Semi-supervised Graph Anomaly Detection via Robust Homophily Learning
Authors: GuoguoAi
Hezhe Qiao
Hui Yan
Guansong Pang
Venue: NeurIPS 2025 poster ()
Keywords: Graph Anomaly Detection
Adaptive Graph Neural Networks
Frequency
One-class classification
Primary area: deep_learning
TL;DR: we propose RHO, namely Robust Homophily Learning, to adaptively learn such homophily patterns. RHO consists of two novel modules, adaptive frequency response filters (AdaFreq) and graph normality alignment (GNA).

Abstract: Current semi-supervised graph anomaly detection (GAD) methods utilizes a small set of labeled normal nodes to identify abnormal nodes from a large set of unlabeled nodes in a graph. These methods posit that 1) normal nodes share a similar level of homophily and 2) the labeled normal nodes can well represent the homophily patterns in the entire normal class. However, this assumption often does not hold well since normal nodes in a graph can exhibit diverse homophily in real-world GAD datasets. In this paper, we propose RHO, namely Robust Homophily Learning, to adaptively learn such homophily patterns. RHO consists of two novel modules, adaptive frequency response filters (AdaFreq) and graph normality alignment (GNA). AdaFreq learns a set of adaptive spectral filters that capture different frequency components of the labeled normal nodes with varying homophily in the channel-wise and cross-channel views of node attributes. GNA is introduced to enforce consistency between the channel-wise and cross-channel homophily representations to robustify the normality learned by the filters in the two views. Experiments on eight real-world GAD datasets show that RHO can effectively learn varying, often under-represented, homophily in the small labeled node set and substantially outperforms state-of-the-art competing methods. Code is available at \url{https://github.com/mala-lab/RHO}.

Semi-supervised Graph Anomaly Detection via Robust Homophily Learning
Guoguo Ai Hezhe Qiao 2+, Hui Yan 11, Guansong Pang
'Scoool of Computer Science and Engineering, Nanjing University of Science and Technology School of Computing and Information Systems, Singapore Management University {guoguo, yanhui jenjust edu. en, hezheqiao. 2022@phdcs. smu. edu. gspangêsmu.c edu.
Abstract
Current semi- superviuee graph anomaly detection (GAD) methods utilizes small set of labeled normal nodes to identify abnormal nodes from a large set of unlabeled nodes a graph. These methods posit that 1) normal nodes share similar level of homophily and 2) the labeled normal nodes can well represent the homophily pat- terns in the entire normal class. However, this assumption often does not hold well since normal nodes in a graph can exhibit diverse homophily in real-world GAD datasets. In this paper, we propose RHO, namely Robust Homophily Learning, adaptively learn such homophily patterns. RHO consists of two novel modules, adaptive frequency response filters (AdaFreq) and graph normality alignment of adaptive spectral filters that capture different frequency components of the labeled normal nodes with varying homophily in the channel-wise and cross-channel views of node attributes. GNA is introduced to enforce consistency between the channel-wise and cross-channel homophily representations to robustify the normality learned by the filters in the two views. Experiments eight real-world GAD datasets show that RHO can effectively learn varying, often under-represented, homophily in the small labeled node set and substantially outperforms state-of-the-ard competing methods. Code is available at https://gitub..com//e lab/RHO.
(GNA). AdaFreq learns
Introduction
Graph anomaly detection (GAD) has received increasing attention in recent years due to its wide range of applications, such as fraud detection in finance, review spam detection, and abusive user detection 27,37]. Semi-supervised GAD, which aims to leverage small set of labeled normal nodes to identify abnormal nodes from a large of unlabeled nodes in graph, is among the most realistic problem settings in real-life applications [33, 37,337,, This is because GAD datasets are typically dominated by normal nodes, which can greatly facilitate the annotation of few normal nodes, e.g., if we randomly sample a few nodes from a large graph, most nodes would be normal. Current supervueeee GAD methods are generally built upon two key assumptions: (1) all normal nodes exhibit similar level of homophily. and (2) the labeled normal nodes can well represent the overall homophily pattern the entire graph [6. 7, 12, 3..42.. However, these two assumptions often do not hold well in the real-world GAD datasets. This is because although normal nodes generally exhibit high homophily, there can large variations homophily across the normal nodes, some of which can possess relatively low homophily (see Fig. la for visualization of this observation on the
"Equal Contribution. The work was done when Guoguo Ai visited Singapore Management University. Corresponding Authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Amazon [10] and Elliptic [43]. Additional visualizations can be found in App. D.1). As result, the GAD methods that rely on the aforementioned assumptions learn inaccurate homophily patterns of the normal class, leading to the misclassificaiton of low-homophily normal nodes as abnormal. This particularly true when they use the popular neighborhood aggregation mechanisms that are prone to produce oversmooth homophily representations [37].
To address this issue, we propose RHO, namely Robust Homophily Learning, to adaptively learn heterogeneous normal patterns from the small set of normal nodes with diverse homophily.
RHO consists of two novel modules, including adaptive freqency response filters (AdaFreq) and graph normality alignment (GNA). Unlike existing filters that are bounded by prior knowl- edge of specific homophily, AdaFreg learns set of adaptive filters with learnable parameters to robustly fit normal nodes of varying levels of homophily. For example, as illustrated in
50000
Number of Normal Node:
20000
10000
600
Areeac
Ellipto
200
Fig. lb and Ic, conventional GCN filters [42] rely on a low-frequency assumption of normal nodes, while the spectral filters in BWGNN [41] learns pre-defined type of frequency response, both of which fail to learn generalized represen- tations for normal nodes across three different homophily variations; AdaFreq is instead op- timized with learnable parameters to fit adap- tively to the normal nodes with three differ- ent homophily cases, showing much more ro- bustness than the other two methods. In RHO, AdaFreq is utilized to learn such adaptive filters in both channel-wise and cross-channel views of node attributes, enabling the learning of het- erogeneous normal patterns in diverse feature channels of the labeled normal nodes.
1090.11
(0.0
(01033)
(0 (0505)
(0 (07.88)
Rango Homophilly Rate
(2)
100
Bilyne
on
AUROC
80
AUROIC
75
-70
90
GCN Beipee
-0
Adaiie
Addic
40
Cass3
Cases
Hoooppily Desrrbuttion Casse
Hamophily Disriiution
(b)
(c)
Figure 1: (a) Homophily distribution of normal nodes in Amazon [10] and Elliptic [43]. (b) and (c) AUC comparison of GCN filter [42]. BWGNN filter [41], and AdaFreq filter on the two datasets under three cases of homophily levels. The three cases are normal nodes sampled from the sets of how-hoompphily, (ighhhomophily) normal nodes, where nodes with homophily greater than 0.9 for Amazon and 0.7 for Elliptic are considered as high homophily. The datasets in the three cases are sampled using a ratio of (80%, 20%), (50%, 50%), ) and (20% 80%) to the two node sets, respectively.
The set of filters learned in AdaFreq can dif- fer from each other substantially. To obtain ro- bust yet consistent representations of the nor- mal nodes, the GNA module is introduced to enforce consistency of homophily representa- tions learned across the filters. To this end, GNA constructs positive pairs using the repre- sentations from the respective shannel-wise and cross-channel views the corresponding nodes, and then maximizes the similarity between the representations learned from these two views while minimizing the similarity of negative pairs composed by representations from different nodes.
In doing so, RHO learns heterogeneous normal representations of normal nodes with varying degrees of homophily via AdaFreq, while ensuring the consistency of the learned normality via GNA. In summary, this work makes the following three main contributions.
We reveal the failure of fitting existing graph filters to a subset of normal nodes with varying levels of homophily and introduce RHO to mitigate this issue. RHO is a novel approach for the semi-supervised GAD that can learn robust and consistent normal patterns for a given set of labeled normal nodes with diverse levels of homophily.
We further introduce two novel modules, AdaFreq and GNA, to implement RHO. AdaFreq learns set of complementary adaptive filters over channel and cross cross-channel repre- sentations to effectively capture heterogeneous normal patterns from the labeled normal nodes of different levels of homophily. GNA robustifies these representations by aligning the normality representations learned by the filters in channel-wise and cross-channel views.
Extensive experiments on eight real-world GAD datasets indicate that RHO performs significantly better than the oftt--of-theanne (SotA) semi-supervised GAD methods.

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2 Related Work
2.1 Graph Anomaly Detection (GAD)
Non-spectral GAD Methods. Many non con-sepetral GAD methods apply traditional anomaly detection techniques GNN backbone, such as reconstruction [7, 12], adversarial learning [5,6], one-class classification [2, 42, 48). Other methods are based on measures designed for GAD, such as the recently proposed affinity-based methods [30-32,36) These methods are typically designed for the unsupervised setting, where no labeled nodes are available. They can be easily adapted to the semi-supervised setting by refining their objective to the labeled normal nodes. GGAD [38] is the first method specifically designed for the semi-supervised setting by generating the outliers and training discriminative one-class classifier with the given labeled normal nodes. These methods failed to consider the homophily discrepancy among the normal nodes. Our proposed RHO addresses this by employing an adaptive filter that learns normal representations from both cross-channel and chamnel-wise views through a one-class optimization objective.
Spectral GAD Methods. The spectral GAD methods focus on the filter design in GNNs to have more effective learning the representations of nodes [3, 13,401) 40]. Among them, AMNet [3], BWGNN [40], and GHRN [13] aim to utilize the frequency signal using different spectral-based GNNs to distinguish between normal and abnormal nodes. Although these spectral-based GAD methods have achieved remarkable success [9, 19,24, they are typically designed for fully supervised setting where both labeled normal and abnormal nodes are available during training. On the other hand, the filters they utilize learn only predefined frequency information, making them difficult learn expressive representations for normal nodes of different frequency components. RHO leverages AdaFreq to assign learnable parameters different feature channels to effectively learn heterogeneous normal representations from the set of normal nodes of varying homophily.
2.2 Graph Homophily Modeling for GAD
Graph homophily modeling methods aim to address the homophily discrepancy in graphs, where target nodes often connect to nodes from different classes [14, 24,266,55, This inter-class connectivity negatively affects the quality of node representations within each class. The problem becomes more pronounced in GAD, where the homophily gap is typically large due to the inherent imbalance in GAD datasets [25,37, 45]. Existing studies Graph homophily modeling for GAD primarily focus on neighbor selection during message propagation by adding or cutting edges, or by assigning different weights to each edge [10, 21,36. In addition, several methods employ advanced strategies to address the homophily discrepancy in message passing, such as gradient- based filtering [13], meta-learning [8, 391, and data augmentation [4,29, 47]. These methods overlook the homophily differences among the labeled normal nodes in semi-supervised settings. our proposed RHO the first work to reveal this issue and address it by robust homophily learning approach.
3 Problem Statement
Notations. Given an attributed graph 9 = (V, X), where V denotes the node set with 0; E V and [V] = N, & denotes the edge set, and X [x1 X2...., . xN] R. Nx is a set of node attributes. Each node v; has --dimensional feature representation x,. The topological structure of 9 is represented by an adjacency matrix A. D denotes a degree matrix which is a diagonal matrix with
D Σ, Aij- Normalized Laplacian matrix L is defined by L IN D 1 AD- 2, where IN NNN denotes identity matrix. Å is the normalized adjacency matrix. The corresponding degree matrix D and Laplacian matrix are defined as D D + II and = IN D A/D 1/2 respectively. The node homophily [34] for a node v defined as HC = HalueN u. b where d, is d, the number of neighbors of node u, N2 is the set of adjacent nodes of u, and ye represents the label of node
Semi-supervised GAD. Let Va. V,, be two disjoint subsets V. where V., represents abnormal node set and V,, represents normal node set, and typically the number of normal nodes is significantly greater than the abnormal nodes, ie., |Val > [Vll then the goal of semi-supervised GAD is to learn the mapping function o R. such that Φ(v) < o(v'), where Ve E Vn. V. given set of
3

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Graph
(a) Adaptive Frequency response Filter (AdaFreeq)
Cross-chamall representation (oer)
Leer
High homophilic nodes homophalic nodes
((A;;
Absormal nodes
(b) Gruph Normality Aligament (GNA)
Anomaly Scoring
D
Channel-wise representation (twr)
t((A)
LoN +p
Unlabeled Nodes
Laheled Normal Nodes
Pull
Inter-Push
Inter-Pull
Decision
Channel
Lyer
Intra-Push
boundary
Figure 2: Overview of the proposed RHO framework. The input graph consists of labeled normal nodes and unlabeled normal/abnormal nodes, where nodes Ui, fi E V2 represent normal nodes with high and low homophily, respectively. (a) AdaFreq learns adaptive filters on both cross-channel and channel-wise representations with learnable parameters to different feature channels. (b) GNA aligns the heterogeneous normal patterss learned from the adaptive filters in the two views. (c) The normal nodes with diverse homophily are enforced to project closer to the center of hypersphere via widely-used one-class loss, while anomaly nodes being distant from the center.
labeled normal nodes V, C Vn and no access to any labels of the abnormal nodes. V. = V//V set of the unlabeled nodes and used as test data.
Graph Filter. The Laplacian matrix L can be decomposed as L UAUT. where U (u,, u2...., Uyy ) is a complete set of orthonormal eigenvectors known as graph Fourier modes and A = diag (A:) is a diagonal matrix of the eigenvalues of L, where \ S )2 < ... < Av and A E 1.5] [44]. Taking the eigenvectors of normalized Laplacian matrix as a set of bases, graph Fourier transform of signal R graph 9 is defined X (f1,... _ and the inverse graph Fourier transform x Ux. The convolution between the signal x and convolution kemel| is follows:
U
- ( o (U*x)) = U*x
(1)
where @ is element-wise product and is diagonal matrix representing the convolution kemel the spectral domain, serving as a substitute of U f.
4 Methodology
4.1 Overview of the Proposed Approach RHO
As shown in Fig. 2, RHO consists of three main components: (1) Adaptive frequency response (AdaFreeq filters for heterogeneous normal pattern learning, which is simultaneously applies to both cross-channel and channel-wise view; (2) Graph normality alignment (GNA) for aligning the learned normality from two views using contrastive learning objective; and (3) the model is lastly optimized with a widely-used one-class objective, along with a contrastive loss in GNA, to learn robust homophily patterns GAD datasets with diverse levels of homophily. We also summarize the workflow RHO and provide a detailed algorithmic description App. E.
AdaFreq: Adaptive Frequency-response Filters
The conventional graph filter used in existing GAD methods is typically fixed, which fails to capture diverse homophily relations of normal nodes. To this end, we are dedicated to learning adaptive frequency response filters that dynamically adjusts the response strength of different frequency components, effectively preserving the consistent frequency components of labeled normal nodes with diverse homophily. A straightforward strategy to control the varying effects of different frequency

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components is to allocate learnable parameter for each frequency component. However, this solution requires explicit cigen-decommositton which is too expensive. To avoid eigen-decomposition of the graph Laplacian, we propose a simple yet effective filter by introducing trainable parameter & to adjust the frequency response, as follows:
(2) where the learnable parameter & governs the degree and type of frequency response imposed by the filter at each layer of GNNs. We show theoretically below that this design allows our filter g(\) to adaptively capture the consistent frequency patterns of normal nodes, regardless of having low- or high-homophily in the normal nodes.
= -1 Kd,
Theorem Let {Am} and {um] be the graph frequencies and frequency components respectively, 3m is the projection coefficient of signal x onto the m-th eigenvector Um, then we have g (Am) Exev, Mmn(i)
Bm Ligr, umilff for the filter (A), indicating that frequencies where normal nodes show coherent spectral behavior (i.e., (i) values agree in sign/magnitude) are amplified, while inconsistent frequency components are suppressed.
The proof can be found in App. A. According Theorem 1, the designed adaptive filter g (A) can preserve the frequency components with consistent spectral behavior of labeled normal nodes by training the parameter k. When k > 0, g()) decreases with respect X, meaning that frequency components associated with larger eigenvalues (i.e., seegffryyy components) are suppressed, while fow-frequeney components are preserved. Conversely, when k 0, g()) emphasizes high- frequency components. The magnitude of k controls the strength of frequency responses. When k 0, the filter g\) acts an all-pass filter preserving the raw graph signal. When K layers are stacked, g(A) = - k;)), a set trainable parameters {ki, ... .x)) can produce more complex frequency responses. Therefore, g() can dynamically learn the parameter & to capturing the consistent frequency components of normal nodes with diverse homophily, enabling robust homophily representation learning for GAD.
Motivated by the intuition that heteroge- neous homophily information is embedded in different feature channels, we leverage this adaptive frequency response function to learn heterogeneous patterns of the nor- mal nodes in the cross-channel and channel- wise views as follows.
1000
130
14
5
Awaeb Suee (b) Chasnel-wii wiie
(a) Cross-chantel
(c) RHO
Homophily Learning across the Chan- nels. According to Eq. (1) and (2), AdaFreq can be utilized to learn a sin- gle learnable parameter k E shared across all node attributes in the cross- channel view. The learned cross- channel representations, denoted by (!) 1-th layer, can be defined as:
(d) Crese-chancel
(1) RHO
cer for the
Figure: 3: Anomaly score distributions and t-SNE visu- alizations of node embeddings generated by the cross- channel view ((a) and (d)), the channel- -onnel-wise view ((b)
(()) = a(I HL) ) eer
(3)
and (e)), and the full RHO model ((c) and (f) on Ama- zon [10].
where o-)) is a non- non-linear activation func- tion.
fo(X) with fo be a two-layer MLP, and w:()) cer the learnable weight matrix.
To focus our filter on normal patterns, we map the representations of the normal nodes, H close to one-class hypersphere center Ceer in the embedding space. To achieve this, the training objective minimizes the average squared distance between the embeddings and the center:
Eeer
1
Σ (1) - cear VEV.
I|
+ Σ
[ 12
(4)
|V
where h (T) (i) denotes the representation of i-th node, i.e. i-th row of H and Coer = N e
Σ h (i). Optimizing Beer enables the filter to leam the heterogencous normal patterns
(2)
5

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from the cross coas-channee perspective, encouraging normal nodes of different homophily to cluster around the center Ceer: while anomalous nodes are distant from the center, as illustrated in Fig. 3a and the t-SNE visualization in Fig. 3d. However, relying solely on eeer may result in some anomalous nodes being located near the center, likely because they are camouflaged anomalies exhibiting distribution patterns similar to those of normal nodes in the cross-channel view.
Homophily Learning in Individual Channels. To capture normal patterns complementary those in H we utilize AdaFreq in individual feature channels that can contain complementary homophily the cross-channel view. To this end, learn d adaptive filters with each equipped with learnable
frequency response parameter, K = [k1, k2 _kkd] E R where d is the number of channels in the
feature layer of the MIP network in fo, learn the channel-wise representations H H
H (T L. h (t-1) (T- k2I) h2 (c-1)
kaL() h (e-1)
W(O CWTT
(5)
where [h] (t)1)
h2 (1-1) h 1 - E Rxd, and H = (X).
These d learnable frequency parameters allow the model to capture various homophily variations our homophily pattern learning in individual channels. Alternatively, the filtering process can be compactly expressed using the Hadamard product as:
H o a(I - É) (H((")) @ K)W c.I.).
(6)
The same one-class objective is applied to the channel wise view, with its loss Cow defined as
OUT
M
Σ
our (i)
Cowr
2
Σ (
LEV
where h (2) is the i-th row of HT, Tis the number of layers, and Ceur = 1 Σ (1). By V
optimizing Lowr, the model captures normal representation patterns that differ from those in the cross-channel view, as exemplified in Fig. 3b and Fig. 3e. However, placing much emphasis on channel-wise features may cause certain normal nodes to drift away from the center, leading to some false positive detections.
Inspired by the complementary homophily information, AdaFreq learns the adaptive filters in both the cross-channel and channel-wise views model the heterogeneous normal patterns. As shown in Fig. 3c and Fig. 3f, when we jointly leverage the two complementary views, normal nodes are more effectively clustered, and misclassified anomalies in Fig. 3d successfully detected. Note that there are two centers since we apply AdaFreq to both views, and the GNA module we introduce in the next section is designed to align the heterogeneous normality represented by the two centers.
4.3 GNA: Graph Normality Alignment
There can be heterogencous homophily representations learned in the two views above. We thus introduce GNA to mitigate this issue in AdaFreq. Specifically, for each node, we consider the representations generated by the adaptive filters as a positive pair, while representations from other nodes within the same batch are treated as negative pairs. GNA then promotes the normality alignment by maximizing the similarity of positive pairs and minimizing the similarity of negative pairs. Let Zeer (i), Zewr (1) E Rd be the representations of node v, under the two views, which can be obtained using Zeer (i) &peer * (i)) and Zewr (i) = Poorr (heur (i)). where (.)r(.) and Powr (-) are two MLP functions, then given the positive pair (Zeor (i), Zewr (i)), the alignment is formulated as
e (sim (Zeer (i))
[(Zeer (i), Bewr (i)) -log Eiys (Siim((eer (1) (i)T)) + Ein e (sim((acr (0))7)
(8)
where sim(..) denotes cosine similarity, and T is temperature hyperparameter.
To better learn similarity of two views, we further designed an auxiliary loss module CC Zeeerr (i), Zee (i)) treating channel-wise view anchor, contrasting to the L (Zeewr()) (i), Zeer (i)) that treats the cross-channel view as the anchor. Therefore, the overall objective of LANA is defined
LONA 2N
(C (C (1), Zozer (1)) + L L(Zeourr (i), Zeer (i)))
1=1

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By minimizing LGNA, the model is encouraged to capture consistent normality patterns across the cross-channel and channel-wise views, achieving robustness homophily representation learning, with better ability to detect the abnormal nodes, e.g., the hard anomalies in Fig. 3c and 3f.
4.4 Training and Inference
Training. During training, RHO is guided by the one-class loss to learn heterogeneous normal patterns through the filters in AdaFreq and the alignment in GNA. To be specific, the total loss function total formulated combination of the one-class classification loss, applied to the two views in AdaFreg, and the alignment loss LGNA A in GNA.
1 (Eeer + Lowr) ) + aLGNA,
Lotaal
(10)
2
where a E [0, 1] is a hyper-parameter to adjust the influence of LGNA in the overall optimization.
Inference. During inference, the anomaly score of node v; E V.. is defined as the squared Euclidean given node and the one-class center. In RHO, we
distance between the learned representations compute the average distances of each node to both the cross-channel and channel- -wine centers. The anomaly score S; for node 02 is thus defined as
1 S. 2 cer (1)
||²
Cewr |3)
Coor
+ 1 (1)
(11)
The abnormal nodes are located farther from the centers compared to the normal nodes and are therefore expected to have higher anomaly scores than the normal nodes.
4.5 Complexity Analysis
The computational complexity of RHO consists of three parts: (1) The two-layer MLP used for initial feature transformation has a complexity of O(Nd(M + d)), where denotes the number of nodes, M is the input feature dimension, and d represents the hidden dimension. (2) RHO employs two independent views, including the cross-channel and channel-wise views. Each view incurs a computational cost of O(EE d + Nd where & is the number of edges. The overall complexity of this component remains O ([Eld+ N) in practice. (3) In GNA, the projection head has complexity of d(Nd)). and the alignment loss computation has complexity of O(Nbd), where b denotes the batch size. Therefore, the overall time complexity is O + 2 Eld 4Nd Nbd)). Empirical results for running time can be found in App. D.4.
5 Experiments
Datasets.
We evaluate RHO eight real-world GAD datasets of diverse size from different domains, including social networks Reddit [18] and Questions [35], co-review network Amazon [10], co-purchase network Photo [28], collaboration network Tolokers [28], financial networks T-Finance [41], Elliptic [43], and DGraph [15]. See App. B for more details about the datasets.
Competing Methods. The competing methods can be categorized into reconstruction methods (in- cluding DOMINANT [7] and AnomalyDAE [12]), one-class classification OCGNN [42], adversarial learning (including AEGIS [6] and GÀAN [5]), affinity maximization method TAM [36], generative method GGAD [38], partitioning message passing method (PMP) [49], and data augmentation method CONSISGAD [4]. Except for GGAD that is specifically designed for the semi-supervised setting, the fully supervised GAD approaches, which are adapted to the semi-supervised scenario by refining the training set to the labeled normal nodes, following [38]. We also compare RHO against conventional GCN filter and an advanced spectral-based filter used in BWGNN [41]. Note that for the fully supervised methods, we utilize only their proposed filter as the encoder and apply it within our semi-supervised setting by optimizing the encoder using the one-class loss. See App. C for more details about the competing methods. Note that some unsupervised methods, such as CoLAA [23], SL-GAD [46], and GRADATE [11], are designed with proxy tasks and cannot be adapted to our setting, which are excluded from our comparison.
other methods are originally unsupervised
7

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Table 1: AUROC and AUPRC on eight GAD datasets. The best performance is boldfaced, with the second-bes best underlined. 'f" indicates that the model cannot handle DGraph.
Datasets
Metric
Methods
Reddit
Tolokers
Photo
Amazon
Elliptic
Question
T-Finance
DGraph
DOMINANT
0.5194
0.5121
0.5314
0.8867
0.3256
0,5454
0,6167
0.5851
Ano
0.5280
0.6074
0.5272
0,9171
0.5409
0.5347
0,6027
0.5866
OCGNN
0,5622
0.4803
0.6461
0.8810
0.2881
0.5578
0,5742
/
AEGIS
0,5605
0.4451
0.5936
0.7593
0,5132
0.5344
0,6728
0.4450
GAAN
0.5349
0.3578
0.4355
0.6531
0.2724
0.4840
0,3636
/
AUROC
TAM
0.5829
0.4847
0.6013
0.8405
0.4150
0.5222
0,5923
/
GGAD
0.6354
0.5340
0.6476
0.9443
0.7290
0.5122
0,8228
0.5943
GCN
0.5523
0,4954
0.5727
0.7345
0.6463
0,4556
0.7262
0.5112
BWGNN
0,5580
0.5821
0.6861
0.8312
0.7241
0.5740
0.7683
0.4958
PMP
0.5472
0.5815
0,5844
0,8329
0,5617
0.5790
0.8321
0.5376
CONSISGAD
0,5347
0,5974
0.5859
0.8715
0.7354
0.5737
0.8277
0.5735
RHO
0.6207
0.6255
0.7129
0,9302
0.8509
0.5833
0.8623
0.6033
DOMINANT
0.0414
0.2217
0.1238
0.7289
0.0652
0.0314
0.0542
0.0076
AnomalyDAA
0.0362
0.2697
0.1177
0.7748
0.0949
0.0317
0.0538
0.0071
OCGNN
0.0400
0.2138
0.1501
0.7538
0.0640
0.0354
0.0492
/
AEGIS
0.0441
0.1943
0.1110
0.2616
0.0912
0.0313
0.0685
0.0058
GAAN
0.0362
0.1693
0.0768
0.0856
0.0611
0.0359
0,0324
/
AUPRC
TAM
0.0446
0.2178
0.1087
0.5183
0.0552
0.0391
0,0551
GGAD
0.0610
0.2502
0. 0.1442
0,7922
0.2425
0.0349
0,1825
0.0082
GCN
0.0420
0.2351
O. 0.1257
0.1617
0.1176
0.0405
0.1387
0.0040
BWGNN
0.0360
0.2687
0.2202
0.3797
0.1869
0.0410
0.1436
0.0041
PMP
0.0388
0.2939
0.1254
0,3582
0.1006
0.0383
0,4084
0.0046
CONSISGAD
0.0360
0.3059
0. 0.1319
0.6523
0.1765
0.0424
0.4152
0.0050
RHO
0.0616
0.3256
0.2337
0.7879
0.5095
0.0430
0,4893
0.0059
Evaluation Metries. Following previous studies [19, 22,38,, evaluate the detectors using two
popular metries: AUROC (Area Under the Receiver Operating Characteristic curve) and AUPRC (Area Under the Recall Curve). AUROC assesses the model's overall ability distinguish between normal and anomalous nodes across varying thresholds. AUPRC measures the precision- recall tradeoff. For each method, report the average results over five independent runs.
Implementation Details. The RHO model is implemented in PyTorch 2.0.0 with Python 3.8 and executed on GeForce RTX 3090 GPU (24 GB). RHO is trained using the Adam optimizer [16] with weight decay of 5e 55 We set the default learning rate to 5e -3 Nevertheless, owing to the variations in edge density across differentt graphs, we find that models trained on sparser graphs are generally more sensitive to large learning rates and thus benefit from smaller ones to ensure stable convergence. Therefore, we use learning rate of 54 for the Elliptic and Question datasets, and further decreased to 5e-6 for the extremely sparse dataset, DGraph. The hyperparameter a is set to 1.0 for all datasets except the small datasets Reddit and Photo, which require less regularization. It is set 0.1 in these two datasets. A detailed analysis of this hyperparameter is presented in Sec. 5.4. Following previous work [38], we randomly sample 12% of the normal nodes labeled data for training, where R E (5. 10, 15, 20). To ensure fair comparison, we obtain the publicly avviile-availabll official source code of all competitors and execute these models using the parameter settings suggested by their authors.
5.1 Main Comparison Results
The main comparison results are shown in Table where all models use 15% labeled normal nodes during training. RHO outperforms the semi-supervised methods on six datasets having maximally 12.19% AUROC and 30.68% AUPRC improvement over the best competing method GGAD. GGAD achieves the highest AUROC on Amazon and Reddit; however, it underperforms RHO on the other datasets. This suggests that generative approaches may not generalize well across different datasets where normal nodes have varying levels of homophily. In contrast, RHO achieves SotA performance on a variety of graph datasets without relying on any generated anomalies, by learning the varying normal patterns from different datasets. The fully supervised methods PMP and CONSISGAD
8

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perform less effectively when applied in the remi-supervised setting, since the lack of labeled anomalies prevents them from leveraging crucial abnormality information. The reconstruction-bast method, AnomalyDAF, yields good performance on Tolokers and Amazon, but it still underperforms methods are prone to overfitting
RHO on most datasets. The main reason that reconstruction- part of the prevalent homophily patterns (e.g., low-frequency nodes) and struggle to handle hard normal samples (e.g., those that have high homophily). In contrast, RHO leverages AdaFreq to effectively learn robust normal patterns from the normal nodes with diverse homophily, leading to substantially improved performance. As for one-class methods, OCGNN is based on non-adaptive filters in learning the normal patterns; it underperforms RHO across all datasets. Moreover, RHO also consistently outperforms GCN and BWGNN under the semi-supery supervised setting due to its superior capability in learning heterogeneous frequency components.
5.2 Analysis of Adaptive Filters in AdaFreq
We perform a filter analysis examine the contribution of different frequency compo-
1.4
nents in RHO. In Fig. 4, we provide a qualitative comparison between the filter in
1.2
12
1.0-
1.0
0.8
GCN and the d adaptive filters learned
160.6
0.6
by RHO in the two views. We observe that: (i) AdaFreq in cross-channel view flexi- bly retains varying levels of low-frequency signals across two datasets with diverse homophily patterns in the normal nodes, thereby enabling more effective learning of the heterogeneous normal representations. (ii) The channel-wise learnable filters, on the other hand, retain enhance the dis- criminability of the frequency components different levels, allowing RHO to capture important homophily patterns specific to each feature channel. These two capabilities sharply contrast to the simple, fixed frequency pattern in GCN and its uniform treatment of all feature channels.
0.4
0.4
0.0
0.0
0.0
0.5
1.0
1,5
0.0
0.5
1.0
1,5
A
(a) Amazon
(b) T-Finance
Figure 4: The filter curves learned by RHO on Amazon and T-Finance, where the black dashed line represents the baseline filter response from GCN, the black solid line indicates the cross-channel response, and the col- ored solid lines are the channel-wise responses.
5.3
Ablation Study
In this section, we perform ablation study to evaluate the contribution of each component in RHO.
Table 2: AUROC and AUPRC results of our ablation study.
Component
Datasets
Metric
Leer
hewr
EGNA
Reddit
Tolokers
Photo
Amazon
Elliptic
Question
T-Finance
DGraph
0,5756
0,5712
0.6322
0.7436
0.7371
0.5704
0,7682
0.5542
0.6095
0.5878
0.6144
0.8905
0.7718
0.5811
0.8123
0.5215
AUROC
V
/
0.6117
0.5727
0.6361
0.8692
0.7954
0.5774
0.7756
0.5637
d
V
0.6207
0.6255
0.7129
0.9302
0,8509
0.5833
0.8623
0.6033
0.0431
0.3052
0.1390
0,2423
0.1862
0.0424
0.3282
0.0049
0.0481
0.2923
0.1451
0.5895
0.2287
0.0420
0.5061
0.0043
AUPRC
0.0556
0.3024
0.1398
0,5443
0.2657
0.0426
0.3365
0.0051
0.0616
0.3256
0.2337
0.7879
0.5095
0.0430
0.4893
0.0059
To evaluate the effectiveness of GNA in RHO, we assess the variant of RHO with LGNA removed and train the model using only Leer and Lowr. As shown in Table 2, removing GGNA consistently decreases performance in terms of both AUROC and AUPRC. This indicates that GGNA is crucial for bridging the gaps among the heterogeneous normal representations learned by the chie and cross-channel views.
Effectiveness of GNA.
Effectiveness of AdaFreq. To verify the effectiveness of the proposed AdaFreq, we present the results of using AdaFreq in only one the two views (i.e., using solely Leer or Lewr). Table 2 shows that AdaFreq applied to both views consistently outperforms the variant that applies AdaFreq in only one of these views. This performance improvement is primarily attributed to the complementary nature of the two views. The homophily patterns captured in the channel-wis view contain distinct

---

information that cannot be fully learned from the cross-channel view alone, as exemplified in Fig. 4. The full model of RHO effectively integrates both views, resulting in improved performance across the datasets.
5.4 Hyperparameter Sensitivity Analysis
We evaluate the sensitivity of RHO w.r.t. the hyperparameter a and the training size R. Detailed results on more datasets can be found in App. D.
Performance w.r.t. Hyperparameter a.
As shown in Fig. 5, with increasing a, the perfor- mance on Elliptic, Tolokers, Amazon, Question and T-Finance has different degrees of improve- ment, suggesting that stronger consistency con- straint between cross-channel and channel-wise representations is beneficial. In contrast, on Red- dit and Photo, the performance slightly declines as a increases. This is primarily because these small datasets with high average feature sim- ilarity among the nodes. The representations learned from the two views are already highly similar, and imposing excessive regularization through the normality alignment may lead to overly smoothing features, which may lead to negative effects.
Roddit -+-
Photo
-*
Elliptic
T-Finance
Tolokers
Amazon
Question
1.0
0.8
0.9
0.5
0.4
0.5
04
0.3
0.1
0.4
0.7
1.0
0.0
0.1
0.4
0.7
1.0
Panimoter (40 Value
Prommcter (00 Volte
(a)
(b)
Figure 5: AUROC and AUPRC results of RHO w.r.t. hyperparamer a.
Performance w.r.t. Training Size R.
To explore the impact of training size R. we compare RHO with GGAD, OCGNN, and BWGNN, using varying numbers of training normal nodes, with the results reported in Fig. 6. As the number of the labeled normal nodes increases, RHO generally improves across the datasets, which is consistent with other semi- supervised methods. Notably, RHO demon- strates stable and remarkable improvement trend on the Elliptic dataset, even when the amount of labeled data limited. This observa- tion indicates that our model can effectively exploit the normality patterns under sparse supervision. Under different label rates, our RHO consistently maintains the best performance, showing the superiority of our method.
0.9
0.75
A0300
D9XGNN
ADROC
GGAD
0.5
OCGNX
RH1
0.85
0.10
0.15
0.20
Training Size (R)
Triiiin
(a) Elliptic
(b) Photo
Figure 6: AUROC w.r.t. data size R.
Conclusion and Future Work
In this paper, we propose RHO, the very first GAD approach designed learn heterogeneous normal patterns on a set of labeled normal nodes. RHO is implemented by two novel modules, AdaFreq and GNA. AdaFreq learns view of node attribute to capture the heterogeneous normal patterns from the given limited labeled normal nodes, while GNA designed enforce the consistency of the learned normal patterns, thereby facilitating the learning of robust normal representations on datasets with different levels of homophily in the normal nodes. This robustness is comprehensively verified by results on eight GAD datasets. A potential limitation of RHO is that it is transductive method and thus cannot be directly applied in inductive settings. This limitation is shared with most existing GAD methods and is left. for future exploration.
of adaptive spectral filters in both the cross-channel and channel-wise
10

---

Acknowledgments
This research supported by the Ministry of Education, Singapore under its Tier- Academic Research Fund 14-SSISSSMU-O0S A*STAR under its MTC YIRG Grant (M24N8c0103), and the Lee Kong Chian Fellowship.
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A Theoretical Analysis
graph filter operation on graph signal E IN can be defined as Ug(A) x, where U = (uo u1..... -1) is a complete set of orthonormal eigenvectors of the Laplacian matrix, g(A) is an adaptive filter with diagonal matrix and g(>o). g(1).. (\x=11) its main diagonal. In graph signal processing (GSP), {Ar} and {u;) | are called frequencies and frequency components of graph 9. Let V, the labeled normal nodes our training data and e be a center of a one-class classifier, then the objective function of the one-class classification under semi-supervised graph anomaly detection (GAD) can be defined as:
Definition A.1 (Spectral One-class Loss) Let 3 U x, then in the one-class classification scenario, the loss of filter (()) on graph can be formulated as follows:
L Σ (I(Ug(A)3) -dl Σ 1| -cl;
(12)
iEV.
iEV.
where B -Bu, B1, 11 with B... be the projection coefficient of on the m-th eigenvector um (i.e,, x -N-11 3m Um). and the adaptive filter g(A) operates on each spectral component, L.., (AA)3 (o. (\1)) ... 9(AN-1)8N-1) Then, through the inverse transformation, Em=d (g(An, )m with 2 Σ -m= (Am 3m Jum (i),
obtain the node representation where um (i) is the value the eigenvector u.m at node i.
Theorem Let {mm} and {um} be the graph frequencies and frequency components respec- tively, Bm is the projection coefficient of signal x onto the m-th eigenvector ums then (Am) Diev, (in(f)
Pm Liev, unliy holds for the filter g (A), indicating that frequencies where normal nodes show co-
herent spectral behavior (i.e., um (i) values agree sign/nagninude) are amplified, while inconsistent
frequency components are suppressed.
Proof 1 Starting from the one-class classification loss over the labeled normal nodes Vt:
N-1
L=
Σ
Σ
Bm um (1) -
iEV:
m==
compute the gradient of L with respect to g(Am):
N-1
OC =
Σ
Σ
((3))8,u5(1)- c)
Bm Bm Um (1).
0g(Am)
iEVI
Setting the derivative to zero for optimality and assuming orthogonality between eigenvectors (L.e., dropping cross terms for m + j), we obtain:
g(Am)
Σ Um
(1)2 = Σ Uml()) (i).
iEV;
iEV:
Solving for g(>m) yields the closed-form by assuming the center c = 1:
Eiev, Um
g(Am) = 3m Eiev, Unn (i)2
In the theorem, the denominator Bm. Eiey, (1)2 represents the weighted total energy of the m-th frequency component distributed over the normal nodes. It serves as a normalization factor, reflecting how uniformly the m-th frequency component is distributed across the normal nodes. When the m-th frequency components exhibit high consistency across the labeled normal nodes, e.g., um (i) have the same sign (all positive or all negative) and similar amplitude for E Vi, then the ratio Eier, (in(
Bm Enev, um(t)7 will be either much greater than (case I: 8m > and Evey, (m)
Σ.ev, (2) >> case Ever, (.. or much less than 0 (case 1: 8m > and Eier, um(1)
2: Bm < 0 and Eiey, (x)
Liev, (i(
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case 2: 3m < and Exer, um (1)
>> 0), resulting in a frequency positively or negatively enhancing
Liev, Mm (1)2
response g( (m))) > or g(Am) < O. Only when the frequency components are highly inconsistent, e.g., Um (i) exhibits opposite signs across the labeled normal nodes, these inconsistent components will cancel each other out (i.e., Eiey, um + 0), resulting in a frequency suppressing response g(Am)) + O. To summarize, minimizing the one-class loss enables an adaptive filtering mechanism to automatically enhance frequency components that are consistent across normal nodes while suppressing the inconsistent components.
Table 3: Key statistics of GAD datasets.
Datasets
Reddit
Tolokers
Photo
Amazon
Elliptic
Question
T-Finance
DGraph
#Nodes
10,984
11,758
7,484
11,944
203,769
48,921
39,357
3,700,550
#Edges
168,016
519,000
119,043
4,398,392
234,355
153,540
21,222,543
4,300,999
#Attributes
64
10
745
25
166
301
10
17
Anomaly
3.3%
21.8%
4.9%
9.5%
9.8%
2.98%
4.6%
1.3%
B Detailed Description of Datasets
The key statistics of the datasets are presented in Table 3. A detailed introduction of these datasets is given follows.
Reddit [18]: It a user-subreddit graph which captures one month's 's worth of posts shared across various subreddits at Reddit. The node represents the users, and the text of each post is transformed into a feature vector and the features of the user and subreddits are the feature summation of the post they have posted. The anomalies the used who have been banned by the platform.
obtained from the Toloka crowdsoureing platform, where the node represents the person who has participated in the selected project, and the edge represents two workers work on the same task. The attributes of the node are the profile and task performance statistics of workers.
Tolokers [28]:
Photo [28]: It is obtained from an Amazon co-purchase network where the node repre- sents the product and the edge represents the co-purchase relationship. The bag-of-words representation of the user's comments is used as the attribute of the node.
Amazon [10]: It is co-review network obtained from the Musical Instrument category on Amazon.com com. There are also three relations: U-P-U (users reviewing least one same product), U-S-U (users having at least one same star rating within one week), and U-V-U (users with top-5% % mutual review similarities).
Elliptic [43]: It is Bitcoin transaction network in which each node represents a transaction and an edge indicates a flow of Bitcoin currency. The Bitcoin transaction is mapped to real-world entities associated with licit categories in this dataset.
Question [35]: It is question answering network which is collected from the website Yandex Q where the node represents the user and the edge connecting the node represents the user who has answered other's questions during a one-year period. The attribute of nodes is the mean of FastText embeddings for words in the description of the user. For the user without description, the additional binary features are employed the feature of the user.
T-Finance [41]: It is financial transaction network where the node represents anonymous account and the edge represents two accounts that have transaction records. Some attributes of logging like registration days, logging activities, and interaction frequency, etc, are used as the features of each account. Users are labeled as anomalies if they fall into categories such as fraud, money laundering, or online gambling.
DGraph [15]: It is large-scale attributed graph with millions of nodes and edges where the node represents a user account financial company and the edge represents that the user was added to another account as an emergency contact. The feature of each node represents the user's profile information, including age, gender, and other demographic attributes. Users with history of overdue payments are labeled as anomalies.
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C Description of Baselines
more detailed introduction of the nine competing GAD models is given follows.
DOMINANT [7] applied the conventional autoencoder
graph for GAD. It consists of an encoder layer and a decoder layer, which are designed to reconstruct the features and structure of the graph. The reconstruction errors from the features and the structural modules are combined as an anomaly score.
DAAE [12] leverages the advanced autoencoder and an attribute autoencoder to learn both node embeddings and attribute embeddings jointly in latent space by assigning the corresponding weight in the loss function. In addition, an attention mechanism is employed in the structure encoder capture normal structural patterns more effectively.
OCGNN [42] combines one-class SVM and GNNs, aiming at leveraging one-class classifiers and the powerful representation of GNNs. A one-class hypersphere learning objective is used drive the training of the GNN. The samples that fall outside the hypersphere, meaning they deviate from the normal pattern, are defined as anomaly.
AEGIS [6] designs a new graph neural layer to learn anomaly- node representations and further employ generative adversarial networks detect anomalies among new data. The generator takes noise sampled from prior distribution as input and aims to produce informative pseudo-anomalies. Meanwhile, the discriminator seeks to determine whether a given input is the representation of normal node or a generated anomaly.
GAAN [5] is based on generative adversarial network where fake graph nodes generated by a generator. To encode the nodes, they compute the sample covariance matrix for real nodes and fake nodes, and discriminator is trained to recognize whether two connected nodes are from a real or fake node.
TAM [36] adopts the local affinity-based approach as an anomaly measure. It learns tailored node representations for
anomaly measure by maximizing the local affinity of nodes to their neighbors. TAM is optimized on truncated graphs where non-homophily edges are removed iteratively. The learned representations result in significantly stronger local affinity for normal nodes than abnormal nodes.
GGAD [38] generates pseudo anomaly nodes based on two important priors, including asymmetric local affinity and egocentric closeness, and trains a discriminative one-class classifier for semi-supervised GAD.
GCN [17] obtains the node representations by aggregating information from neighbors. which can be seen as special form of low-pass filter. The low-pass filter primarily preserves the similarity of node features during graph representation learning.
BWGNN [41] reveals the 'right shift' phenomenon that the spectral energy distribution con- centrates more on the node with high frequencies and less on the node with low frequencies, and proposes band-pass filter to learn the effective node representation for supervised GAD.
PMP [49] introduces a partitioning message passing scheme that separately aggregates information from homophilic and heterophilic neighbors using node-specific functions. This design enables the adaptive adjustment of information flow from different neighbor types, thereby effectively capturing structural heterogeneity and enhancing robustness to heterophily in graph-based fraud detection.
CONSISGAD [4] introduces a consistency- based framework for GAD under limited su- pervision. It leverages unlabeled data through a Iearnable data augmentation mechanism that injects controlled noise for consistency training. Additionally, by exploiting the vari- ance homophily distributions between normal and anomalous nodes, CONSISGAD employs a simplified GNN backbone to enhance discriminability and robustness against class imbalance.
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Photo
Tolokers
T- T-Finance
Question
Reckit
40000
20000-
Number of Normal Nodes
2000 -
1500-
1000
500-
[0.0..1]]
(0.102)
(0.20.3))
(0.3.04)
(0.4 4.0.5
(0.5...)
(0..0.7)
(0.7.08)
(0.8.099)
(0.9.1.))
Range of Homophily Rate
Figure Homophily distributions of normal nodes in five datasets.
D Additional Experimental Results
D.I Homophily Distribution Results for the Remaining Datasets
In Fig. 7, visualize the homophily distributions of normal nodes the rest of five datasets. As shown in Fig. 1, the observation is consistent with the distributions on Amazon and Elliptic, i... although most normal nodes exhibit high homophily, some normal nodes show relatively low homophily, resulting in significant variations in homophily levels among the normal nodes. This further indicates that this phenomenon commonly observed in the GAD datasets, and existing models based on the aforementioned assumptions may learn inaccurate homophily patterns of the normal class on these datasets, while RHO adaptively learning heterogeneous normal patterns from the small set of normal nodes with diverse homophily can well address this challenge.
SPCC
0005
620
033
(a) Reddit
(b) Tolokers
(c) Amazon
(d) Question
(e) T-Finance
Figure 8: AUROC w.r.t. training data size R
03
SPCCO
Labe
Lid
(a) Reddit
(b) Tolokers
(c) Amazon
(d) Question
(e) T-Finance
Figure 9: AUPRC w.r.t. training data size R
D.2 Performance w.r.t. Different Training Size
The AUROC and AUPRC results under varying proportions of training normal nodes are shown in Fig. 8 and Fig. 9, respectively. The results further demonstrate that as the number of labeled normal nodes increases, the performance of RHO generally improves across the datasets, which is consistent with the behavior observed in other semi-supervised methods. Besides, RHO consistently outperforms the competing methods across different numbers of training normal nodes on Tolokers, Question, and T-Finance in terms of AUROC. RHO also achieves the best AUPRC performance across all datasets under varying training data sizes, further demonstrating its effectiveness in varying homophily pattern learning in the small normal node set.
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06
0.0
0.7
07
0.6-
06
y05
203
044
3.4
55
135
5%
15%
(a) Photo
(b) Amazon
(c) Elliptic
(d) Question
Figure 10: AUROC results w.r.t. different anomaly contamination rates.
0:30
6.7
0.7
0:04
0:05
S 6:23
0.00
6002
0:00
6:00
06+
(a) Photo
(b) Amazon
(c) Elliptic
(d) Question
Figure 11: AUPRC results w.r.t different anomaly contamination rates.
D.3
Performance w.tt. Anomaly Contamination
In real applications, labeled normal nodes are often susceptible to contamination by anomalies due to factors such as annotation errors. To address this issue, introduce controlled ratio of anomaly contamination into the training set of labeled normal nodes. Specifically, we randomly sample abnormal nodes from the unlabeled nodes in the test set and incorporate them normal nodes into the training set of normal node set. All sampled abnormal nodes are excluded from the test set.
As shown in Fig. 10 and Fig. 11, the performance of all models declines the level of contamination increases, particularly on the AUPRC. RHO achieves the best performance among all competing methods and maintains consistent results across varying contamination rates, demonstrating its robustness in representation learning for semi-supervised GAD.
D.4 Runtime Results
The running time, including both training and inference time, of
Table 4: Runtimes (in seconds) on the six datasets on CPU.
RHO and six competing meth- ods are shown Table 4. Al- though our method can run all datasets using an RTX 3090 GPU, some baseline methods may encounter out-of-memory (OOM) issues on the same GPU due to their different compu- tational mechanisms, highlight- ing that RHO is more memory- efficient. To ensure fair com- parison, we report the runtime performance on the CPU for all methods. The runtime results demonstrate that RHO remains efficient processing in the
Datasets
Methods
Reddit
Photo
Amazon
Question
T-Finance
DOMINANT
437
1592
740
10721
OCGNN
162
125
765
973
5717
AEGIS
166
417
660
15258
TAM
432
165
4516
29280
17360
GGAD
368
136
1020
1125
9345
CONSISGAD
483
681
3259
483
20172
RHO
398
201
3358
675
5184
even on the targe-scale graph T-Finance, primarily due to the use of GNA module. By adopting mini-batch strategy, the complexity of GNA reduces to ((b²d) for a batch size of b. and the total per-epoch complexity becomes ((Nbd) when averaged across all batches, which significantly reduces memory usage and computational overhead, thereby improving the scalability. In contrast, methods such as DOMINANT, TAM, and GGAD involve reconstruction or affinity calculations across nodes, which typically incur a time complexity of O(N2), causing significant computational overhead on large-scale graphs.
25

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Algorithm RHO: Semi-supervised Graph Anomaly Detection via Robust Homophily Learning
Input: Graph 9 = (V, &), node features X, labeled normal nodes Vi, unlabeled nodes V.. number of training epochs E, temperature 7, trade-off weight a, batch size b, propagation depth T Output: Anomaly scores S; for each E V..
1: Initialize parameters 0, {Weet W(() ben,
eve } frequency coefficients l (shared in cross-channel),
K [k...., ka] (channel-wise)
2: Set initial features: (U) cer
(X(X),
H
(0) cur fo(X);
Ceer
0, Cour
3: for to 7 do
4: Update cross-channel view: cer
t
HL)H W
5: Update channel-wise view: t [I (H
K
our
6: end for
7: Compute centers: Ceor + M Eiev b (i), Cowr + [8] EiEv
hoar (1)
8: Compute projections: Zeer Peor
(T) Zewr foorr (hew))
9: for epoch = to E do
Compute one-class losses Coers hewr U2 1i
11: if b > then
Sample batch B V of size
13: else
Set V
15: end if
for each v; E B do
17: LGNA t- 2| E, (E(cer (i), Bewr (1)) + L [(Zoor (i), Zeer (1))
18:
end for
19: Ltotal f (f + Lour) + aLGNAA
20: Coer f- কo. Eier (n07 (i), Covor fo 4 Eiev TTT (i)
21: Update all parameters via backpropagation
22: end for
23: for each ts E V. do
24: Anomaly scoring: S. 1 (1 (T) (i)
Coer 11 2 + | (i) Cewr 113)
end for
26: return Anomaly scores , Spv.l
E Algorithm
The algorithm of RHO summarized in Algorithm 1.
26


Reviews:
[{"review_id": "GaUISCA5JS", "reviewer": "Reviewer_yntr", "summary": "This paper introduces RHO, a robust homophily learning framework for semi-supervised graph anomaly detection that adaptively captures diverse normal patterns by learning spectral filters tailored to varying homophily levels. It proposes two key modules—AdaFreq, which learns channel-wise and cross-channel adaptive frequency-response filters, and GNA, which aligns their representations via contrastive learning to ensure consistency. Extensive experiments on eight real-world datasets show that RHO consistently outperforms state-of-the-art methods in detecting anomalies across heterogeneous homophily settings.", "questions": "See Weaknesses.", "limitations": "See Weaknesses.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The RHO framework’s adaptive spectral filters effectively capture diverse normal patterns across varying homophily levels.\n- The integration of AdaFreq and GNA modules ensures both frequency-wise adaptation and representation consistency through contrastive learning.\n- Evaluation on eight real-world datasets demonstrates RHO’s superior anomaly-detection performance.\n\nWeaknesses:\n- The paper is missing recent baselines such as PMP [1] and ConsisGAD [2].\n- The term \"cross-channel\" is confusing: in this paper it refers to sharing across channels, but readers may interpret it as fusion among channels.\n- Important sections, such as the complexity analysis, should be included in the main text rather than relegated to the appendix.\n\n[1] Wei Zhuo, Zemin Liu, Bryan Hooi, Bingsheng He, Guang Tan, Rizal Fathony, and Jia Chen. \"Partitioning Message Passing for Graph Fraud Detection.\" In The Twelfth International Conference on Learning Representations.\n\n[2] Nan Chen, Zemin Liu, Bryan Hooi, Bingsheng He, Rizal Fathony, Jun Hu, and Jia Chen. \"Consistency training with learnable data augmentation for graph anomaly detection with limited supervision.\" In The Twelfth International Conference on Learning Representations. 2024.", "was_revised": true, "final_justification": "The rebuttal has addressed all my concerns.", "rebuttal": "We sincerely appreciate your constructive and positive comments on our design and empirical justification. Please see our response to your comments one by one below.\n\n>**Weakness #1:** The paper is missing recent baselines such as PMP [Ref1] and ConsisGAD [Ref2].\n\nFollowing your suggestion, we include two additional baselines, PMP and ConsisGAD, both published in 2024.  It's worth mentioning that both methods are fully supervised, assuming both the normal label and the abnormal label are known.  To ensure a fair comparison, similar to the adaptation of BWGNN, we adapt these methods to a semi-supervised setting by replacing the cross-entropy loss with the one-class classification loss. The experimental results are shown in Tables A1 and A2.\n\n```\nTable A1. AUROC results of comparing RHO with two additional baselines on eight datasets.\n```\n|Datasets|Reddit|Tolokers|Photo|Amazon|Elliptic|Question|T-Finance|DGraph|\n|---|---|---|---|---|---|---|---|---|\n|PMP|0.5472|0.5815|0.5844|0.8329|0.5617|0.5790|0.8321|0.5376|\n|ConsisGAD|0.5347|0.5974|0.5859|0.8715|0.7354|0.5737|0.8277|0.5735|\n|RHO|**0.6207**|**0.6255**|**0.7129**|**0.9302**|**0.8509**|**0.5833**|**0.8623**|**0.6033**|\n\n```\nTable A2. AUPRC Results of comparing RHO with two additional baselines on eight datasets.\n```\n|Datasets|Reddit|Tolokers|Photo|Amazon|Elliptic|Question|T-Finance|DGraph|\n|---|---|---|---|---|---|---|---|---|\n|PMP|0.0388|0.2939|0.1254|0.3582|0.1006|0.0383|0.4084|0.0046|\n|ConsisGAD|0.0360|0.3059|0.1319|0.6523|0.1765|0.0424|0.4152|0.0050|\n|RHO|**0.0616**|**0.3256**|**0.2337**|**0.7879**|**0.5095**|**0.0430**|**0.4893**|**0.0059**|\n\nFrom the results, we can observe that RHO consistently outperforms the two additional baselines across all benchmark datasets, reinforcing the better effectiveness and generalization of RHO against various competing methods. PMP and ConsisGAD are designed for fully supervised GAD settings where labeled anomalies can offer critical abnormality information during the optimization; they become less effective when applied in the semi-supervised setting where no labeled anomalies are unavailable and only a small portion of labeled normal nodes is available.\n\n- [Ref1]  Zhuo, W., et al. Partitioning message passing for graph fraud detection, In ICLR 2024.\n\n- [Ref2] Chen, N., et al. Consistency training with learnable data augmentation for graph anomaly detection with limited supervision. In ICLR 2024.\n\n>**Weakness #2:** The term \"cross-channel\" is confusing: in this paper it refers to sharing across channels, but readers may interpret it as fusion among channels.\n\nThank you very much for the comment. We understand that the term \"cross-channel\" may have different interpretations in the literature. In our work, we use it to specifically denote the shared information across the channels, rather than channel fusion. ``Inter-channel'' may be a more precise term. We will use this alternative term and clearly define it within the context of our paper.\n\n>**Weakness #3:** Important sections, such as the complexity analysis, should be included in the main text rather than relegated to the appendix.\n\nThank you for your valuable suggestion. We will move the complexity analysis to the main paper in the final version."}, {"review_id": "g8I9R5aZwo", "reviewer": "Reviewer_Yk4n", "summary": "This paper proposes a novel semi-supervised graph anomaly detection framework named RHO, which aims to address the challenge of homophily among normal nodes—i.e., the fact that normal nodes may have either high or low homophily in graph structure. To tackle this, the authors introduce AdaFreq, an adaptive frequency-response filter that adjusts graph signal propagation strengths based on learnable frequency parameters, applied in both a cross-channel view (aggregating all features) and a channel-wise view (learning distinct filters per feature channel). To unify these two complementary views, the paper further proposes a Graph Normality Alignment (GNA) module that encourages consistent representations across views through a contrastive alignment objective. Experimental results and visualizations demonstrate that RHO effectively captures diverse normal patterns and improves anomaly detection robustness compared to existing methods.", "questions": "1. In the channel-wise view, each feature dimension is propagated independently using a dedicated frequency filter. Have the authors considered modeling interactions between feature dimensions (e.g., pairwise or grouped channels)? Would such multi-dimensional interactions further enhance the model’s capacity to capture complex homophily patterns?\n\n2. In the experiments, most baseline methods are from earlier years, with only a few from 2024. \n\n3. The GNA module focuses on aligning representations across the cross-channel and channel-wise views. Within the channel-wise view, however, each feature dimension is processed independently. Is there any mechanism to encourage consistency or coordination among individual feature channels?", "limitations": "See above.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. **Novel problem formulation**: The paper identifies an important yet underexplored issue in graph anomaly detection named *heterogeneous homophily among normal nodes*, and it designs the model explicitly to address this challenge.\n\n2. **Methodological originality**: The introduction of AdaFreq, a learnable frequency-response filter applied to both cross-channel and channel-wise views, is a compelling design. Especially, using channel-wise frequency modulation to model per-feature homophily variation is a fresh perspective.\n\n3. **Effective view alignment strategy**: The proposed Graph Normality Alignment (GNA) module employs a contrastive objective to reconcile representations from different views, leading to more robust normality modeling.\n\n4. **Strong empirical results**: The paper presents comprehensive experiments with both quantitative metrics and qualitative visualizations (e.g., t-SNE, anomaly score histograms), which together provide strong support for the method’s effectiveness.\n\n5. **Clarity of exposition**: The paper is clearly written and logically structured. Design motivations are well-justified, and ablation studies are provided to isolate the impact of each module.\n\n**Weaknesses:**\n\n1. While the paper focuses on per-feature adaptive filtering, it does not explore potential interactions across multiple feature combinations (e.g., pairwise or group-wise channel dependencies), which may limit its capacity to capture complex homophily patterns.\n\n2. Some modules, such as GNA, introduce additional complexity. A more detailed runtime or scalability analysis would further strengthen the paper.", "was_revised": true, "final_justification": "We thank the authors for their detailed rebuttal, which clearly addressed the concerns we raised—especially Question 1 regarding channel interactions. \n\nWe encourage the authors to ensure that all discussed revisions and clarifications are carefully integrated into the final version.\n\nGood luck!", "rebuttal": "We sincerely appreciate your constructive and positive comments on the presentation, visualization, method, and empirical justification. Please see our response to your comments one by one below.\n\n>**Weakness #1 and Question #1:** While the paper focuses on per-feature adaptive filtering, it does not explore potential interactions across multiple feature combinations (e.g., pairwise or group-wise channel dependencies), which may limit its capacity to capture complex homophily patterns.\n\nThanks for your valuable suggestion. The cross-channel filter was designed in RHO to capture the interactions among part/all of the channels. To examine the necessity of including additional interactions beyond our cross-channel design,  we add group-wise channels by enumerating all possible groups of a fixed size to explore more potential interactions. We then learn a dedicated filter for each group. Note that  when the group size is set to two, the group-wise view essentially reduces to the pairwise view. To ensure that the group-wise channel dependencies align well with the RHO framework, we concatenate the representations obtained from both the pairwise/group-wise and cross-channel views and feed it to the one-class classification loss $\\mathcal{L}_{ccr}$. The fused representation is then aligned with the channel-wise representation using the proposed GNA module. Tables A1 and A2 present the performance of two group configurations with different group sizes (gs) across five benchmark datasets.\n\n\n```\nTable A1. AUROC results under varying group sizes on five datasets.\n```\n||Reddit|Tolokers|Amazon|Elliptic|Question|\n|---|---|---|---|---|---|\n|gs=2|0.5976|0.5199|0.9149|0.8469|0.5514|\n|gs=3|0.5813|0.5451|0.9055|0.8195|0.5338|\n|RHO|**0.6207**|**0.6255**|**0.9302**|**0.8509**|**0.5833**|\n\n```\nTable A2. AUPRC results under varying group sizes on five datasets.\n```\n||Reddit|Tolokers|Amazon|Elliptic|Question|\n|---|---|---|---|---|---|\n|gs=2|0.0505|0.2690|0.7478|0.4615|0.0423|\n|gs=3|0.0473|0.2759|0.6038|0.3577|0.0321|\n|RHO|**0.0616**|**0.3256**|**0.7879**|**0.5095**|**0.0430**|\n\nThe experimental results indicate that incorporating the group-wise learning does not bring additional benefits to the model’s performance. Due to the lack of an informative grouping strategy, it can be difficult for our group-wise views to provide additional important information that our cross-channel view does not have for the model. By contrast, these group-wise views may contribute noisy or redundant information into the model optimization, which leads to less effective performance than the original RHO. \n\nIn addition, please note that the use of these additional group-wise views can add a significant computational burden since there are a large number of such views in each dataset.\n\n>**Weakness #2:** Some modules, such as GNA, introduce additional complexity. A more detailed runtime or scalability analysis would further strengthen the paper.\n\nThank you very much for the valuable suggestion. As discussed in Appendix E regarding time complexity,  although GNA may bring some additional complexity,  this overhead is acceptable given the performance gains compared with methods like DOMINANT (due to reconstruction) and TAM or GGAD (due to affinity calculations) that typically incur a time complexity of  $O(N^2)$,  causing their runtime to increase significantly as the number of nodes grows. To demonstrate this, we further provide a running time comparison result in Table A3. Although our method can run on all datasets using an RTX 3090 GPU, some baseline methods may encounter out-of-memory (OOM) issues on the same GPU due to their different computational mechanisms, highlighting that RHO is more memory-efficient. To ensure a fair comparison, we report the runtime performance on the CPU for all methods. The runtime results demonstrate that RHO remains efficient even on the large-scale graph T-Finance, primarily due to the use of mini-batch processing in the GNA module. By adopting a mini-batch strategy, the complexity of GNA reduces to $O(b^2d)$ for a batch size of $b$, and the total per-epoch complexity becomes $O(Nbd)$ when averaged across all batches, which significantly reduces memory usage and computational overhead, thereby improving the scalability. In contrast, methods such as DOMINANT, TAM, and GGAD involve reconstruction or affinity calculations across all nodes, which typically incur significant computational overhead on large-scale graphs.\n\n```\nTable A3. Runtimes (in seconds, including both training and testing runtimes) on the five datasets.\n```\n||Reddit|Photo|Amazon|Question|T-Finance|\n|---|---|---|---|---|---|\n|DOMINANT|125|437|1592|740|10721|\n|OCGNN|162|125|765|973|5717|\n|AEGIS|166|417|1121|660|15258|\n|TAM|432|165|4516|29280|17360|\n|GGAD|368|136|1020|1125|9345|\n|ConsisGAD|483|681|3259|483|20172|\n|RHO|398|201|3358|675|5184|\n\n\n\n>**Question #2:**  In the experiments, most baseline methods are from earlier years, with only a few from 2024.\n\nThank you for your valuable suggestion. In response to **Reviewer yntr**’s comment, we have incorporated two additional baseline methods published in 2024. For detailed information regarding these newly added comparison, we kindly refer you to our response to **Reviewer yntr** – **Weakness #1**.\n\n>**Question #3:** The GNA module focuses on aligning representations across the cross-channel and channel-wise views. Within the channel-wise view, however, each feature dimension is processed independently. Is there any mechanism to encourage consistency or coordination among individual feature channels?\n\nThank you for the valuable comments. We would like to clarify that the cross-channel is specially designed to explore the consistency and coordination among the feature channels by learning a shared filter for them. \n\nIn addition, we have also explored the group-wise and pair-wise feature channels in our response to **Question #1**, and we empirically found that they do not bring additional benefits to RHO, indicating that the current cross-channel design can provide sufficient channel-interaction-based information that is complementary to the channel-wise view."}, {"review_id": "xeuPvXWuH3", "reviewer": "Reviewer_pqei", "summary": "The paper proposes RHO, a semi‐supervised graph anomaly detection (GAD) method that learns heterogeneous homophily patterns from a small set of labeled normal nodes. RHO combines two novel modules:  \n1. AdaFreq: adaptive spectral filters with trainable frequency‐response parameter $k$, applied in both cross‐channel and channel‐wise views to capture diverse low‐ and high‐frequency components of normal nodes.  \n2. Graph Normality Alignment (GNA): a contrastive alignment loss that enforces consistency between the two views’ representations.", "questions": "See weaknesses above", "limitations": "The proposed method only applies to node-level anomaly detection. Also, the authors do not provide interpretation of the learned $k$.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### **Strengths**\nThe proposed method:\n1. Leveraging both cross‐channel and channel‐wise perspectives captures complementary homophily patterns.\n2. The GNA module effectively aligns the two views, improving robustness.\n\n### **Weaknesses**\n\n#### **1. The analyses are confusing at times:**\n\n**In Theorem 1:**\n\n1.  **\"assuming $β_m > 0$\":** It is unclear why all projection coefficients \\(\\beta_m\\) must be positive. What if \\(\\beta_m<0\\)? Does the result still hold or collapse? Any justification or empirical evidence for this assumption is missing. \n2.  **Motivation for the filter design:** The manuscript states “According to Theorem 1, … by training the parameter $k$.” However, Theorem 1’s statement involves $g(\\lambda_m)$ under an assumption on $\\beta_m$ The definition of $g(\\lambda)$in Theorem 1 differs from Eq. (2) (where $g(\\lambda)=1-k\\lambda$), so it is unclear how Theorem 1 *motivates* the choice of this linear filter form.  \n3. **\"When k = 0, g(λ) retains raw frequency signals.\":** The claim “When $k=0$, $g(\\lambda)$ retains raw frequency signals” seems contradictory: if $g(\\lambda)=1$, then the output is constant across all eigenvalues, not the original signal $\\lambda$. Should raw frequency signal be $\\lambda$ rather than 1?  \n\n#### **2. Adaptive filters are well-studied in the graph learning field. What's new (advantages) in your method over existing ones?**\n\n### Minor Issues  \n1. “which is simultaneously applies on” → should be “which is simultaneously applied to” (Sec. 4.1).  \n2. “AUROCassesses the model’s overall ability...” → missing space: “AUROC assesses…” (Sec. 5.1).", "was_revised": true, "final_justification": "The rebuttal has addressed most of my concerns, thus I am inclined to weak accept this paper.", "rebuttal": "We sincerely appreciate your constructive and positive comments on our model design. Please see our response to your comments one by one below.\n\n>**Weakness #1.1:** \"assuming $\\beta_m>0$\": It is unclear why all projection coefficients ($\\beta_m$) must be positive. What if ($\\beta_m$<0)? Does the result still hold or collapse? Any justification or empirical evidence for this assumption is missing.\n\n\nThank you for raising the question regarding the assumption of $\\beta_m>0$. For simplicity, we only analyze one side of the symmetric case in the derivation; when the $m$-th frequency components exhibit high consistency across the labeled normal nodes, e.g., $u_m(i)$ have the same sign (all positive or all negative) and similar amplitude for $i\\in \\mathcal{V}\\_l$, then the ratio $\\frac{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)}{\\beta_m\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)^2}$ will be either much greater than 0 (case 1: $\\beta_m>0$ and $\\frac{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)}{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)^2} \\gg 0$, case 2: $\\beta_m<0$ and $\\frac{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)}{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)^2} \\ll 0$)\nor much less than 0 (case 1: $\\beta_m>0$ and $\\frac{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)}{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)^2} \\ll 0$, case 2: $\\beta_m<0$ and $\\frac{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)}{\\sum_{i \\in \\mathcal{V}\\_l }u_m(i)^2} \\gg 0$), resulting in a frequency positively or negatively enhancing response $g(\\lambda_m) \\gg 0$ or $g(\\lambda_m) \\ll 0$.\nOnly when the frequency components are highly inconsistent, e.g., $u_m(i)$ exhibits opposite signs across the labeled normal nodes, these inconsistent components will cancel each other out (i.e., $\\sum_{i \\in \\mathcal{V}\\_l }u_m(i) \\rightarrow 0$), resulting in a frequency suppressing response $g(\\lambda_m) \\to 0$. To summarize, our theoretical conclusion—minimizing the one-class loss enables an adaptive filtering mechanism to automatically enhance frequency components that are consistent across normal nodes while suppressing the inconsistent components—does not depend on the sign of $\\beta_m$. Instead, it is solely determined by the consistency of the frequency components on the labeled normal nodes. We will clarify this in the final version.\n\n>**Weakness #1.2:** Motivation for the filter design: The definition of $g(\\lambda)$ in Theorem 1 differs from Eq. (2) (where $g(\\lambda) = 1-k\\lambda$), so it is unclear how Theorem 1 motivates the choice of this linear filter form.\n\nThank you very much for the comment. $g(\\lambda)$ is a graph spectral filter function defined with respect to the eigenvalue $\\lambda$. In Theorem 1, we unify the graph spectral filter with the one-class learning objective function to theoretically characterize the spectral behavior of the filter $g(\\lambda)$ during optimization. So, $g(\\lambda)$ in Theorem 1 is not a definition of the filter function itself, but rather a theoretical description of its desired frequency response. Theorem 1 motivates us to design an adaptive filter that dynamically adjusts the response strength of different frequency, effectively preserving the consistent frequency components of labeled normal nodes with diverse homophily in the optimization. In our implementation, we adopt a linear filter formulation due to its simplicity and effectiveness. Specifically, we parameterize the filter as $g(\\lambda) = 1-k\\lambda$ where the parameter $k$ is learned to ensure that the filter adheres to the underlying homophily pattern. Moreover, when stacking $K$ layers, the proposed filter becomes $g(\\lambda) = \\prod_{i=1}^{K} (1 - k_i \\lambda)$, which can be further expressed as a polynomial expansion, i.e., $g(\\lambda) = \\sum_{l=0}^{K} \\alpha_l \\lambda^l$, where $\\alpha_l$ are learnable parameters derived from $k_i$. Therefore, our filter also possesses nonlinear expressive capability when the network layer $K>1$. \n\nWe further compared our filter with an adaptive nonlinear spectral filter $h(\\lambda) = a_0 \\lambda + \\sum_{i=1}^{d/2} a_i sin(\\frac{\\epsilon \\lambda}{10000^{2i/d}}) + \\sum_{i=1}^{d/2} b_i  cos(\\frac{\\epsilon \\lambda}{10000^{2i/d}})$ based on Fourier series, as used in the SpecFormer method [Ref1]. In the experiments, we replaced the filter function $g(\\lambda)$ in the RHO framework with SpecFormer’s nonlinear filter $h(\\lambda)$ and conducted the experiments under the same settings. The results are presented in Tables A1 and A2.\n\n```\nTable A1. AUROC results of our filter and SpecFormer’s nonlinear filter.\n```\n|Datasets|Reddit|Tolokers|Amazon|\n|---|---|---|---|\n|SpecFormer|0.5923|0.6017|0.9039|\n|RHO|**0.6207**|**0.6255**|**0.9302**|\n\n```\nTable A2. AUPRC results of our filter and SpecFormer’s nonlinear filter.\n```\n|Datasets|Reddit|Tolokers|Amazon|\n|---|---|---|---|\n|SpecFormer|0.0493|0.3073|0.7110|\n|RHO|**0.0616**|**0.3256**|**0.7879**|\n\nFrom the results, our proposed filter demonstrates superior performance compared to the complex nonlinear spectral filter used in SpecFormer. The results indicate that our simple linear design exhibits stronger adaptability and better generalization in capturing homophilic patterns among normal nodes. In contrast, the complex filter design in SpecFormer is likely to overfit the limited number of normal samples in the training set,  resulting in less effective generalization. These findings further validate the effectiveness and generality of our filter in graph anomaly detection. In addition, the nonlinear filter in SpecFormer is a type of filter that typically requires explicit spectral decomposition, which incurs a computational cost of $O(N^3)$, making it not scalable to large-scale graphs.\n\n- [Ref1] Bo, D., et al.  Specformer: Spectral graph neural networks meet transformers. ICLR 2023.\n\n>**Weakness #1.3:** \"When $k$ = 0, $g(\\lambda)$ retains raw frequency signals.\": The claim ''when $k=0, g(\\lambda)$ retains raw frequency signals'' seems contradictory: if $g(\\lambda)=1$, then the output is conatant across all eigenvalues, not the original signal $\\lambda$. Should raw frequency signal be $\\lambda$ rather than 1?\n\nWe apologize for the misunderstanding caused by this concept. We would like to clarify that the \"signal\" in this sentence refers specifically to the graph signal $\\mathbf{x} \\in \\mathbb{R}^{N}$. As stated in the first sentence of our theoretical analysis, ''A graph filter operation on a graph signal $\\mathbf{x} \\in \\mathbb{R}^{N}$ can be defined as $\\mathbf{z} = \\mathbf{U}g(\\mathbf{\\Lambda})\\mathbf{U}^\\top\\mathbf{x}$.''  When $k$ = 0, the filter $g(\\lambda) = 1$, which is equivalent to $g(\\mathbf{\\Lambda}) = \\mathbf{I}$. As a result, we obtain $\\mathbf{z} = \\mathbf{U}\\mathbf{I}\\mathbf{U}^\\top\\mathbf{x}=\\mathbf{x}$, indicating that the filter $g(\\lambda)$ leaves the signal unchanged and thus preserves the raw input $\\mathbf{x}$. We will revise the content and provide this detailed clarification in the final version.\n\n>**weakness #2:** Adaptive filters are well-studied in the graph learning field. What's new (advantages) in your method over existing ones?\n\nAdaptive filters are well-studied for addressing the heterophily issue in node classification tasks. But for graph anomaly detection, i) its one-class classification nature poses significant challenges for the application of conventional spectral methods; and ii) moreover, we empirically reveal that the normal nodes can have diverse homophily in the GAD datasets, which has been overlooked by existing methods. The adaptive filter we propose has a unique advantage that it adaptively captures a wide range of homophily patterns among the nodes in the normal class. Such a filter is specifically designed to address the two aforementioned issues in graph anomaly detection, offering important insights into the one-class homophily learning problem.\n\n>**Minor Issues**\n“which is simultaneously applies on” → should be “which is simultaneously applied to” (Sec. 4.1).\n“AUROC assesses the model’s overall ability...” → missing space: “AUROC assesses…” (Sec. 5.1).\n\nThanks for pointing out these typo errors. We will correct them in the final version.\n\n>**Limitation:** The proposed method only applies to node-level anomaly detection. Also, the authors do not provide an interpretation of the learned $k$.\n\nThank you very much for the comments. Node-level anomaly detection is among the most popular and challenging GAD tasks due to its broad real-world applications in diverse domains. Other GAD tasks, such as edge-level or graph-level anomaly detection, require very different designs, since we are required to deal with different types of graph instances, such as a set of edges or full graphs. Thus, similar to numerous existing studies, we focus on node-level anomaly detection in this work and plan to explore the proposed methodology in edge- or graph-level anomaly detection in our future work.\n\nTo enhance the interpretation of the learnable $k$ in our adaptive filter, we have provided an intuitive visualization in Figure 4 in Section 5.2. The results show that in real-world datasets, the cross-channel view learns positive $k$ values, which approximate the low-pass filtering curve (inversely proportional to the frequency $\\lambda$), supporting the modeling of low-frequency signals. Meanwhile, in the channel-wise view, negative $k$ values  are learned for different feature channels, which approximate high-pass filter curves (proportional to the frequency $\\lambda$) capturing the high-frequency signals; in some cases we obtain $k=0$, which approximates horizontal all-pass filter curve, preserving the raw signals. We will clarify this interpretation of $k$ in the final version."}, {"review_id": "OU11t5evLn", "reviewer": "Reviewer_tFPZ", "summary": "This work proposes a novel framework, Robust Homophily Learning (RHO), for semi-supervised graph anomaly detection (GAD). RHO incorporates two novel designs: AdaFreq, which captures diverse frequency components of labeled normal nodes across channel-wise and cross-channel views of node attributes; and GNA, which enforces consistency between two views to robustly model node normality, accounting for varying levels of homophily among normal nodes. Experiments on eight real-world GAD datasets demonstrate the effectiveness of RHO.", "questions": "Please refer to Weaknesses.", "limitations": "Please refer to Weaknesses.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n \n1. The paper is well written and easy to follow.\n\n2. The motivation for the proposed RHO is novel, and theoretical analysis supports the design of the adaptive filter. The visualization in Figure 3 is intuitive.\n\n3. The experiments and theory are sufficient to demonstrate the effectiveness.\n\n**Weaknesses and Questions:**\n\n1. According to the algorithm design of RHO, the centers $c$ in the two views are dynamically updated during training, which differs from the static center design used in previous works. What is the motivation behind this dynamic update? If the centers are set as static, how would it affect the model's performance?\n\n2. The description of symbols are unclear. What is the difference between the convolution kernel $g$ in Line 145 and convolution kernel $g\\_\\theta$ in Line 146.\n\n3. Figure 1a presents the statistics of the homophily rate for normal nodes. The formula provided for computing node-level homophily (Lines 128–129) is more appropriate than using graph-level homophily.\n\n4. RHO achieves significant improvements on some datasets, but demonstrates competitive performance on Reddit and Amazon. What could be the possible reasons behind this?", "was_revised": true, "final_justification": "The authors' responses have addressed my concerns and further improved the paper's quality. Therefore, I would like to increase my rating.", "rebuttal": "We sincerely appreciate your constructive and positive comments on the presentation, visualization, and empirical justification.  Please see our response to your comments one by one below.\n\n>**Weakness #1:** According to the algorithm design of RHO, the centers in the two views are dynamically updated during training, which differs from the static center design used in previous works. What is the motivation behind this dynamic update? If the centers are set as static, how would it affect the model's performance?\n\nThank you very much for your comment. The motivation of using dynamic centers $c$ stems from our observation that the frequency signals of normal nodes may shift during training due to model updates and evolving spectral representations. A static center $c$, which remains fixed throughout the training, would fail to capture the evolving distribution of spectral embeddings, leading to suboptimal performance. In contrast, dynamically updating the center enables the model to adaptively refine the target frequency region in response to changes in the spectral representations, resulting in more effective feature learning. The collaborative mechanism between the filter and the center allows the learnable spectral filter to more effectively preserve the consistent frequency components associated with normal nodes.\n\n\nTo validate effectiveness, we add and evaluate the results of RHO-Static, a variant of RHO that uses a fixed center. The experimental results are shown in Tables A1 and A2 below. From the results, we can observe that RHO with a dynamic center achieves the best results on almost all datasets, which validates the effectiveness of dynamically adapting the center.\n\n```\nTable A1. AUROC results of RHO and RHO-Static\n```\n|Datasets|Reddit|Tolokers|Photo|Amazon|Elliptic|Question|T-Finance|\n|---|---|---|---|---|---|---|---|\n|RHO-Static|0.5676|0.5988|**0.7234**|0.9153|**0.8509**|0.5729|0.8208|\n|RHO|**0.6207**|**0.6255**|0.7129|**0.9302**|**0.8509**|**0.5833**|**0.8623**|\n\n```\nTable A2. AUPRC results of RHO and RHO-Static\n```\n|Datasets|Reddit|Tolokers|Photo|Amazon|Elliptic|Question|T-Finance|\n|---|---|---|---|---|---|---|---|\n|RHO-Static|0.0468|0.2865|0.2311|0.7341|0.5092|0.0393|0.3023|\n|RHO|**0.0616**|**0.3256**|**0.2337**|**0.7879**|**0.5095**|**0.0430**|**0.4893**|\n\n \n>**Weakness #2:** The description of symbols are unclear. What is the difference between the convolution kernel $g$ in Line 145 and the convolutional kernel $g_\\theta$ in Line 146.\n\nThank you for pointing this out. We apologize for the confusion caused by this notation. $g$ and $g_\\theta$ denote convolution kernels in the spatial and spectral domains, respectively. \n\nTo provide a clearer explanation, we would replace $g$ with $f$, and revised the description of Equation (1) as follows.\n\n*The convolution between the signal $x$ and convolution kernel $f$ is:\n    $ f *  \\mathbf{x}=\\mathbf{U}\\left(\\left(\\mathbf{U}^{T} f\\right) \\odot\\left(\\mathbf{U}^{T} \\mathbf{x}\\right)\\right)=\\mathbf{U} g_{\\theta} \\mathbf{U}^{T} \\mathbf{x}$,\nwhere $\\odot$ is an element-wise product and $g_{\\theta}$ is a diagonal matrix, which represents the convolutional kernel in the spectral domain, replacing $\\mathbf{U}^{T} f$.*\n\nWe will clarity this and add this revised description in the final version.\n\n>**Weakness #3:** Figure 1a presents the statistics of the homophily rate for normal nodes. The formula provided for computing node-level homophily (Lines 128–129) is more appropriate than using graph-level homophily.\n\nThank you very much for your valuable suggestion.  We will provide the definition of the node-level homophily metric used in Figure 1a, rather than the graph level. The formula for computing node-level homophily is $\\mathcal{H}_{node}^{v}= \\frac{|\\\\{u \\mid u \\in \\mathcal{N}\\_{v}, y\\_{u}=y\\_{v}\\\\}\\|}{d\\_{v}}$.\n\n>**Weakness #4:** RHO achieves significant improvements on some datasets, but demonstrates competitive performance on Reddit and Amazon. What could be the possible reasons behind this?\n\n\nFor the Amazon dataset, which exhibits strong one-class homophily (i.e., nearly all normal nodes tend to have consistently high homophily while anomalies have low homophily, see Figure 1a), competing methods like GGAD excel at modeling such evident cues of normal nodes, and so they can achieve very strong detection performance.  Owing to the robust homophily learning capability, our method RHO is more advantageous in GAD datasets with complex homophily relations in the normal nodes, such as Elliptic that shows diverse homophily strengths in the normal nodes (see Figure 1a).  As a result, RHO can significantly outperform GGAD on datasets like Elliptic, but it may not be able to outperform GGAD on the datasets like Amazon that have monotonous homophily in the normal nodes.  For the Reddit dataset, anomaly detection is particularly challenging due to the extremely high global average similarity among the nodes in this dataset, which is close to 0.99, making it very difficult to differentiate anomalies from normal nodes from various aspects. As a result, most baseline methods achieve an AUROC of only around 0.55; although GGAD and RHO perform better on this dataset, both still find it difficult to put anomalies at the top in the anomaly ranking. Modeling more discriminative anomaly/normality priors may be needed for more promising performance on Reddit."}]

Author rebuttal:

Meta-review: This paper proposes RHO (Robust Homophily Learning), a framework for semi-supervised graph anomaly detection that tackles the challenge of heterogeneous homophily among normal nodes. The method combines AdaFreq, adaptive frequency-response filters applied at both channel-wise and cross-channel levels, with Graph Normality Alignment (GNA), which enforces consistency across views via contrastive learning. Experiments on eight real-world datasets demonstrate consistent improvements over strong baselines, including newly added 2024 methods.

Strengths: The paper addresses an important and underexplored problem in GAD, introduces a novel and well-motivated methodological design, provides solid theoretical insights, and shows strong empirical performance with comprehensive ablations and runtime analysis. The writing is generally clear, and visualizations aid intuition.

Weaknesses: The initial version missed recent baselines and runtime details, though these were added in rebuttal. Some theoretical explanations and notations were confusing, and exploration of richer feature interactions was limited. The scope remains restricted to node-level detection.

Rebuttal/Discussion: Reviewers’ main concerns (theory clarity, missing baselines, feature interactions, runtime analysis) were carefully addressed with new experiments and clarifications. Two reviewers explicitly increased their ratings after rebuttal, and others confirmed that their concerns were resolved.

The paper makes a solid, original contribution to semi-supervised GAD with convincing results. While not strong enough for spotlight/oral due to scope and generality limitations, it is a clear accept.
Decision: Accept (poster)