Anomaly Subgraph Detection through High-Order Sampling Contrastive Learning

1 Introduction

Anomaly Detection (AD) is a vital application in real-world datasets. AD typically involves two primary tasks: Anomalous Node Detection (AND) and Anomaly Subgraph Detection (ASD). Existing AD approaches are usually categorized into two main groups: graph-based and deep learning-based approaches. Early traditional graph-based ASD approaches can obtain high-precision detection results in scenes with small-scale graphs and obvious anomaly features[Akoglu et al., 2015]. As the demand for large-scale data applications grows and AI technology advances, numerous deep learningbased AD approaches continue to be proposed. These approaches learn node representations through deep and shallow machine-learning models. Some of them project the node embeddings into Euclidean or hyperbolic space and identify outliers based on their distances. Others define anomaly-measuring functions to calculate anomaly scores for all nodes, with the top-K anomalous nodes identified as the detection result[Pang et al., 2021]. In summary, most recent AD approaches based on deep learning techniques primarily focus on identifying anomalies at the node level rather than detecting anomalous subgraphs with their internal structures.

∗Corresponding authors: Nannan Wu, Wenjun Wang

Graph anomalies often appear as structural anomalies and contextual anomalies. A structural anomaly refers to an anomalous pattern or behavior in the overall structure of a system or dataset. It involves deviations from the expected relationships, connections, or configurations within the data. A contextual anomaly refers to an anomalous behavior or pattern that is considered unusual within a specific context or environment, such as a specific time period. Anomaly subgraphs are clusters of nodes exhibiting anomaly features along with associated structures. Typically, an anomaly subgraph consists mostly of anomalous nodes, with only a minority being normal in it. However, detecting anomaly subgraphs faces multiple challenges: (1) The rarity of anomalous nodes in large graphs may result in ineffective representations of anomaly features for subgraphs. (2) The challenge of deriving anomaly features for subgraphs leads existing deep learning-based AD approaches to neglect the overall anomalous attributes of interconnected nodes (i.e., subgraphs) within graphs. (3) The limited capacity to represent subgraphs within existing deep-learning models hampers the development of an end-to-end ASD algorithm. (4) Detecting subgraph is an optimization problem with exponential growth, posing a significant challenge in terms of time cost. With the expansion of the graph's size, the number of all possible subgraphs increases exponentially, leading to a significant rise in computational costs.

Our paper aims to detect the maximum connected anomaly subgraph in a graph and proposes a novel end-to-end ASD approach. Initially, we investigate a high-order sampling strategy to build node-subgraph instance pairs for contrastive learning, as shown in Fig.1. Next, we utilize a GNN module to capture the representation embeddings of instance pairs within the graph. Additionally, we apply a self-supervised contrastive learning method to reveal the anomalous characteristics of nodes and subsequently rank them based on their anomaly scores. Furthermore, we establish an evaluation function utilizing Non-Parametric Scan Statistics (NPSS) to gauge the level of anomalous significance in a subgraph. Finally, we employ a suboptimal procedure to detect the maximum anomaly subgraph within the graph.

The main contributions of this paper are as follows.

• We propose a novel end-to-end algorithm ASD-HC designed for detecting anomaly subgraphs. ASD-HC employs a novel high-order neighborhood sampling and contrastive learning method to extract the anomaly features of nodes within the graph comprehensively.
• We present a suboptimal detecting anomaly subgraph method with non-parameter scan statistics assessing function, effectively reducing the run-time cost.
• Our approach is evaluated against five state-of-the-art deep learning-based anomaly detection algorithms on five real-world benchmark datasets, confirming its effectiveness and accuracy.

2 RelatedWorks

Traditional graph-based approaches achieve detection capabilities by extracting features from graph data[Wu et al., 2019; Sun et al., 2020; Sun et al., 2022]. However, they encounter limitations in scalability due to the constraints imposed by the size of the graphs on their algorithms. Most, if not all, recent anomaly detection approaches are based on deep learning techniques, with a predominant focus on anomalous node detection(AND). There are also a few approaches that tackle anomaly subgraph detection(ASD).

Contrastive learning-based AND: Contrastive learning, a self-supervised approach drawing insights from unlabeled data, has become popular in anomaly detection. It eliminates the need for manual labeling, reducing associated costs. [Jin et al., 2021] estimates node anomalous scores through the contrast of node & node and node& ego-net multi-scale instance pairs. In [Liu et al., 2022], contrastive learning is applied to capture anomaly features and calculate the anomaly scores of nodes. Its instance pairs consist of the node and its K-hop neighborhood which is constructed through a random walk procedure. Another approach is presented by [Hu et al., 2023], which detects anomalous nodes via the subgraphaligned contrastive learning across multiple views of the graph. [Duan et al., 2023] calculates the anomaly scores of nodes via contrastive learning among node-node,nodesubgraph, and subgraph-subgraph multi-scale instance pairs between the original view and the augmentation view.

Deep learning-based ASD: Recently, several deep learning-based ASD approaches have been proposed for different scenarios. For instance, [Wu et al., 2022] focuses on detecting anomaly subgraphs in multiple attributed networks within the federated learning framework. [Zhang et al., 2023] employs the k-hop Breadth-First Search strategy and Triadic closure to identify anomaly subgraphs in dynamic graphs. [Huang et al., 2023] proposes a Hybird-Order GAT model to detect anomalous nodes and anomalous motif instances within an attributed network. [Zhang and Zhao, 2022] employs a location-aware graph autoencoder to reconstruct the given graph and identify anomalous areas through a residual graph, which reveals disparities between the original and the reconstructed graphs. Hence, [Zhang and Zhao, 2022] is the only one among the above approaches, which detects the anomaly subgraph in scenarios most similar to our approach.

3 Preliminaries

In this paper, we first define an attributed graph as G = (V,E,A,X), where V and E are the node and edge sets, A ∈ Rn×n denotes the adjacency matrix of G, X ∈ Rn×m is the observed m dimensional attribute sets of nodes in G, E ⊆ V × V , n = |V | denotes the number of nodes (n ∈ R), S = (VS, ES, AS, XS) is a connected subgraph of G, where Vs ⊆ V and Es ⊆ E.

Definition 3.1 (Anomaly subgraph detection). Given G = (V,E,A,X) and S = (VS,ES,AS,XS), S ⊆ G is a connected subgraph of an attributed graph G, the function f (S) ∈ R measures the significantly anomalous lever of a subgraph S. If f (S) ≥ K (K is a constant), then S is defined as "Anomaly Subgraph".

To evaluate the significantly anomalous lever of a subgraph within a graph, we introduce the Higher Criticism(HC) statistic to define our objective function f (S). Notably, HC is one of the Non-Parametric Graph Scan statistics(NPGS) that have been utilized in road congestion detection, event detection, and other real-world applications. Our approach can also be generalized to other NPGSs.

Definition 3.2 (Maximum anomaly subgraph). Given the significant level of anomaly feature α (e.g.,0.25), the maximum anomaly subgraph of the graph G, denoted as ˆ S, is defined as follows:

where Nα(S) = Σvi∈VS δ(p(ϖi) ≤ α) denotes the number of anomalous nodes of S, ϖi is the anomaly score of node vi, and p(ϖi) denotes the p-value associated with ϖi. As for the function δ(·) = 1 if its input is True, otherwise δ(·) = 0. N(S) = Σvi∈VS δ(1) is the total number of nodes in S.

4 Methodology

4.1 High-Order Neighborhood Sampling Strategy

This paper proposes a high-order neighborhood sampling method to find the k-order neighbor subgraph of each node. Firstly, the high-order neighborhood weight of the graph is represented by matrix B = (⃗b(1) , ⃗b(2) , .., ⃗b(k) ) ∈ Rn×k , where vector ⃗b(k) ∈ Rn denotes the number of k-order neighbors for each node. Let ⃗b(1) i = ∑nj=1 a(i, j), 1 ≤ i ≤ n, where a(i, j) denote the elements of the adjacency matrix A. Secondly, the k-order neighborhoods of nodes are calculated as the following equation:

where k is a hyperparameter. ⃗b(k) is also represented as a column vector, where each element quantifies the influence of the k-order neighborhood of each node. Therefore, the korder maximum influential neighbor ˆ υi of the current node vi is defined in Eq.(3).

where Ni denotes the neighbors of the node vi. Our highorder neighborhood sampling strategy sets ˆ υi as the next node of vi in the k-order neighbor subgraph.

4.2 Contrastive Learning of Anomaly Features

To enhance the efficiency of our algorithm, we initially preprocess the data and subsequently generate the candidate sets of positive and negative samples, i.e., positive and negative neighbor subgraphs, respectively. In our paper, we extract k-order maximal/minimal neighbor-subgraphs from the high-order neighborhood to construct contrastive instance pairs. EPi denotes the positive instance pair comprising a target node vi and its corresponding k-order maximal neighborsubgraph, denoted as Pi (initiated with vi). On the other hand, the negative instance pair ENi consists of vi and the korder minimal neighbor-subgraph derived from another node. They are defined as follows:

To facilitate the comparison of nodes and subgraphs in positive and negative instance pairs, it is essential to execute the training process for nodes and subgraphs within the same parameter space.

1). Node Embedding: Our approach trains the node embeddings using the weight matrix W and the activate function σ(·), consistent with the GCN module, the input is the attribute of nodes without the structural data. Therefore, the module is similar to the MLP:

where h(0) vi = X(i) is the attribute vector of node vi, i.e., the i-th row of the matrix X associated with the graph G.

2). k-Order Neighbor-subgraph Embedding: As the GNN[Scarselli et al., 2009] method is widely used for processing graph data, our paper leverages this model to learn embeddings of nodes in the k-order neighbor-subgraph from the graph's structure and attribute information. In this paper, we conduct experiments using the GCN [Kipf and Welling, 2017] model. Additionally, the GCN module can be substituted with other GNN models, such as GAT[Veliˇckovi´c et al., 2018], GraphSAGE[Hamilton et al., 2018], etc. For an attributed graph G = (V,E,A,X), our GCN module calculates the following iteration equation:

where ˜ A = A + I, I is an identity matrix, ˜D = D + I, D is the degree matrix of graph G, σ(·) is an activation function(e.g. ReLU), W (l) is the l -layer trainable parameter matrix, H(0) = X is the initial input matrix. Therefore, the korder neighbor-subgraph Pi, being a set of connected nodes, undergoes its representation learning through the GCN module, utilizing the adjacency matrix APi and the attribute matrix XPi , then Eq.(6) is rewritten as the following equation:

For computational convenience, a readout layer follows the output of Eq.(7) to transform the matrix into a vector.

where t = |Pi| is the number of nodes in the k-order neighbor-subgraph Pi, HPi (j) is the j-th row of matrix HPi .

• 3). Loss of Contrastive Learning Module We employ the Bilinear function to evaluate the relationships x between node embeddings hv and k-order neighbor-subgraph embeddings hP , and the relationships of the positive and negative instance pairs x(+) , x(-) are denoted by Eq.(9), respectively.

The loss function in our contrastive learning module utilizes the Binary Cross-Entropy Loss function in conjunction with a Sigmoid layer, shown in Eq.(10).

where nB is the batch size, σ() denotes the Sigmoid function, y represents the labels of positive(EP ) and negative(EN) instance pairs, defined as follows.

• 4). Anomaly Score Function: The intuitive concept of contrastive learning is to train the correlation value of a node's positive instance pair, x(+) , to approach 1, and the node's negative instance pair, x(-) , to approach 0. The majority of nodes in the graph are expected to be normal and adhere to the rule stated above. However, for the anomalous node, the values of x(+) and x(-) deviate from this expected pattern. Like the other contrastive learning model, we introduce the anomaly score function (shown in Eq.(12)) to derive the anomaly features for nodes in the graph.

where i denotes the node vi. When vi is a normal node, it can be inferred from the above equation that ϖi is a negative number, ideally set to -1. Conversely, vi is an anomalous node, ϖi will be greater than -1. The greater the anomaly of the node, the farther the value of ϖi deviates from -1. The vector ϖ represents the anomaly features of each node learned through our approach.

Algorithm 1 Anomaly Subgraph Detection With High-Order Neighborhood Sampling Contrastive Learning (ASD-HC)

Input: G = (V,E,A,X), k (k-order), t (the k-order neighborsubgraph size), E (number of epoch), nB (the batch size), the anomalous threshold α

Output: Maximal connected anomaly subgraph ˆ S

• 1: Randomly initializing the parameters of model W;
• 2: Initialization: ⃗b(1) = ∑ a(i, j), i, j ≤ |V |; e ← 0;
• 3: Calculate the k-order vector ⃗b(k) = Ak-1⃗b(1) ;
• 4: for all node i ∈ V do
• 5: Generating positive subgraphs {Pi} with max(⃗b(k) i );
• 6: Generating negative subgraphs {Ni} with min(⃗b(k) i );
• 7: end for
• 8: while e ≤ E do
• 9: Selecting the batch set {v0, v1, ..., vnB } ∈ V
• 10: for vi ∈ {v0, v1, ..., vnB } training do

11: EP ← (vi, Pi); EN ← (vi, Pj), s.t. i ̸= j;

• 12: Embedding by GCN : hvi , hPi as Eq.(5), (8);
• 13: Quantifying EPi and ENi : x(+) i , x(-) i as Eq.(9);
• 14: Defining labels of EPi and ENi : y(+) i ← 1, y(-) i ← 0;
• 15: TrainingW as loss: L←BCEWithLogitsLoss(x, y);
• 16: end for
• 17: end while
• 18: for all nodes testing do
• 19: Calculating anomaly score ϖi ← x(-) i -x(+) i , vi ∈ V ;
• 20: end for
• 21: Filtering out anomalous nodes: Ω ← {vi|p(ϖi) ≤ α}
• 22: RWR start with Ω in G: → {S0, S1, S2, ...}
• 23: while S ∈ {S0, S1, S2, ...} do
• 24: ˆ S ← arg max f (S)
• 25: end while
• 26: return ˆ S;

4.3 Anomaly Subgraph Detection Algorithm

To create an efficient algorithm for detecting connected anomaly subgraphs, we adopt the following steps:

• 1 Assuming that the ϖ values follow a uniform distribution, then calculate the probability of ϖ, i.e. p(ϖ).
• 2 Filtering out the anomalous nodes with p(ϖ) values less than a specified threshold α.
• 3 Constructing connected anomaly subgraphs within a larger graph based on the filtered anomalous nodes via the Random Walk with Restart (RWR) algorithm.
• 4 Assessing the candidate connected anomaly subgraphs with NPGS in Eq.(1).
• 5 Iterative optimization processing.

Based on hypothesis testing and statistical theory, if the null hypothesis is valid and the distribution of ϖ should theoretically be uniform, the p-values are expected to follow a normal distribution within the interval [0, 1]. The p-value of the observed data is relatively low under the null hypothesis, which means that the observed data represent an even more extreme scenario. This provides evidence for rejecting the null hypothesis. Consequently, in our paper, we establish a predefined threshold α (e.g., α = 0.25), then when the probability of a node's anomalous score ϖi, represented as p(ϖi), is less than or equal to α, it indicates that the node vi has a high likelihood of being an anomalous node. A collection of anomalous nodes Ω is filtered based on the condition of p(ϖi) ≤ α.

As we know, the computational complexity of graph detection increases exponentially with larger graph sizes. The computational cost is one of the challenges of the graph detection process. Our approach employs the RWR algorithm [Tong et al., 2006] to construct candidate connected anomaly subgraphs. The induced subgraphs are constructed around a set of anomalous nodes Ω = {vi|p(vi) ≤ α} but are not confined solely to these anomalous nodes. Therefore, this strategy addresses the challenges associated with traditional anomaly subgraph detection algorithms and significantly mitigates the computational demands for subgraph identification in large graphs.

5 Experiments

Our approach and baselines are all evaluated on the five benchmark datasets, as shown in Table 1. We explore all experiments on the GPU: GeForce GTX 1080 Ti (11GB) ×2, with a total memory of 128GB.

5.1 Dataset

Our paper leverages five widely recognized datasets to assess the performance of our algorithm and baselines. These datasets are commonly employed in other deep learningbased approaches[Zheng et al., 2023; Liu et al., 2022; Jin et al., 2021], and their key information is outlined in Table 1. These datasets are categorized into two types:

Citation Networks: Cora 1 [McCallum et al., 2000], CiteSeer 2 [Lawrence et al., 1999], ACM[Sen et al., 2008][Tang et al., 2008]. In these graphs, nodes represent corresponding documents, edges denote links between those documents, and their node features are extracted from the text contents of each document.

Social Networks: BlogCatalog 3 and Flickr[Tang and Liu, 2009]. BlogCatalog data is extracted from the blog-sharing website BlogCatalog, while Flickr is sourced from the imagesharing platform Flickr. In these two graphs, nodes represent users of the respective websites, and edges signify the relationships between users. Additionally, the features of nodes are derived from the tag contents of the posted blogs and photos, respectively.

1 www.cora.justresearch.com

2 www.scienceindex.com

3 http://www.blogcatalog.com

5.2 Experiment Setting

Metrics: We use the AUC metric to evaluate both anomalous node detection and anomaly subgraph detection, which is commonly used to evaluate the performance of a binary classification model. AUC quantifies the model's ability to distinguish between positive and negative instances. A higher AUC value, closer to 1, indicates better performance.

Baselines: Our experiments explore five state-of-the-art deep learning-based baselines for anomalous node detection and one method (AS-GAE) for anomaly subgraph detection.

1. DOMINANT[Ding et al., 2019] employs a deep graph autoencoder method and utilizes graph structure and features for detecting anomalous nodes in a graph.
2. ANEMONE[Jin et al., 2021] is an anomalous node detection method based on Graph Neural Networks (GNN), aiming to identify graph anomalies using multiscale patch and context-level contrastive learning.
3. CoLA[Liu et al., 2022] is an anomaly detection algorithm focusing on nodes, utilizing a GNN-based contrastive learning method at the node-subgraph level. It computes anomaly scores for nodes by evaluating representations derived from nodes and subgraphs in positive and negative instance pairs.
4. GRADATE[Duan et al., 2023] is an anomalous node detection approach based on node-node, node-subgraph, and subgraph-subgraph multi-view contrastive learning.
5. AS-GAE[Zhang and Zhao, 2022] is an anomaly subgraph detection algorithm utilizing a graph autoencoder. It generates a residual graph by reconstructing the given graph and detects anomaly subgraphs by identifying different areas between the original graph and its reconstructed graph.

Anomaly injection: To evaluate the detecting ability of our algorithm, having anomaly ground truth in datasets is essential. However, all experimental datasets lack anomaly labels. We explore two types of anomaly injection: structural anomaly injection and attribute anomaly injection. We randomly select q nodes and induce q connected subgraphs using a random walk approach and then transform them into q fully connected subgraphs. Similarly, we randomly sample q connected subgraphs with the same number of nodes as Set T, and select k nodes as Set C. The attribute of each node in T is perturbed based on the Euclidean distance between it and the randomly selected node in C. The number of anomalies in each dataset can be found in Table 1.

5.3 Experimental Result Analysis Anomaly Detection Evaluation

In this section, we focus on evaluating the performance of our approach in detecting anomalous nodes within the graph and comparing it with other baseline methods. Due to the prevalent use of the ROC curve and the Area Under the Curve (AUC) metric for evaluating binary classification algorithms, our experiments utilize these two metrics to assess our algorithm's capability in distinguishing anomalous nodes from normal nodes within a graph. The figures from our experiments, as indicated in Table 2, illustrate that our approach outperforms the baselines across five benchmark datasets. In the Cora dataset, our approach, ASD-HC, achieves the highest AUC score of 0.9217, surpassing the competitive baseline ANEMONE by 2.42%. ASD-HC also attains the highest AUC score of 0.8786 in ACM, outperforming the competitive baseline ANEMONE by 3.88%. Similarly, in the Flickr dataset, ASD-HC achieves a score of 0.7719, consistently surpassing the baselines by at least 2.51%. Moreover, in the BlogCatalog dataset, ASD-HC and ColA both attain the highest score of 0.7800, exceeding other baselines by 13.35%. Although our approach does not maintain the highest score in the Citeseer dataset, it is only 0.8% less than ANEMONE.

Based on the experimental data depicted in Figure 3, when the average degrees of nodes are small, such as in Cora, ACM, and Citeseer as shown in Figure 3(f), where the degrees are all less than 65.5, ANEMONE serves as the competitive baseline for our approach. When the degrees of graphs increase, CoLA becomes the primary competitive baseline for ASDHC. In both scenarios, whether compared with ANEMONE or CoLA, ASD-HC either surpasses them or closely matches their performances. Therefore, our focus is directed toward analyzing the distinctions between ASD-HC and them. ANEMONE learns node embeddings through node&node and node&ego-net contrastive instance pairs within multiviews. It considers the influence from the node's direct neighbor to the target node. CoLA's result is derived from its nodesubgraph contrastive instance pairs by employing the random walking method to sample 4 nodes for each subgraph. The choice of 4 is the default setting in its original algorithm and serves as the walking path length in RWR, which aligns with the ideal value described in its paper. We attribute the superior performance of ASD-HC over ANEMONE and CoLA to our high-order neighbor-subgraph sampling strategy, which enables a more comprehensive capture of the characteristics of node structure and relationships between key neighbors.

Compared to the state-of-the-art algorithms in deep learning-based anomaly detection within graph structural data, these results highlight the excellent ability of our approach to detect anomalous nodes in a graph. Figure 3

presents detailed data from our experiments on ROC curves. The ROC curve is generated by calculating True Positive Rate (TPR) and False Positive Rate (FPR) scores at various thresholds between predicted anomalous and normal instances. These curves are then plotted to illustrate the performance characteristics of the models. Figure 3 reveals that the AUC score corresponds to the values presented in Table 2. The ROC curves represented by the red lines in each dataset also demonstrate the effectiveness of our approach. However, in these experiments, we only assess the performance of detecting anomalous nodes for ASD-HC and all baselines. Anomalous node detection is not really our ultimate goal. Therefore, there is a need to further assess our algorithm and AS-GAE in the context of anomaly subgraph detection.

Anomaly Subgraph Detection Evaluation

After anomaly features have been extracted from each node through contrastive learning, we predict the distribution of the anomalies within the entire graph. Under the Uniform distribution, we calculate the p-values (probabilities) for all nodes' anomalous scores p(ϖ). Anomalous nodes are identified by filtering out those with p-values lower than the specified significantly anomalous threshold (i.e., p(ϖi) ≤ α). In our experiments, the threshold α is set to 0.25. Additionally, our approach employing the RWR algorithm is designated as "ASD-HC(RWR)." To enable a thorough comparison, we replace RWR in our approach with the Breadth-First Search (BFS) algorithm, creating a variant denoted as "ASDHC(BFS)." Additionally, AS-GAE serves as another baseline in the anomaly subgraph detection experiment. We evaluate the nodes in the maximum anomaly subgraph detected by our two variants ASD-HCs and AS-GAE, calculating AUC metrics, as shown in Table 3. The AUC metrics of both our ASD-HCs consistently outperform the AUC of AS-GAE, demonstrating an improvement of at least 13.01%. Furthermore, the detection result of AS-GAE is a set of nodes, not refined into anomalous and normal nodes. Hence, we represent its node count in the "N(S)" column. When comparing the two ASD-HCs, the subgraphs detected by ASD-HC(BFS) are notably larger and almost equal to the entire graph. Although ASD-HC(BFS)'s AUC slightly outperforms that of ASD-HC(RWR), this advantage is not deemed satisfactory. According to the values of f( ˆ S) in Table 3, the subgraph detected by ASD-HC(RWR) demonstrates the most significantly anomalous levers on all benchmark datasets. In summary, ASD-HC(RWR) exhibits the most optimal capability in detecting anomaly subgraphs. Furthermore, ASD-HC(BFS)'s running time is longer compared to ASD-HC(RWR).

Parameter Studying

In our experiment, the batch size is set to 200, and the number of epochs is set to 100 for smaller graphs (e.g., Cora and Citeseer). For larger graphs with a substantial number of nodes or edges (e.g., ACM, Flickr, and BlogCatalog) reflecting more complex graph structures, additional training time is required. In such cases, we set the number of epochs to 400. The learning rate is set in the range of [0.001, 0.0035]. In addition to these above common parameters, our algorithm introduces three other crucial parameters: the order of neighbors (denoted as k) that we consider, the size of the neighbor-subgraph (denoted as t), and the significant level of the anomaly α. Typically, we set k equal to 3, t to 15, and assign a value to α within the range of [0.15, 0.25].

5.4 Ablation Studies

Effect of the Size of Neighbor-Subgraph

In Figure 4(a), with the order of neighbors k set to 3, the size of the neighbor-subgraph t increases accordingly. AUC shows a rising trend, reaching optimal values when t is in the range of [15, 20], and then followed by a decline. This pattern is consistent across most datasets. However, in the BlogCatalog dataset, the AUC does not exhibit a downward trend until t > 35. After analysis, BlogCatalog has the highest edgenode ratio, indicating a tighter and more complex graph structure with stronger correlations between nodes. Enlarging the subgraph size t is beneficial for capturing node features with greater accuracy.

Effect of the High-Order of Neighbors

Taking the Cora dataset as an example, although specifying four different orders of neighbors (k = 1, 2, 3, 4), the AUC value achieves its optimal value when t is set to 15. However, when increasing the value of k, the AUC exhibits an upward trend, as shown in Figure 4(b).

6 Conclusion

Detecting anomaly subgraphs is vital for numerous realworld applications. Most existing anomaly detection methods overlook the identification of anomaly subgraphs in the graph data. In this paper, we present ASD-HC, a novel end-to-end Graph Neural Network (GNN)-based contrastive learning approach designed to detect the maximum connected anomaly subgraph within graph-structured data. ASD-HC employs a high-order neighborhood sampling strategy to construct node-subgraph instance pairs. It extracts anomaly features of nodes and defines an evaluation function based on the Non-Parametric Graph Scan statistics to detect the maximum connected anomaly subgraph. Our approach highlights its capabilities in detecting anomalous nodes and anomaly subgraphs, especially, outperforming baselines by over 13.01% in detecting the maximum anomaly subgraph. Experiment results show our approach effectively detects anomaly subgraphs in large-scale graphs.

Acknowledgements

This work is supported by the National Key Research and Development Program of China(31400) and the Natural Science Foundation of Inner Mongolia(2022LHMS06008). It is also partially sponsored by the Key R&D and Transformation Plan of Qinghai Province(2022-QY-218).