h -1.42271pt ] ) )+t ( -1.42271pt [ x_i^1\|h_r|c^iW_h -1.42271pt ] ), (11) where Θ is a attribute partitioning function [4] that splits x~i x_i into x~i1∈ℝdx~/2 x_i^1 ^d_ x/2 and x~i2∈ℝdx~/2 x_i^2 ^d_ x/2, respectively. s(⋅)s(·) and t(⋅)t(·) are transformation coefficients predicted by a learnable neural network, and is implemented using a MLPs [12, 59]. Wh∈ℝdh×dx~/2W_h ^d_h× d_ x/2 is a learnable parameters, and ⊙ represents the element-wise product. Based on Eq.(3), the robust rationale conditional density of x~i x_i can be defined as: logp(x~i|hr|ci)=logpZ(ℱθ(x~i,hr|ci))+log|det∇xℱθ(x~i,hr|ci)|. -11.38092ptlogp_X( x_i|h_r|c^i) -2.84544pt= -2.84544ptlogp_Z -2.84544pt ( -2.84544ptF_θ( x_i;h_r|c^i) -2.84544pt ) -2.84544pt+ -2.84544ptlog -2.84544pt -1.42271ptdet _xF_θ( x_i;h_r|c^i) -1.42271pt . (12) Similarly, conditioned on the fragile rationale representation hf|cih^i_f|c and the non-rationale representation hoih_o^i, we can obtain the latent conditional embeddings zf|ciz_f|c^i and zoiz_o^i for node viv_i, along with the corresponding fragile rationale conditional density logp(x~i|hf|ci)logp_X( x_i|h_f|c^i) and non-rationale conditional density logp(x~i|hoi)logp_X( x_i|h_o^i). For simplicity, we omit the superscript i in the subsequent sections without loss of generality. b) Rationale-Non-Rationale Gaussian Mixture Modeling. Based on NCNF (IV-B), we can derive the conditional densities logp(x~i|hr|ci)logp_X( x_i|h_r|c^i), logp(x~i|hf|ci)logp_X( x_i|h_f|c^i), and logp(x~i|hoi)logp_X( x_i|h_o^i) for x~ x of node viv_i under the representations hr|cih_r|c^i, hf|cih_f|c^i, and hoih_o^i, respectively. However, existing normalizing flow models typically adopt a standard multivariate normal distribution (0,)N(0,I) as the prior and assume that all latent conditional embeddings z follow an identical distribution. This assumption is inherently restrictive in UGAD, as it fails to capture the distinct characteristics of normal and abnormal nodes in the latent space, thereby weakening the ability to distinguish between normal and abnormal patterns. Therefore, to better fit the normal distribution in the presence of anomalies, we introduce rationale-non-rationale Gaussian mixture modeling (RRGM), which models rationale and non-rationale representations as distinct latent classes. Specifically, RRGM formulates a conditional Gaussian mixture model based on the dependencies among representation classes, where rationale representations correspond to the normal class (i.e., label Y=0Y=0) and non-rationale representations as the abnormal class (i.e., label Y=1Y=1). Consequently, RagGAD uses class-dependent mean uyu_y and covariance Σy _y for each class as the prior distribution for z, defined as: pZ(z)=∑y∈0,1p(y)(z,uy,Σy),pZ|Y(z|y)=(z,uy,Σy), split -11.38092pt&p_Z(z)= -8.5359pt _y∈\0,1\ -5.69046ptp(y)N (z;u_y, _y ),\\ -11.38092pt&p_Z|Y(z|y)=N(z;u_y, _y), split (13) where uyu_y and Σy _y are the mean and covariance matrix of class y, respectively. p(y)=softmax(WyTz+by)p(y)=softmax (W_y^Tz+b_y ) represents the class prior distribution of z, where Wy∈ℝdzW_y ^d_z and by∈ℝb_y are the learnable parameters. RRGM setting the covariance matrix Σy _y is the identity matrix I for better convergence. Then, we can calculate the log-likelihood for latent conditional embedding z as follows: logpZ(z)=log[∑y∈0,1p(y)(z,uy,Σy)] _Z(z)=log [ _y∈\0,1\p(y)N(z;u_y, _y) ] (14) =log[∑y∈0,1p(y)(z,uy,)] =log [ _y∈\0,1\p(y)N(z;u_y,I) ] (15) ⇓pZ(z)=(2π)−dz2e−12zTz, p_Z(z)=(2π)^- d_z2e^- 12z^Tz, (16) =−dz2log(2π)+log(∑y∈0,1eϵy⋅e−‖z−uy‖222) =- d_z2log(2π)+log ( _y∈\0,1\e _y· e^- z-u_y ^2_22 ) (17) =−dz2log(2π)+log(∑y∈0,1e−‖z−uy‖222+ϵy) =- d_z2log(2π)+log ( _y∈\0,1\e^- z-u_y ^2_22+ _y ) (18) =−dz2log(2π)+logsumexpy∈0,1(−‖z−uy‖222+ϵy), =- d_z2log(2π)+ y∈\0,1\logsumexp (- z-u_y ^2_22+ _y ), (19) where ϵy=logp(y) _y=logp(y) denotes the logarithmic class weight, scaling the class prior distribution. Further, we bring logpZ(z)logp_Z(z) into Eq. (12), and reformulate the conditional log-density logp(x~|h)logp_X( x|h) of x~ x as follows: logp(x~|h)=logsumexp(−‖ℱθ(x~,h)−uy‖222+ϵy)−dz2log(2π)+log|detJ|, split -8.5359ptlogp_X( x|h)&= ylogsumexp (- _θ( x;h)-u_y ^2_22+ _y )\\ -8.5359pt&- d_z2log(2π)+log|detJ|,\\ split (20) where J=∇xℱθ(x~,h)J= _xF_θ( x;h). The loss function of RRGM is defined as follows: ℒrrgm=x~∼p()[−logp(x~|h)]=x~∼p()[−logsumexp(−‖ℱθ(x~,h)−uy‖222+ϵy)−log|detJ|+dz2log(2π)]. split -11.38092ptL_rrgm&=E_ x p(X)[-logp_X( x|h)]\\ -11.38092pt&=E_ x p(X)[- ylogsumexp -2.84544pt ( -2.84544pt- _θ( x;h)-u_y _2^22+ _y -2.84544pt )\\ -11.38092pt&-log + d_z2log(2π)]. split (21) c) Robust-Fragile Rationale Mixture Learning. In real-world large-scale graphs, the diversification of normal patterns among nodes is a critical problem that must be considered for UGAD. If a normalizing flow model can learn the diverse fine-grained normal distributions of nodes, it can help suppress false detections for divergent patterns within normals. Consequently, we propose RFRM strategy, which captures the diversity among normals nodes through fine-grained rationale Gaussian mixture learning. Specifically, we first extends the Gaussian prior pZ|Y(z|y)=(z,uy,Σy)p_Z|Y(z|y)=N(z;u_y, _y) to mixture Gaussian prior pZ|Y(z|y)=∑k=1Kpk(y)(z,uyk,Σyk)p_Z|Y(z|y)= _k=1^Kp_k(y)N(z;u_y^k, _y^k), where K is the number of Gaussian components [21, 59]. If, we further assume Σyk= _y^k=I, then the likelihood of latent conditional embedding z can be calculate as follows: pZ(z)=∑yp(y)(∑k=1Kpk(y)(z,uyk,Σyk)).p_Z(z)= _yp(y) ( _k=1^Kp_k(y)N(z;u_y^k, _y^k) ). (22) When pZ(z)p_Z(z) is obtained, the log-likelihood logpZ(z)logp_Z(z) of z can be defined as follows: logpZ(z)=log(∑yp(y)sumexp[−‖z−uyk‖222+ϵyk−dz2log(2π)]), -11.38092ptlogp_Z(z)=log -2.84544pt ( -2.84544pt _yp(y) ksumexp -2.84544pt [- z-u^k_y ^2_22+ε^k_y -2.84544pt- -2.84544pt d_z2log(2π) ] -2.84544pt ), (23) where the latent conditional embedding z=ℱθ(x~,h)z=F_θ( x;h). ϵyk=logpk(y)=logsoftmaxk(ψy)ε^k_y=logp_k(y)=logsoftmax_k( _y) denotes the logarithmic component center weights, and ψy _y is a learnable vector specify for class y, with ψy∈ℝK _y ^K, to adaptively learn the component weights. Further, we can reformulate Eq. (20) to get the conditional log-density logp(x~|h)logp_X( x|h) for any x~ x on condition h when upon mixture Gaussian prior: logp(x~|h)=log(∑yp(y)sumexp[−‖z−uyk‖222+ϵyk−dz2log(2π)])+log|detJ|. split -12.80365ptlogp_X( x|h)& -2.84544pt= -2.84544ptlog -2.84544pt ( -2.84544pt _yp(y) ksumexp -2.84544pt [- z-u^k_y ^2_22+ε^k_y- d_z2log(2π) ] -2.84544pt )\\ -11.38092pt&+log|detJ|.\\ split (24) Furthermore, in RFRM, for each class y, we learn a central component uycu_y^c and a set offsets Δuykk=1K\ u_y^k\_k=1^K, used to calculate the centers uyk=uyc+Δuykk=1Ku_y^k=\u_y^c+ u_y^k\_k=1^K for other components. We can directly optimize the central component uycu_y^c using Eq. (21) for each class y. However, when optimizing uyku_y^k for other components, we detach the gradient of uycu_y^c and optimize the offset Δuyk u_y^k by the following loss function: ℒrfrm=(x~,y)∼p(,Y)[−logp(x~|h)]=(x~,y)∼p(,Y)[−log|detJ|−logsumexp(−‖z−([uyc]+Δuyk)‖222+ϵyk)], split -11.38092ptL_rfrm&=E_( x,y) p(X,Y)[-logp_X( x|h)]\\ -11.38092pt&=E_( x,y) p(X,Y) [-log .\\ -12.80365pt& .- klogsumexp -2.84544pt ( -2.84544pt- z- ( G -1.42271pt [u_y^c ]+ u^k_y ) _2^22+ _y^k ) ], split (25) where [⋅] G [· ] is stop gradient back-propagation. dz2log(2π) d_z2log(2π) is a constant term that does not affect the gradient during model optimization, so it is omitted. For normal class (i.e., Y=0Y=0), to model the diverse combinations of robust rationales r|cG_r|c and fragile rationales f|cG_f|c, we assign rationale latent conditional embedding zr|cz_r|c as central component and fragile rationale latent conditional embedding zf|cz_f|c as a complementary component. Specifically, we initialize the central component as ur|cu_r|c for zr|cz_r|c, and the learnable offset Δuf|c u_f|c for zf|cz_f|c is defined as follows: Δuf|c=fΔ(αzf|c;θΔ),α=σ(fα([zf|c||zr|c];θα)), u_f|c=f_ (α z_f|c; _ ), 9.24994ptα=σ (f_α ( [z_f|c||z_r|c ]; _α ) ), (26) where fαf_α computes the similarity α between zr|cz_r|c and zf|cz_f|c, parameterized by θα _α, while fΔf_ generates the offset Δuf|c1 u_f|c^1 based on αzf|cα z_f|c, parameterized by θΔ _ . Δuf|c u_f|c quantifies the offset of the fragile rationale component relative to robust rationale component, and the center of fragile component is uf|c=ur|c+Δuf|cu_f|c=u_r|c+ u_f|c. Formally, the loss function of RFRM for ur|cu_r|c is defined as ℒrrgmcL_rrgm^c directly based on Eq. (21), while the loss function ℒrfrmcL_rfrm^c for uf|cu_f|c can be defined by reformulating Eq. (25), as follows: ℒrrgmc=x~∼p()[−logsumexpy=0(−‖ℱθ(x~|hr|c)−ur|c‖222+ϵy)−log|detJ|+dz2log(2π)], split -14.22636ptL_rrgm^c -2.84544pt&=E_ x p(X) [ y=0-logsumexp -2.84544pt ( -2.84544pt- _θ( x|h_r|c)-u_r|c _2^22+ _y -2.84544pt ) .\\ -11.38092pt& .-log + d_z2log(2π) ], split (27) ℒrfrmc=(x~,y)∼p(,Y)[−log|detJ|−logsumexpy=0(−‖ℱθ(x~|hf|c)−([ur|c]+Δuf|c)‖222+ϵyk)]. split -14.22636pt&L_rfrm^c=E_( x,y) p(X,Y) [-log .\\ -14.22636pt& . -2.84544pt- y=0logsumexp -2.84544pt ( -2.84544pt- _θ( x|h_f|c) -2.84544pt- -2.84544pt ( G -2.84544pt [u_r|c ] -2.84544pt+ -2.84544pt u_f|c ) _2^22 -2.84544pt+ -1.42271pt _y^k -2.84544pt ) -2.84544pt ]. split (28) For the abnormal class (i.e., Y=1Y=1), we employ a two-component Gaussian mixture model. The class central uou_o and the learnable offset Δuo u_o are optimized by reformulating Eq. (21) and Eq. (25). The corresponding loss functions ℒrrgmoL_rrgm^o and ℒrfrmoL_rfrm^o are defined as follows: ℒrrgmo=x~∼p()[−logsumexpy=1(−‖ℱθ(x~|ho)−uo‖222+ϵy)−log|detJ|+dz2log(2π)], split -14.22636ptL_rrgm^o -2.84544pt&=E_ x p(X) [ y=1-logsumexp -2.84544pt ( -2.84544pt- _θ( x|h_o)-u_o _2^22+ _y -2.84544pt ) .\\ -14.22636pt& .-log + d_z2log(2π) ], split (29) ℒrfrmo=(x~,y)∼p(,Y)[−log|detJ|−logsumexpy=1(−‖ℱθ(x~|ho)−([uo]+Δuo)‖222+ϵky)]. split -14.22636ptL_rfrm^o&=E_( x,y) p(X,Y) [-log .\\ -14.22636pt& . -2.84544pt- y=1logsumexp -2.84544pt ( -2.84544pt- _θ( x|h_o) -2.84544pt- -2.84544pt ( G -2.84544pt [u_o ] -2.84544pt+ -2.84544pt u_o ) _2^22 -2.84544pt+ -1.42271ptε^k_y -2.84544pt ) -2.84544pt ]. split (30) Finally, the overall loss function of our RGMN model is defined as: ℒrgmn=γ(ℒrrgmc+ℒrfrmc)+(1−γ)(ℒrrgmo+ℒrfrmo),L_rgmn=γ(L_rrgm^c+L_rfrm^c)+(1-γ)(L_rrgm^o+L_rfrm^o), (31) where hyperparameter γ controls the balance between the rationale and non-rationale Gaussian mixture normalizing flow losses. IV-C The Overall Objective and Anomaly Scoring a) Overall Objective. In the training phase, the overall objective of RagGAD is formulated as follows: ℒ=λ1ℒrgmn+λ2ℒrcon+λ3ℒspr,L= _1L_rgmn+ _2L_rcon+ _3L_spr, (32) where ℒrcon=‖−^‖22L_rcon=\|X- X\|_2^2 is the attribute reconstruction loss. ℒspr=|c|L_spr=|G_c| is the rationale sparsity loss, which encourages RagGAD to automatically discard weak non-rationale correlations during the optimization process. The hyperparameters λ1 _1, λ2 _2, and λ3 _3 control the relative contributions of each loss term and are tuned via grid search. b) Anomaly Scoring. Since anomalies deviate significantly from the majority of data instances, we hypothesize that their densities are low. Therefore, for each node viv_i, based on Eq. (24), the overall log-density conditioned on the rationale and non-rationale representations hoh_o, hr|ch_r|c, and hf|ch_f|c is computed as follows: logp(x~|h)=logp(x~|ho)+logp(x~|hr|c)+logp(x~|hf|c). _X( x|h) -1.42271pt= -1.42271ptlogp_X( x|h_o) -1.42271pt+ -1.42271ptlogp_X( x|h_r|c) -1.42271pt+ -1.42271ptlogp_X( x|h_f|c). (33) During testing, the anomaly score (xi)S(x_i) for each node viv_i is defined as the negative log-density, i.e., (xi)=−logp(x~|h)S(x_i)=-logp_X( x|h). A higher (xi)S(x_i) indicates a higher likelihood of viv_i being abnormal. TABLE I: AUROC and AUPRC results (in percentage) on six real-world UGAD datasets with injected/real anomalies. Higher AUROC/AUPRC indicates better performance. The best results are shown in bold, and the second-best results are bold. Dataset Metric Method BlogCatalog ACM Amazon Facebook Reddit YelpChi Anomalous[39] 56.52±2.5056.52_± 2.50 68.56±6.3068.56_± 6.30 44.57±0.3044.57_± 0.30 90.21±0.5090.21_± 0.50 53.87±1.2053.87_± 1.20 49.56±0.3049.56_± 0.30 DOMINANT[10] 75.90±1.0075.90_± 1.00 85.69±2.0085.69_± 2.00 59.96±0.4059.96_± 0.40 56.77±0.2056.77_± 0.20 55.55±1.1055.55_± 1.10 41.33±1.0041.33_± 1.00 CoLA[32] 77.46±0.9077.46_± 0.90 82.33±0.1082.33_± 0.10 58.98±0.8058.98_± 0.80 84.34±1.1084.34_± 1.10 60.28±0.7060.28_± 0.70 46.36±0.1046.36_± 0.10 SL-GAD[65] 81.23±0.2081.23_± 0.20 84.79±0.5084.79_± 0.50 59.37±1.1059.37_± 1.10 79.36±0.5079.36_± 0.50 56.77±0.5056.77_± 0.50 33.12±3.5033.12_± 3.50 AUROC HCM-A[20] 79.80±0.4079.80_± 0.40 80.60±0.4080.60_± 0.40 39.56±1.4039.56_± 1.40 73.87±3.2073.87_± 3.20 45.93±1.1045.93_± 1.10 45.93±0.5045.93_± 0.50 ComGA[34] 76.83±0.4076.83_± 0.40 82.21±2.5082.21_± 2.50 58.95±0.8058.95_± 0.80 60.55±0.0060.55_± 0.00 54.53±0.3054.53_± 0.30 43.91±0.0043.91_± 0.00 GADAM[7] 79.92±0.0079.92_± 0.00 94.66±0.00 [rgb]0.0195,0.3672,0.636794.66_± 0.00 61.71±0.0061.71_± 0.00 94.03±0.0094.03_± 0.00 56.54±0.0056.54_± 0.00 41.77±0.0041.77_± 0.00 TAM[40] 82.48±0.30 [rgb]0.0195,0.3672,0.636782.48_± 0.30 88.78±2.4088.78_± 2.40 70.64±1.0070.64_± 1.00 91.44±0.8091.44_± 0.80 60.23±0.4060.23_± 0.40 56.43±0.7056.43_± 0.70 HUGE[36] 62.13±0.0062.13_± 0.00 83.27±0.0083.27_± 0.00 85.16±0.0085.16_± 0.00 97.60±0.00 [rgb]0.0195,0.3672,0.636797.60_± 0.00 59.06±0.0059.06_± 0.00 60.13±0.0060.13_± 0.00 CoCo[54] 79.10±0.9379.10_± 0.93 88.37±1.5688.37_± 1.56 88.96±3.11 [rgb]0.0195,0.3672,0.636788.96_± 3.11 96.75±0.0196.75_± 0.01 61.92±0.51 [rgb]0.0195,0.3672,0.636761.92_± 0.51 63.89±0.0663.89_± 0.06 SmoothGNN[13] – – 83.17±0.8883.17_± 0.88 44.46±1.0844.46_± 1.08 56.87±1.6956.87_± 1.69 57.48±0.0657.48_± 0.06 GCTAM[61] 62.47±0.7062.47_± 0.70 90.77±1.5090.77_± 1.50 84.38±1.4084.38_± 1.40 92.38±0.3092.38_± 0.30 59.21±0.6059.21_± 0.60 79.00±0.50 [rgb]0.0195,0.3672,0.636779.00_± 0.50 FreeGAD[64] 74.84±0.0074.84_± 0.00 84.84±0.0084.84_± 0.00 88.57±0.0088.57_± 0.00 91.51±0.0091.51_± 0.00 57.21±0.0057.21_± 0.00 78.55±0.0078.55_± 0.00 RagGAD 87.32±0.67 [rgb]0.0625,0.4883,0.320387.32_± 0.67 97.24±0.34 [rgb]0.0625,0.4883,0.320397.24_± 0.34 92.04±0.21 [rgb]0.0625,0.4883,0.320392.04_± 0.21 98.62±0.27 [rgb]0.0625,0.4883,0.320398.62_± 0.27 63.71±0.71 [rgb]0.0625,0.4883,0.320363.71_± 0.71 81.13±0.14 [rgb]0.0625,0.4883,0.320381.13_± 0.14 Anomalous[39] 6.52±0.506.52_± 0.50 6.35±0.606.35_± 0.60 5.58±0.105.58_± 0.10 18.98±0.4018.98_± 0.40 3.75±0.403.75_± 0.40 5.19±0.205.19_± 0.20 DOMINANT[10] 31.02±1.1031.02_± 1.10 44.02±3.6044.02_± 3.60 14.24±0.2014.24_± 0.20 3.14±4.103.14_± 4.10 3.56±0.203.56_± 0.20 3.95±2.003.95_± 2.00 CoLA[32] 32.70±0.0032.70_± 0.00 32.35±1.7032.35_± 1.70 6.77±0.106.77_± 0.10 21.06±1.7021.06_± 1.70 4.49±0.204.49_± 0.20 4.48±0.204.48_± 0.20 SL-GAD[65] 38.82±0.7038.82_± 0.70 37.84±1.1037.84_± 1.10 6.34±0.506.34_± 0.50 13.16±2.0013.16_± 2.00 4.06±0.404.06_± 0.40 3.50±0.003.50_± 0.00 AUPRC HCM-A[20] 31.39±0.1031.39_± 0.10 34.13±0.4034.13_± 0.40 5.27±1.505.27_± 1.50 7.13±0.407.13_± 0.40 2.87±0.502.87_± 0.50 2.87±1.202.87_± 1.20 ComGA[34] 32.93±2.8032.93_± 2.80 28.73±1.2028.73_± 1.20 11.53±0.5011.53_± 0.50 3.54±0.103.54_± 0.10 3.74±0.103.74_± 0.10 4.23±0.004.23_± 0.00 GADAM[7] 28.14±0.0028.14_± 0.00 33.77±0.0033.77_± 0.00 11.74±0.0011.74_± 0.00 24.06±0.0024.06_± 0.00 4.62±0.004.62_± 0.00 4.23±0.004.23_± 0.00 TAM[40] 41.82±0.50 [rgb]0.0195,0.3672,0.636741.82_± 0.50 51.24±1.8051.24_± 1.80 26.34±0.8026.34_± 0.80 22.33±1.6022.33_± 1.60 4.46±0.104.46_± 0.10 7.78±0.907.78_± 0.90 HUGE[36] 7.44±0.007.44_± 0.00 32.93±0.0032.93_± 0.00 66.72±0.0066.72_± 0.00 36.74±0.00 [rgb]0.0195,0.3672,0.636736.74_± 0.00 5.11±0.00 [rgb]0.0195,0.3672,0.63675.11_± 0.00 7.08±0.007.08_± 0.00 CoCo[54] 37.53±0.3637.53_± 0.36 40.19±2.4440.19_± 2.44 39.20±0.0639.20_± 0.06 36.31±0.0836.31_± 0.08 5.02±0.175.02_± 0.17 7.18±0.017.18_± 0.01 SmoothGNN[13] – – 39.05±2.3339.05_± 2.33 1.95±0.041.95_± 0.04 4.17±0.164.17_± 0.16 18.18±0.03 [rgb]0.0195,0.3672,0.636718.18_± 0.03 GCTAM[61] 12.90±0.6512.90_± 0.65 52.10±0.70 [rgb]0.0195,0.3672,0.636752.10_± 0.70 50.69±8.1050.69_± 8.10 22.81±0.6022.81_± 0.60 4.17±0.104.17_± 0.10 16.04±1.0016.04_± 1.00 FreeGAD[64] 34.03±0.0034.03_± 0.00 37.15±0.0037.15_± 0.00 75.06±0.00 [rgb]0.0195,0.3672,0.636775.06_± 0.00 22.50±0.0022.50_± 0.00 3.85±0.003.85_± 0.00 15.80±0.0015.80_± 0.00 RagGAD 46.32±1.21 [rgb]0.0625,0.4883,0.320346.32_± 1.21 55.27±0.21 [rgb]0.0625,0.4883,0.320355.27_± 0.21 77.67±0.11 [rgb]0.0625,0.4883,0.320377.67_± 0.11 39.33±0.42 [rgb]0.0625,0.4883,0.320339.33_± 0.42 5.45±0.20 [rgb]0.0625,0.4883,0.32035.45_± 0.20 19.38±0.43 [rgb]0.0625,0.4883,0.320319.38_± 0.43 V Experiments V-A Experimental Setup a) Datasets. We evaluate RagGAD on ten benchmark datasets for UGAD, following TAM [40] and HUGE [36]. These datasets comprise six real-world datasets, including BlogCatalog [50], ACM [49], Amazon [14], Facebook [58], Reddit [24], and YelpChi [24]. Additionally, four large-scale datasets are used, namely Amazon-all [40], YelpChi-all [6], T-Finance [48], and OGB-Proteins [18]. Among these datasets, BlogCatalog, ACM, and OGB-Protein contain two types of injected anomalies, contextual and structural anomalies. Following [40, 36], all graphs are converted to homogeneous undirected graphs for consistency during evaluation. b) Baselines. We compare RagGAD with a comprehensive set of state-of-the-art baselines, including Anomalous [39], DOMINANT [10], CoLA [32], SL-GAD [65], HCM-A [20], ComGA [34], GADAM [7], TAM [40], HUGE [36], SmoothGNN [13], GCTAM [61], CoCo [54], and FreeGAD [64]. Following [40, 31], two evaluation metrics, Area Under the Receiver Operating Characteristic Curve (AUROC) and Area Under the precision recall curve (AUPRC), are used for UGAD. The reported AUROC and AUPRC results are averaged over five independent runs with different random seeds. Reported performance for all baselines is obtained from publicly available results or reproduced using their official source code. c) Implementation Details. RagGAD adopts a consistent strategy for hyperparameter tuning, early stopping, and model selection, following TAM [40] and HUGE [36]. All experiments are performed using the PyTorch framework on an NVIDIA GeForce RTX 3090 (24GB) GPU. The model is optimized with the Adam optimizer over 500 epochs with a learning rate of 1e-4. The projector (⋅)P(·) employs the PCA method, while the graph convolution layer GCN(⋅)GCN(·) is implemented as a two-layer GCN. The reconstructor ℛ(⋅)R(·) is designed as a two-layer MLP to reconstruct node attributes. V-B Results a) Comparison with Baselines. Table I presents the AUROC and AUPRC results for all baselines across six real-world UGAD datasets with both real and injected anomalies. Overall, RagGAD consistently achieves the best performance across all datasets under both metrics, demonstrating its strong effectiveness and robustness. In terms of AUPRC, which is more sensitive to class imbalance, RagGAD also achieves the best performance across all datasets. RagGAD surpasses TAM on BlogCatalog and ACM by +4.50% (46.32 vs. 41.82) and +3.17% (55.27 vs. 52.10), respectively. On Amazon, RagGAD improves over FreeGAD by +2.61% (77.67 vs. 75.06), while on Facebook, it exceeds HUGE by +2.59% (39.33 vs. 36.74). Although the absolute gains on Reddit and YelpChi are relatively smaller (+0.34% and +1.20%), RagGAD still consistently ranks first, demonstrating stable performance under highly imbalanced and noisy conditions. In terms of AUROC, RagGAD ranks first on all six datasets, outperforming the second-best methods by clear margins. Specifically, it improves over TAM on BlogCatalog by +4.84% (87.32 vs. 82.48), over GADAM on ACM by +2.58% (97.24 vs. 94.66), and over CoCo on Amazon by +3.08% (92.04 vs. 88.96). On Facebook, RagGAD achieves 98.62%, exceeding the strong baseline HUGE (97.60) by +1.02%. Even on more challenging datasets such as Reddit and YelpChi, RagGAD still attains the highest scores (63.71% and 81.13%), surpassing the second-best methods by +1.79% and +2.13%, respectively. Moreover, compared with recent strong methods such as HUGE, TAM, and FreeGAD, which exhibit dataset-dependent performance variations, RagGAD maintains consistently high performance across all scenarios. TABLE I: Results on large-scale graphs demonstrate the effectiveness of each baseline for handling large numbers of nodes and edges. OOM indicates out-of-memory on a 24GB GPU. Dataset Metric Method Amazon-all YelpChi-all T-Finance OGB-Proteins DOMINANT[10] 69.37 53.90 53.80 72.67 ComGA[34] 71.54 53.52 55.42 71.34 CoLA[32] 26.14 48.01 48.29 71.42 AUROC SL-GAD[65] 27.28 55.51 46.48 73.71 GADAM[7] 41.55 47.95 25.38 73.38 TAM[40] 84.76 58.18 61.75 74.49 GCTAM[61] 88.89 59.57 OOM OOM CoCo[54] 85.61 OOM OOM OOM HUGE[36] 88.92 57.67 OOM OOM FreeGAD[64] 87.83 54.17 92.13 66.51 RagGAD 93.15 63.12 92.98 78.83 DOMINANT[10] 10.15 16.38 4.74 22.17 ComGA[34] 18.54 16.58 4.81 15.54 CoLA[32] 5.16 13.61 4.10 13.49 GADAM[7] 5.78 13.79 3.04 21.04 AUPRC SL-GAD[65] 4.44 17.11 3.86 17.71 TAM[40] 43.46 18.86 5.47 21.73 GCTAM[61] 67.18 20.13 OOM OOM CoCo[54] 70.36 OOM OOM OOM HUGE[36] 76.68 18.69 OOM OOM FreeGAD[64] 66.32 16.96 73.71 25.21 RagGAD 80.02 23.25 74.81 31.44 TABLE I: Ablation studies for key components in RagGAD (in percentage). We report the average AUROC and AUPRC over five independent random runs. IMP: the average improvement of each variant over the rest (in percentage). Amazon Facebook Reddit YelpChi-all T-Finance Avg (IMP) Variant AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC w/o ARD 75.49 61.35 93.15 19.88 58.01 4.27 54.52 17.84 78.81 66.08 72.00 (0.00) 33.88 (0.00) w/o RGMN 73.23 60.07 95.36 22.68 59.36 4.30 55.68 18.09 77.80 66.02 72.29 (0.29) 34.23 (0.35) w/o RRGM 86.53 66.88 91.08 23.03 60.37 4.58 57.21 19.60 84.23 66.79 75.88 (3.60) 35.82 (1.94) w/o RFRM 90.58 76.03 95.85 33.03 61.04 4.80 61.57 21.94 90.84 70.23 79.97 (4.09) 38.91 (5.03) RagGAD 92.04 77.67 98.62 39.33 63.71 5.45 63.12 23.25 92.98 74.81 82.09 (2.21) 44.10 (2.90) Fig. 3: Visualization of (a) homophily distribution of normal and abnormal nodes, and (b)-(e) the anomaly score of TAM (local affinity) [40], HUGE (label-free heterophily measure) [36], GCTAM (contextual and global affinity) [61], and RagGAD (rationale-aware conditional log-density) on YelpChi [24], Facebook [58], and Amazon-all [40] datasets. Fig. 4: The visualization of inter-node interrelationships, disentangled non-rationale correlations and rationales (robust and fragile) relationships by RagGAD. The deeper the color, the stronger the relationship between nodes. The red box □ indicates correlations between nodes but are not rationales and can thus be removed. Fig. 5: The t-SNE visualization shows the robust rationale representation (■ ), fragile rationale representation (▲ ), and non-rationale representation (★ ) learned on four benchmark datasets. b) Performance on Large-scale Graphs. We evaluated RagGAD on four large-scale graph datasets with substantial node and edge counts to assess its effectiveness in handling complex graph structures. The results in Table I demonstrate that RagGAD maintains high effectiveness despite the increased complexity of anomalies, delivering consistent and significant performance across all four datasets. In terms of AUROC, RagGAD outperforms the strongest baselines on all datasets, achieving 93.15% on Amazon-all (+4.23% over HUGE), 63.12% on YelpChi-all (+3.55% over GCTAM), 92.98% on T-Finance (+0.85% over FreeGAD), and 78.83% on OGB-Proteins (+4.34% over TAM). On more challenging datasets such as T-Finance and OGB-Proteins, where anomalies are highly subtle and distributions are more complex, RagGAD still maintains clear advantages, indicating its robustness in capturing intricate structural patterns. Consistent improvements are also observed under AUPRC, with notable gains such as +3.34% on Amazon-all and +6.23% on OGB-Proteins, demonstrating its effectiveness under highly imbalanced settings. Moreover, several recent methods encounter out-of-memory (OOM) issues on larger datasets, whereas RagGAD remains stable and achieves strong performance across all settings. These results demonstrate that modeling stable rationales while mitigating spurious correlations enables RagGAD to better generalize across diverse graph structures and anomaly types, leading to superior performance in large-scale graph anomaly detection. c) Visualization of Anomaly Score. Figure 3(a), presents the homophily distribution of normal and abnormal nodes across benchmark datasets. The results indicate that the Facebook and Amazon-all datasets exhibit one-class homophily, characterized by stronger connectivity or affinity among normal nodes compared to abnormal nodes. In contrast, the YelpChi dataset deviates from this assumption, with connectivity or affinity among abnormal nodes being comparable to, or even surpassing, that among normal nodes. Figure 3(b)-(e), depict the anomaly score distributions for TAM, HUGE, GCTAM, and RagGAD. As shown in Figure 3(b)-(e), TAM fails to establish a clear boundary between the distributions of normal and abnormal nodes across all datasets. HUGE demonstrates improved performance on the Facebook and Amazon-all datasets, yet does not differentiate normal and abnormal nodes on the Facebook dataset. Although HUGE identifies the homophily trap and introduces a label-free heterophily measure strategy, it falls short of substantially enhancing the distributional distinction between normal and abnormal nodes. GCTAM extends TAM by incorporating contextual and global affinity to filter anomalous nodes, but it still fails to effectively distinguish between normal and anomalous distributions on the YelpChi dataset. Conversely, RagGAD achieves a clear distributional distinction between normal and abnormal nodes across all datasets. While some overlap in distribution density occurs on the YelpChi dataset, the anomaly score distributions of normal and abnormal nodes remain separated, exhibiting clearly differentiated peaks. d) Visualization of Rationale Graphs. In disentangling non-rationale correlations and rationales between nodes, we visualize the disentangled non-rationale correlations oG_o and rationales (r|cG_r|c and f|cG_f|c) on the Facebook dataset, as shown in Figure 4. The highly challenging Facebook dataset, has the lowest anomaly ratio (2.49%\%) and the smallest number of nodes, making it an ideal choice for this validation. To improve visualization clarity, we randomly extract 20×20 subgraphs from the 1,080-node of Facebook for display. As shown in Figure 4(b), the disentangled rationales cG_c are sparse, yet they retain most of the node correlations while eliminating certain spurious correlations (highlighted by the red box) from A. Figure 4(c)-(e) demonstrate that robust rationales r|cG_r|c dominate the graph, accounting for approximately 80%\% of the relationships, compared to fragile rationales f|cG_f|c. However, despite their lower proportion, fragile rationales are crucial for diversifying normal patterns. When extended to the entire graph (1081 nodes), the number of fragile rationales becomes non-negligible. Essentially, the presence of fragile rationales greatly expands the diversity of normal node patterns. Therefore, as illustrated in Figure 2, RagGAD further decomposes the rationales into robust and fragile components and performs robust-fragile rationale mixture learning to enhance generalization for anomaly detection. e) Visualization of Representation. Figure 5 illustrates the t-SNE visualization of robust rationale, fragile rationale, and non-rationale representations extracted by RagGAD from benchmark datasets. The visualization demonstrates that RagGAD effectively disentangles rationale and non-rationale representations, further decomposing the rationale components into robust and fragile components. Within RagGAD, the adaptive rationale disentangler (ARD) disentangles rationale and non-rationale correlations from node interdependencies, subsequently decomposing rationale into robust and fragile components. Leveraging rationales, RagGAD derives robust and fragile rationale representations, while non-rationale representations extracted from non-rationale correlations. This enablesprecise modeling and discrimination of conditional densities for normal and abnormal nodes. As shown in Figure 5, RagGAD achieves a compact intra-class feature distribution while distinctly separating clusters of different class representations. V-C Ablation Study a) Components Analysis. We investigate the effectiveness of each component in RagGAD across five datasets, as detailed in Table I. Table I demonstrates a progressive improvement in accuracy as more components are integrated, highlighting the essential role of each component in the overall effectiveness of RagGAD. To evaluate the effectiveness of our ARD (w/o ARD), we remove rationale disentangler and instead utilize node interrelationships ℐI to guide information propagation between nodes, performing density estimation based on conventional normalizing flow. The results in Table I indicate that compared to w/o RGMN, which incorporates ARD followed by a conventional normalizing flow, ARD improves model performance, further validating its effectiveness. Fig. 6: AUROC and AUPRC of RagGAD w.r.t hyperparameter τ. We compare RagGAD with two variants that exclude the corresponding components, i.e., rationale-non-rationale gaussian mixture modeling (w/o RRGM) and robust-fragile rationale mixture learning (w/o RFRM), to evaluate the effectiveness of these key components. The results in the lower part of Table I confirm that integrating RRGM and RFRM significantly improves model performance, with RRGM alone demonstrating strong competitiveness. Notably, RRGM plays a pivotal role, as its removal results in a substantial drop in AUPRC and AUROC by approximately 1.94% (35.82 vs. 33.88) and 3.88% (75.88 vs. 72.00) across all datasets compared to RagGAD. Fig. 7: Sensitivity analysis of hyperparameter γ on the Facebook dataset. b) Sensitivity Analysis of τ. In order to evaluate the sensitivity of RagGAD to the hyperparameter τ, we vary its value within the range [0.5, 1] in increments of 0.05 for validation. The parameter τ, which governs the selection of edges as rationales (see Eq.(6)), is a crucial hyperparameter for RagGAD. As shown in Figure 6, as τ increases, RagGAD exhibits a tred of initially rising, then flattening, and finally gradually decreaseing. This variation pattern is reasonable, as a very low value of τ forces the model to incorrectly classify non-rationale correlations as rationales, resulting in excessively high false positives. In contrast, as τ increases, the rationale edge selection strategy becomes stricter. When τ is too high, the model may miss relatively fragile rationales, leading to an increase in rationale false negatives. For AUROC, the impact of changing τ is relatively significant, with a noticeable performance growth between 0.5 and 0.8. Whether for AUROC or AUPRC, when τ is around 0.85, the model achieves its best performance across all datasets. Therefore, in all our experiments, we set the default value of τ to 0.85. c) Sensitivity Analysis of γ. The hyperparameter γ controls the contributions of the rationales and non-rationale gaussian mixture normalizing flow loss terms (see Eq.(31)). We adjust its value across [0,1] with increments of 0.1 on the Facebook dataset. When γ is set to 0, it indicates that the rationale loss term in RGMN is inactive. In constast, when γ is set to 1, the model optimizes RGMN based solely on the rationale loss term. As shown in Figure 7, when γ is set to 0, both AUROC and AUPRC perform poorly, indicating that our rationale-aware Gaussian mixture normalizing flow plays a crucial role in RagGAD. However, when γ is set to 1, despite the absence of guidance from the non-rationale loss term, the model still performs relatively well, achieving 93.82 and 29.08 for AUROC and AUPRC, respectively. When all loss terms are enabled and γ is set to 0.7 to balance rationale and non-rationale losses, RagGAD performs optimally, achieving AUROC and AUPRC of 98.62 and 39.33, respectively. TABLE IV: The effect of parameters λ1 _1, λ2 _2, and λ3 _3 on different datasets. BlogCatalog Amazon Facebook Reddit YelpChi-all T-Finance Avg Dataset AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC AUROC AUPRC 0.001 77.10 35.11 88.57 67.94 89.63 25.60 63.33 5.60 62.40 22.37 91.76 59.27 78.30 36.65 0.01 79.50 34.95 90.84 70.80 91.90 37.55 65.62 5.79 62.80 22.79 89.73 60.37 79.56 39.69 λ1 _1 0.1 86.31 45.47 91.85 74.85 98.62 40.26 63.49 5.35 62.86 22.79 92.98 68.37 81.35 39.91 1 87.32 46.32 92.04 77.67 94.10 30.58 63.60 5.59 63.12 23.33 91.20 74.81 80.88 39.17 10 77.04 36.41 91.93 75.32 90.20 22.18 60.75 5.15 62.77 23.36 84.77 47.46 77.12 35.19 0.001 86.74 47.60 93.07 76.20 93.88 26.70 64.28 6.31 63.41 22.63 89.28 55.26 80.96 40.06 0.01 86.88 46.68 91.73 78.64 98.60 34.74 64.53 6.49 64.32 23.55 89.41 56.00 81.66 41.96 λ2 _2 0.1 87.32 46.20 91.92 74.38 99.03 37.36 65.99 6.73 62.94 22.50 91.02 66.99 82.09 41.34 1 86.96 46.70 91.15 74.00 99.08 31.41 63.03 5.58 62.15 22.30 91.50 86.43 81.24 40.69 10 87.13 46.00 92.91 77.01 96.40 31.87 63.04 5.11 62.94 22.45 93.45 67.30 81.51 40.65 0.001 84.55 47.05 92.12 73.02 89.50 20.23 61.27 5.15 63.19 23.06 78.40 38.55 77.80 36.92 0.01 87.01 47.40 91.45 76.40 94.02 24.84 65.26 5.60 64.38 22.70 82.28 53.78 80.26 38.93 λ3 _3 0.1 86.32 47.10 91.62 78.68 95.90 26.00 65.00 6.00 63.04 22.18 93.50 77.00 81.90 40.12 1 86.80 46.30 91.30 73.60 99.20 36.40 63.50 5.80 62.70 22.95 88.10 55.20 80.93 39.71 10 85.60 47.90 92.80 74.60 96.10 26.90 62.00 5.70 62.90 22.50 82.50 47.10 80.00 38.45 Fig. 8: Results of parameters sensitivity analysis. (a) shows the effects of different η, (b) shows the influence of embedding dimension dx~d_ x on the ACM dataset. d) Sensitivity Analysis of η. The hyperparameter η defines the number of coupling layers in NCNF. We evaluate performance of RagGAD under varying η, with the results presented in Figure 8(a). Overall, RagGAD demonstrates robustness to changes in η across both small-scale graphs (BlogCatalog, Facebook, and YelpChi) and the large-scale graph (Amazon-all). However, the highest AUROC is achieved at η=1η=1 for small-scale graphs, while for large-scale graphs, the optimal setting is η=2η=2. To ensure stable performance across different graph sizes while minimizing parameter redundancy, we set η=2η=2 as the default value for all datasets. e) Sensitivity Analysis of dx~d_ x. The hyperparameter dx~d_ x in Eq.(4) denotes the projected attribute dimension for each node. Following GADAM [7], for datasets with low-dimension attributes, we set dx~d_ x to 64. For high-dimension attribute datasets, to investigate the impact of dx~d_ x on the performance of RagGAD, we vary dx~d_ x within the range of 4 to 1024 on the ACM dataset, increasing it in powers of 2. As shown in Figure 8(b), the performance improves as dx~d_ x increases but starts to decline when it exceeds 256. To strike a balance between effectiveness and computation efficiency, we set dx~d_ x to 128 for all high-dimensional attribute datasets, including BlogCatalog, ACM, and Facebook. f) Sensitivity Analysis of λ1 _1, λ2 _2, and λ3 _3. We perform a grid search to tune these hyperparameters separately for each dataset, with the search range set between 0.001 and 1. As shown in Table IV, for λ1 _1, the optimal performance is obtained in the range of 0.01 to 1. γ1 _1 controls the strength of loss term ℒrgmnL_rgmn, which is the fundamental loss of RagGAD, the best performance is obtained when γ1=0.1 _1=0.1. γ2 _2 and λ3 _3 control the contributions of the attribute reconstruction loss ℒrconL_rcon and the rationale sparsity loss ℒsprL_spr, respectively. From Table IV, we observe that λ2 _2 and λ3 _3 exhibit higher sensitivity compared to λ1 _1. Specifically, λ2 _2 remains relatively stable in the range of 0.001 to 0.1, with a notable decrease in accuracy observed when it exceeds 0.1. λ3 _3 remains relatively stable between 0.01 and 1, while the performance of RagGAD is suboptimal when λ3 _3 falls below 0.01. This is mainly because setting λ2 _2 too high may cause the model to prioritize attribute reconstruction at the expense of accurately disentangling rationales. Conversely, setting λ3 _3 too low leads RagGAD to ignore the sparsity of rationales, resulting in the misclassification of weak non-rationale correlations as rationales. TABLE V: The model complexity, parameter count, and training and testing time (in seconds) of different baselines. R denotes the number of evaluation rounds, c denotes the number of nodes in the local subgraph, k is the number of global affinity truncation graph neighbors [61], and ω is the average node degree of G [7]. Method Overall Complexity Parameters Train Test DOMINANT[10] (ℰ+N2)O(E+N^2) 15769 67.549 0.283 CoLA[32] (cNR(c+ω))O(cNR(c+ω)) 5762 285 2.740 GADAM[7] (ℰ+N)O(E+N) 4226 6.418 0.029 TAM[40] (ℰdx+ℰdh+N2)O(Ed_x+Ed_h+N^2) 105091 1039 217 CoCo [54] (ℰ+Ndx~2+N2dx~)O(E+Nd_ x^2+N^2d_ x) 12283 11.26 0.034 HUGE[36] (N2dh)O(N^2d_h) 36355 18.38 13.45 GCTAM [61] (Nd+ℰd+Nk)O(Nd+Ed+Nk) 72320 41.72 0.3876 RagGAD (N2dx~)O(N^2d_ x) 21439 87.72 0.6488 V-D Efficiency and Complexity Analysis We evaluate efficiency from model complexity, parameter count, and empirical runtime, as summarized in Table V. All runtimes are measured on the Amazon dataset, where the training time corresponds to 100 epochs and the testing time reflects standard inference cost. In terms of complexity, most baselines scale linearly with ℰE or N, such as GADAM and GCTAM, while methods modeling pairwise node interactions introduce quadratic terms (N2)O(N^2), including DOMINANT, TAM, and HUGE. RagGAD falls into this category with complexity (N2dx~)O(N^2d_ x), due to global node-wise interaction and rationale disentanglement in the ARD module. Compared with local or sampling-based methods, this design captures long-range dependencies and high-order structural correlations, which are critical for accurate anomaly characterization. Despite the quadratic complexity, RagGAD maintains a moderate parameter size of 21,439, which is significantly smaller than TAM and comparable to GCTAM and HUGE. This indicates that the additional cost stems from dense computation rather than over-parameterization, reflecting efficient parameter utilization. Empirically, RagGAD achieves competitive efficiency. Its training time of 87.72 seconds is much lower than TAM and CoLA, and its inference time of 0.6488 seconds remains practical, outperforming several high-complexity methods. This suggests that the theoretical complexity does not directly translate into prohibitive runtime, as optimized matrix operations and parallelism mitigate the overhead in practice. Overall, RagGAD introduces additional computation for global dependency modeling and rationale disentanglement, leading to improved representation robustness, while maintaining a favorable balance between effectiveness and efficiency without excessive parameter or runtime overhead. VI Conclusion In this paper, we propose RagGAD, a novel unsupervised graph anomaly detection framework that disentangles robust rationales from non-rationale correlations in node interrelationships. By further decomposing rationales into robust and fragile components, RagGAD captures the underlying mechanisms governing normal behavior while identifying anomalies through deviations in non-rationale interactions. To model the complex distributions of both normal and abnormal nodes, we incorporate a node-level rationale-aware conditional Gaussian mixture normalizing flow that enhances sensitivity to the diversity of normal patterns and mitigates the homophily trap. Extensive experiments on benchmark datasets demonstrate that RagGAD outperforms state-of-the-art baselines, confirming its effectiveness in addressing the challenges of unsupervised graph anomaly detection. Future work will focus on optimizing the model to improve efficiency without sacrificing performance, and expanding its applicability to dynamic and heterogeneous graph scenarios. References [1] A. Abdelhamed, M. A. Brubaker, and M. S. Brown (2019) Noise flow: noise modeling with conditional normalizing flows. In Proceedings of the IEEE/CVF International Conference on Computer Vision, p. 3165–3173. Cited by: §I-B. [2] H. Abdi and L. J. Williams (2010) Principal component analysis. Wiley Interdisciplinary Reviews: Computational Statistics 2 (4), p. 433–459. Cited by: §IV-A. [3] G. AI, H. QIAO, H. YAN, and G. PANG (2025) Semi-supervised graph anomaly detection via robust homophily learning.. In Proceedings of the Conference on Neural Information Processing Systems, p. 2–7. Cited by: §I-A. [4] L. Ardizzone, C. Lüth, J. Kruse, C. Rother, and U. Köthe (2019) Guided image generation with conditional invertible neural networks. arXiv preprint arXiv:1907.02392. Cited by: §I-B, §IV-B. [5] R. Cao, S. Xue, J. Li, Q. Wang, and Y. Chang (2024) FANFOLD: graph normalizing flows-driven asymmetric network for unsupervised graph-level anomaly detection. arXiv preprint arXiv:2407.00383. Cited by: §I-B, §IV-B. [6] B. Chen, J. Zhang, X. Zhang, Y. Dong, J. Song, P. Zhang, K. Xu, E. Kharlamov, and J. Tang (2022) Gccad: graph contrastive coding for anomaly detection. IEEE Transactions on Knowledge and Data Engineering 35 (8), p. 8037–8051. Cited by: §I-A, §V-A. [7] J. Chen, G. Zhu, C. Yuan, and Y. Huang (2024) Boosting graph anomaly detection with adaptive message passing. In International Conference on Learning Representations, Cited by: §I-A, TABLE I, TABLE I, §V-A, §V-C, TABLE I, TABLE I, TABLE V, TABLE V. [8] N. Chen, Z. Liu, B. Hooi, B. He, R. Fathony, J. Hu, and J. Chen (2024) Consistency training with learnable data augmentation for graph anomaly detection with limited supervision. In International Conference on Learning Representations, Cited by: §I. [9] E. Dai and J. Chen (2022) Graph-augmented normalizing flows for anomaly detection of multiple time series. In International Conference on Learning Representations, Cited by: §I-B. [10] K. Ding, J. Li, R. Bhanushali, and H. Liu (2019) Deep anomaly detection on attributed networks. In Proceedings of the SIAM International Conference on Data Mining, p. 594–602. Cited by: §I, §I-A, §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I, TABLE V. [11] L. Dinh, D. Krueger, and Y. Bengio (2014) Nice: non-linear independent components estimation. arXiv preprint arXiv:1410.8516. Cited by: §I-B. [12] L. Dinh, J. Sohl-Dickstein, and S. Bengio (2016) Density estimation using real-nvp. arXiv preprint arXiv:1605.08803. Cited by: §I-B, §IV-B, §IV-B. [13] X. Dong, X. Zhang, Y. Sun, L. Chen, M. Yuan, and S. Wang (2025) Smoothgnn: smoothing-aware gnn for unsupervised node anomaly detection. In Proceedings of the ACM on Web Conference 2025, p. 1225–1236. Cited by: §I, §I-A, TABLE I, TABLE I, §V-A. [14] Y. Dou, Z. Liu, L. Sun, Y. Deng, H. Peng, and P. S. Yu (2020) Enhancing graph neural network-based fraud detectors against camouflaged fraudsters. In Proceedings of the ACM International Conference on Information & Knowledge Management, p. 315–324. Cited by: §V-A. [15] Y. Gao, X. Wang, X. He, Z. Liu, H. Feng, and Y. Zhang (2023) Addressing heterophily in graph anomaly detection: a perspective of graph spectrum. In Proceedings of the ACM Web Cconference, p. 1528–1538. Cited by: §I-A. [16] D. Gudovskiy, S. Ishizaka, and K. Kozuka (2022) Cflow-ad: real-time unsupervised anomaly detection with localization via conditional normalizing flows. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, p. 98–107. Cited by: §I-B. [17] J. He, Q. Xu, Y. Jiang, Z. Wang, and Q. Huang (2024) Ada-gad: anomaly-denoised autoencoders for graph anomaly detection. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 38, p. 8481–8489. Cited by: §I, §I, §I-A. [18] W. Hu, M. Fey, M. Zitnik, Y. Dong, H. Ren, B. Liu, M. Catasta, and J. Leskovec (2020) Open graph benchmark: datasets for machine learning on graphs. Advances in Neural Information Processing Systems 33, p. 22118–22133. Cited by: §V-A. [19] M. Huang, Y. Liu, X. Ao, K. Li, J. Chi, J. Feng, H. Yang, and Q. He (2022) Auc-oriented graph neural network for fraud detection. In Proceedings of the ACM Web Conference, p. 1311–1321. Cited by: §I-A. [20] T. Huang, Y. Pei, V. Menkovski, and M. Pechenizkiy (2022) Hop-count based self-supervised anomaly detection on attributed networks. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, p. 225–241. Cited by: §I-A, TABLE I, TABLE I, §V-A. [21] P. Izmailov, P. Kirichenko, M. Finzi, and A. G. Wilson (2020) Semi-supervised learning with normalizing flows. In International Conference on Machine Learning, p. 4615–4630. Cited by: §IV-B. [22] D. P. Kingma and P. Dhariwal (2018) Glow: generative flow with invertible 1x1 convolutions. Advances in Neural Information Processing Systems 31. Cited by: §I-B. [23] T. N. Kipf and M. Welling (2016) Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907. Cited by: §I. [24] S. Kumar, X. Zhang, and J. Leskovec (2019) Predicting dynamic embedding trajectory in temporal interaction networks. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, p. 1269–1278. Cited by: Fig. 3, §V-A. [25] R. Li, Z. Liu, Y. Ma, D. Yang, and S. Sun (2022) Internet financial fraud detection based on graph learning. IEEE Transactions on Computational Social Systems 10 (3), p. 1394–1401. Cited by: §I. [26] X. Li, X. Jiang, H. Wan, and X. Zhao (2025) TeRed: normal behavior-based efficient provenance graph reduction for large-scale attack forensics. IEEE Transactions on Information Forensics and Security. Cited by: §I. [27] X. Li, X. Li, J. Jia, L. Li, J. Yuan, Y. Gao, and S. Yu (2023) A high accuracy and adaptive anomaly detection model with dual-domain graph convolutional network for insider threat detection. IEEE Transactions on Information Forensics and Security 18, p. 1638–1652. Cited by: §I. [28] J. Liu, A. Kumar, J. Ba, J. Kiros, and K. Swersky (2019) Graph normalizing flows. Advances in Neural Information Processing Systems 32. Cited by: §I-B. [29] J. Liu, M. He, X. Shang, J. Shi, B. Cui, and H. Yin (2024) Bourne: bootstrapped self-supervised learning framework for unified graph anomaly detection. In IEEE International Conference on Data Engineering, p. 2820–2833. Cited by: §I. [30] Y. Liu, X. Ao, Z. Qin, J. Chi, J. Feng, H. Yang, and Q. He (2021) Pick and choose: a gnn-based imbalanced learning approach for fraud detection. In Proceedings of the Web Conference, p. 3168–3177. Cited by: §I. [31] Y. Liu, S. Li, Y. Zheng, Q. Chen, C. Zhang, and S. Pan (2024) ARC: a generalist graph anomaly detector with in-context learning. In Advances in Neural Information Processing Systems, Cited by: §I-A, §IV-A, §V-A. [32] Y. Liu, Z. Li, 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 33 (6), p. 2378–2392. Cited by: §I, §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I, TABLE V. [33] J. Lu and S. Sun (2024) CauDiTS: causal disentangled domain adaptation of multivariate time series. In International Conference on Machine Learning, Vol. 235, p. 33113–33146. Cited by: §IV-A. [34] X. Luo, J. Wu, A. Beheshti, J. Yang, X. Zhang, Y. Wang, and S. Xue (2022) Comga: community-aware attributed graph anomaly detection. In Proceedings of the ACM International Conference on Web Search and Data Mining, p. 657–665. Cited by: §I, §I-A, §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I. [35] C. Niu, H. Qiao, C. Chen, L. Chen, and G. Pang (2024) Zero-shot generalist graph anomaly detection with unified neighborhood prompts. arXiv preprint arXiv:2410.14886. Cited by: §I, §I-A. [36] J. Pan, Y. Liu, X. Zheng, Y. Zheng, A. W. Liew, F. Li, and S. Pan (2025) A label-free heterophily-guided approach for unsupervised graph fraud detection. arXiv preprint arXiv:2502.13308. Cited by: §I-A, TABLE I, TABLE I, Fig. 3, §V-A, §V-A, §V-A, TABLE I, TABLE I, TABLE V. [37] J. Pan, Y. Liu, Y. Zheng, and S. Pan (2023) PREM: a simple yet effective approach for node-level graph anomaly detection. In IEEE International Conference on Data Mining, p. 1253–1258. Cited by: §I-A. [38] G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan (2021) Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research 22 (57), p. 1–64. Cited by: §I-B. [39] Z. Peng, M. Luo, J. Li, H. Liu, Q. Zheng, et al. (2018) ANOMALOUS: a joint modeling approach for anomaly detection on attributed networks.. In International Joint Conference on Artificial Intelligence, Vol. 18, p. 3513–3519. Cited by: TABLE I, TABLE I, §V-A. [40] H. Qiao and G. Pang (2024) Truncated affinity maximization: one-class homophily modeling for graph anomaly detection. Advances in Neural Information Processing Systems 36. Cited by: §I, §I, §I-A, TABLE I, TABLE I, Fig. 3, §V-A, §V-A, §V-A, TABLE I, TABLE I, TABLE V. [41] H. Qiao, H. Tong, B. An, I. King, C. Aggarwal, and G. Pang (2024) Deep graph anomaly detection: a survey and new perspectives. arXiv preprint arXiv:2409.09957. Cited by: §I. [42] H. QIAO, Q. WEN, and X. LI Generative semi-supervised graph anomaly detection.. In Advances in Neural Information Processing Systems, p. 10–15. Cited by: §I-A. [43] A. Roy, J. Shu, J. Li, C. Yang, O. Elshocht, J. Smeets, and P. Li (2024) Gad-nr: graph anomaly detection via neighborhood reconstruction. In Proceedings of the ACM International Conference on Web Search and Data Mining, p. 576–585. Cited by: §I, §I-A. [44] M. Rudolph, B. Wandt, and B. Rosenhahn (2021) Same same but differnet: semi-supervised defect detection with normalizing flows. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, p. 1907–1916. Cited by: §I-B. [45] M. Rudolph, T. Wehrbein, B. Rosenhahn, and B. Wandt (2022) Fully convolutional cross-scale-flows for image-based defect detection. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, p. 1088–1097. Cited by: §I-B. [46] G. W. Stewart (1993) On the early history of the singular value decomposition. SIAM Review 35 (4), p. 551–566. Cited by: §IV-A. [47] J. Tang, F. Hua, Z. Gao, P. Zhao, and J. Li (2023) Gadbench: revisiting and benchmarking supervised graph anomaly detection. Advances in Neural Information Processing Systems 36, p. 29628–29653. Cited by: §I-A. [48] J. Tang, J. Li, Z. Gao, and J. Li (2022) Rethinking graph neural networks for anomaly detection. In International Conference on Machine Learning, p. 21076–21089. Cited by: §I, §I-A, §V-A. [49] J. Tang, J. Zhang, L. Yao, J. Li, L. Zhang, and Z. Su (2008) Arnetminer: extraction and mining of academic social networks. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, p. 990–998. Cited by: §V-A. [50] L. Tang and H. Liu (2009) Relational learning via latent social dimensions. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, p. 817–826. Cited by: §V-A. [51] C. Villani (2021) Topics in optimal transportation. Vol. 58, American Mathematical Soc.. Cited by: §I-B. [52] C. Wang and H. Zhu (2022) Wrongdoing monitor: a graph-based behavioral anomaly detection in cyber security. IEEE Transactions on Information Forensics and Security 17, p. 2703–2718. Cited by: §I. [53] Q. Wang, G. Pang, M. Salehi, W. Buntine, and C. Leckie (2023) Cross-domain graph anomaly detection via aanomaly-aware contrastive alignment. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 37, p. 4676–4684. Cited by: §I-A. [54] R. Wang, L. Xi, F. Zhang, H. Fan, X. Yu, L. Liu, S. Yu, and V. C. Leung (2025) Context correlation discrepancy analysis for graph anomaly detection. IEEE Transactions on Knowledge and Data Engineering 37 (1), p. 174–187. Cited by: §I, §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I, TABLE V. [55] R. Wang, K. Nie, T. Wang, Y. Yang, and B. Long (2020) Deep learning for anomaly detection. In Proceedings of the International Conference on Web Search and Data Mining, p. 894–896. Cited by: §IV-B. [56] X. Wang, B. Jin, Y. Du, P. Cui, Y. Tan, and Y. Yang (2021) One-class graph neural networks for anomaly detection in attributed networks. Neural Computing and Applications 33, p. 12073–12085. Cited by: §I-A. [57] C. Wu, J. Sun, J. Chen, M. Alazab, Y. Liu, and Y. Xiang (2025) TCG-ids: robust network intrusion detection via temporal contrastive graph learning. IEEE Transactions on Information Forensics and Security 20, p. 1475–1486. Cited by: §IV-A. [58] Z. Xu, X. Huang, Y. Zhao, Y. Dong, and J. Li (2022) Contrastive attributed network anomaly detection with data augmentation. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, p. 444–457. Cited by: Fig. 3, §V-A. [59] X. Yao, R. Li, Z. Qian, L. Wang, and C. Zhang (2025) Hierarchical gaussian mixture normalizing flow modeling for unified anomaly detection. In European Conference on Computer Vision, p. 92–108. Cited by: §I-B, §IV-B, §IV-B. [60] X. Yao, R. Li, J. Zhang, J. Sun, and C. Zhang (2023) Explicit boundary guided semi-push-ppull contrastive learning for supervised anomaly detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, p. 24490–24499. Cited by: §I-B. [61] X. Zhang, H. Peng, Z. He, C. Xie, X. Jin, and H. Jiang (2025) GCTAM: global and contextual truncated affinity combined maximization model for unsupervised graph anomaly detection. In Proceedings of the International Joint Conference on Artificial Intelligence, p. 3642–3650. Cited by: §I-A, TABLE I, TABLE I, Fig. 3, §V-A, TABLE I, TABLE I, TABLE V, TABLE V. [62] H. Zhao, A. Chen, X. Sun, H. Cheng, and J. Li (2024) All in one and one for all: a simple yet effective method towards cross-domain graph pretraining. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, p. 4443–4454. Cited by: §IV-A. [63] T. Zhao, C. Deng, K. Yu, T. Jiang, D. Wang, and M. Jiang (2020) Error-bounded graph anomaly loss for gnns. In Proceedings of the ACM International Conference on Information & Knowledge Management, p. 1873–1882. Cited by: §I-A. [64] Y. Zhao, Y. Liu, S. Li, Q. Chen, Y. Zheng, and S. Pan (2025) Freegad: a training-free yet effective approach for graph anomaly detection. In Proceedings of the ACM International Conference on Information and Knowledge Management, p. 4379–4389. Cited by: §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I. [65] Y. Zheng, M. Jin, Y. Liu, L. Chi, K. T. Phan, and Y. P. Chen (2021) Generative and contrastive self-supervised learning for graph anomaly detection. IEEE Transactions on Knowledge and Data Engineering 35 (12), p. 12220–12233. Cited by: §I, §I-A, TABLE I, TABLE I, §V-A, TABLE I, TABLE I. [66] Q. Zhou, J. Chen, H. Liu, S. He, and W. Meng (2023) Detecting multivariate time series anomalies with zero known label. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 37, p. 4963–4971. Cited by: §IV-B. [67] Q. Zhou, S. He, H. Liu, J. Chen, and W. Meng (2024) Label-free multivariate time series anomaly detection. IEEE Transactions on Knowledge and Data Engineering 36 (7), p. 3166–3179. Cited by: §I-B.