LineMVGNN: Anti-Money Laundering with Line-Graph-Assisted Multi-View Graph Neural Networks

Academic Editor: Xianrong (Shawn) Zheng Received: 17 February 2025 Revised: 20 March 2025 Accepted: 28 March 2025 Published: 3 April 2025 Citation: Poon, C.-H.; Kwok, J.; Chow , C.; Choi, J.-H. LineMVGNN: Anti-Money Laundering with Line-Graph-Assisted Multi-V iew Graph Neural Networks. AI 2025 , 6 , 69. https://doi.org/10.3390/ai6040069 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.or g/ licenses/by/4.0/). Article LineMVGNN: Anti-Money Laundering with Line-Graph-Assisted Multi-V iew Graph Neural Networks Chung-Hoo Poon 1,2, * , James Kwok 2 , Calvin Chow 1, * and Jang-Hyeon Choi 1 1 Logistics and Supply Chain MultiT ech R&D Centre, Level 11, Cyberport 2, 100 Cyberport Road, Hong Kong 2 Department of Computer Science and Engineering, The Hong Kong University of Science and T echnology , Clear W ater Bay , Kowloon, Hong Kong * Correspondence: chpoonag@connect.ust.hk (C.-H.P .); cchow@lscm.hk (C.C.) Abstract: Anti-money laundering (AML) systems are important for pr otecting the global economy . However , conventional rule-based methods rely on domain knowledge, lead- ing to suboptimal accuracy and a lack of scalability . Graph neural networks (GNNs) for digraphs (directed graphs) can be applied to transaction graphs and capture suspi- cious transactions or accounts. However , most spectral GNNs do not naturally support multi-dimensional edge features, lack interpretability due to edge modifications, and have limited scalability owing to their spectral natur e. Conversely , most spatial methods may not capture the money flow well. Ther efore, in this work, we propose LineMVGNN ( Line -Graph-Assisted M ulti- V iew G raph N eural N etwork), a novel spatial method that considers payment and receipt transactions. Specifically , the LineMVGNN model extends a lightweight MVGNN module, which performs two-way message passing between nodes in a transaction graph. Additionally , LineMVGNN incorporates a line graph view of the original transaction graph to enhance the propagation of transaction information. W e conduct experiments on two real-world account-based transaction datasets: the Ethereum phishing transaction network dataset and a financial payment transaction dataset from one of our industry partners. The results show that our proposed method outperforms state-of-the-art methods, r eflecting the effectiveness of money laundering detection with line-graph-assisted multi-view graph learning. W e also discuss scalability , adversarial robustness, and r egulatory considerations of our proposed method. Keywords: graph neural networks; anti-money laundering; transaction graphs 1. Introduction Money laundering is the process of disguising the origins of illegally obtained funds to make them appear legitimate. Existing anti-money laundering (AML) systems can be generally categorized into two gr oups: rule-based methods and machine learning-based methods. Rule-based methods are popular among commer cial institutions due to simple and easy-to-code rules pr edefined by domain experts [ 1 ]. However , it is time-consuming and labor -intensive to keep updating the rules under ever -evolving data. Machine learning- based anti-money laundering (AML) systems leverage large volumes of historical transac- tion data to learn models. For instance, support vector machine, logistic regression, k-means clustering, k-nearest neighbors, random forests, MLP , etc. [ 2 ]. In recent years, graph neural networks (GNNs) have emerged as the de facto tool for graph learning tasks. Example application domains of GNNs for fraud detection tasks include credit car d transactions, e-payment data, and cryptocurrency transactions [ 3 ]. AI 2025 , 6 , 69 https://doi.org/10.3390/ai6040069 AI 2025 , 6 , 69 2 of 21 Some GNNs for digraphs (directed graphs) ar e suitable for money laundering detec- tion tasks since transaction graphs ar e directed. However , some not naturally supporting edge features ar e unsuitable for node-and-edge-attributed transaction graphs. This type of graph is common for account-based transaction graphs, in which nodes and edges r epresent accounts and transactions, respectively . Recent state-of-the-art (SOT A) GNNs for digraphs can be categorized into spectral methods [ 4 – 10 ] and spatial methods [ 11 – 13 ]. Spectral GNNs perform graph convolutions by applying graph signal filters in the spectral domain, while spatial GNNs aggregate feature information fr om neighboring nodes directly in the spatial domain. Although most of the recent works focus on the spectral methods, spectral GNNs for digraphs may not be suitable for attributed account-based transaction graphs, because (1) they do not naturally support multi-dimensional edge features, and (2) they have limited scalability due to their spectral natur e. Usually , full graph propagation is needed during training, and the number of required graph propagations depends on the degree of the polynomial graph filter . In contrast, spatial GNNs for digraphs may be more suitable for transaction graphs due to the nature of being attributed and dir ected. However , there is a lack of exploration of spatial GNNs for digraphs. Several relevant papers did not empirically validate the extension of GNNs to digraphs [ 13 – 15 ], and some do not apply to transaction graphs due to differ ent reasons, such as graph structural constraints [ 12 ], or aggregation from only out-neighbors [ 13 ]. Dir-GNN [ 11 ] is a generic spatial digraph GNN framework, but the use of separate sets of learnable parameters for in- and out-neighbors may be r edundant for some domains such as transaction data. Our experimental results show a good model performance despite parameter sharing. However , we design our models within this framework due to its genericness. In addition, specifically for AML detection, identifying suspicious accounts by the relevant transactions depends on cash flow information. Nevertheless, such information may not be effectively captured by SOT A GNNs for digraphs such as DiGCN [ 7 ], Mag- Net [ 6 ], SigMaNet [ 5 ], FaberNet [ 4 ], and Dir-GNN [ 11 ]. As an illustrative example shown in Figure 1 , suppose a series of path-like transactions exist in a transaction graph. The corresponding accounts can be considered suspicious when they serve as temporary repos- itories for funds [ 16 ]. Identifying these suspicious accounts requires identifying suspicious transactions, and identifying a suspicious transaction requir es information from (past) receipt and (futur e) payment transactions, such as comparing the transaction time and the transaction amounts, to acquire the money flow information. Although stacking GNN layers allows information propagation, the edge-to-edge (transaction-to-transaction) infor- mation exchange is indirect and less detailed. Also, the first GNN message passing will be meaningless since raw edge (transaction) attributes are aggr egated without comparing with other relevant transactions. Early propagation of edge (transaction) information and edge updates before the first iteration of GNN message passing can ease the learning of a GNN model for suspicious account detection. T o capture inter -edge interactions, line graphs of edge adjacencies L ( G ) can be lever- aged for edge feature propagation. The line graph L ( G ) is transformed from the input graph G , where L ( G ) encodes the directed edge adjacency structure of G using the non- backtracking matrix, allowing information to propagate along the directed edges while preserving orientation. W e propose LineMVGNN which aggr egates and propagates trans- action information in the line graph before node (account) feature updates in the input graph G . Our main contributions are as follows: AI 2025 , 6 , 69 3 of 21 • MVGNN is introduced as a lightweight yet effective model within the Dir-GNN framework due to its genericness. It supports edge features and considers both in- and out-neighbors in an attributed digraph, such as a transaction graph. • LineMVGNN is proposed, extending MVGNN by utilizing the line graph view of the original graph for the effective pr opagation of transaction information (edge features in the original graph). • Extensive experiments are conducted on the Ether eum phishing transaction network and the financial payment transaction (FPT) dataset. The remainder of this paper is organized as follows. Section 2 provides an overview of related work in anti-money laundering (AML) and graph neural networks (GNNs) for directed graphs. Section 3 introduces the problem statement and the mathematical framework for our proposed method, including its two-way message passing mechanism and line graph view . Section 4 presents the experimental setup, datasets, and results, comparing LineMVGNN with SOT A methods. Section 5 discusses the limitations of the proposed method and potential futur e work. Finally , Section 6 concludes the paper with a summary of contributions. Figure 1. Hypothetical examples of benign and suspicious accounts in path patterns. 2. Related W ork 2.1. GNNs for Digraphs 2.1.1. Spectral Methods Recently , r esearchers have extended spectral convolutions to directed graphs [ 8 – 10 ]. In particular , DiGCN [ 7 ] uses an approximate digraph Laplacian based on personalized PageRank [ 17 ] and incorporates kth -order proximity to produce k receptive fields; Mag- Net [ 6 ] utilizes a complex Hermitian matrix, namely the magnetic Laplacian, to encode both undirected structur e and directional information; SigMaNet [ 5 ] unifies the treatment of undirected and directed graphs with arbitrary edge weights by introducing the Sign- Magnetic Laplacian to extend spectral graph convolution theory to graphs with positive and negative weights; FaberNet [ 4 ] utilizes advanced complex analysis and spectral theory to extend spectral convolutions to directed graphs. Nevertheless, spectral GNNs might not be appropriate for edge-attributed transaction graphs, because they do not inher ently accommodate multi-dimensional edge characteristics and their scalability is r estricted by their spectral characteristics. T ypically , full graph pr opagation is needed during training, with the necessary number of propagations contingent on the degree of the polynomial graph filter . AI 2025 , 6 , 69 4 of 21 2.1.2. Spatial Methods Though the extension of spatial models to directed graphs is suggested in several classical papers, empirical experiments are not conducted [ 13 – 15 ]. GGS-NNs [ 13 ] handles directed graphs but only aggregates from out-neighbors. It neglects in-neighbor informa- tion and edge features. Although DAGNN [ 12 ] supports edge features, it is limited to directed acyclic graphs. Extended from the Message Passing Neural Network (MPNN) framework [ 14 ], Dir-GNN [ 11 ] is a spatial GNN that accounts for edge directionality by separately aggregating messages from in-neighbors and out-neighbors. Although the importance of the two types of messages is differ entiated by a hyperparameter , Dir-GNN cannot capture the element-wise interaction between these two types of messages. It also lacks efficiency as it doubles the number of parameters to separately aggregate the two types of messages. Overall, both spectral and spatial methods fail to effectively addr ess the challenges of transaction-based tasks due to their inability to propagate edge-level information ef fectively . This limitation is critical for tasks like AML detection, where transaction-level details, such as cash flow patterns, are essential for identifying suspicious activities. Spectral methods are further hindered by their lack of support for multi-dimensional edge features and scalability issues, while spatial methods may struggle to capture nuanced interactions between in- and out-neighbors, leading to inefficiencies and suboptimal performance. 2.2. Edge Feature Learning and Line Graphs Co mb in in g no de a nd e dg e fe at ure le ar ni ng h as b ee n su gg es te d by s om e re- se arc he rs [ 14 , 18 ]. Recently , LGNN [ 19 ] operates on line graphs of edge adjacencies, lever- aging a non-backtracking operator to enhance performance on community detection tasks. However , the model is r estricted to undirected graphs. Multiple works also leverage line graphs for learning edge embeddings, but only for link prediction [ 20 , 21 ]. Although node and edge featur e information usually complement each other , edge feature pr opagation and learning for node classification tasks remain r elatively unexplored. T o incorporate multi-dimensional edge features for node feature updates, Zhang et al. [ 22 ] proposes a graph representation learning framework that generates node embeddings by aggregating local edge embeddings. Nevertheless, the learned edge embeddings do not effectively capture the interactions between edges, since the edge embeddings are learned from the concatenation of the edge featur es and the features of the corr esponding end nodes. T o capture inter-edge interactions and enhance node embeddings, line graphs of edge adjacencies can be leveraged, but this remains r elatively unexplored for node classification tasks. CensNet [ 23 ] co-embeds nodes and edges by switching their roles using line graphs, but is limited to undirected graphs without parallel edges because of the use of spectral graph convolution. LineGCL [ 24 ] transforms the original graph into a line graph to enhance the repr esentation of edge information and facilitate the learning of node features by contrastive learning, but it assumes no edge features in the original graph. 3. Proposed Method 3.1. Problem Statement Consider a directed transaction graph G = ( V , E ) , where V is the set of nodes { v 1 , . . . , v n } and E is the set of edges { e 1 , . . . , e m } . Each node repr esents an account and each edge represents a transaction. An edge’s source node and destination node repre- sent the payer and payee of the corresponding transaction, respectively . Each node v is attributed with x v and each edge is attributed with e vw . Each node is either licit or illicit, AI 2025 , 6 , 69 5 of 21 and is repr esented by 0 and 1, respectively . A subset of nodes are unlabeled. W e aim to learn a model that predicts the label y v of each unlabeled node v . 3.2. T wo-Way Message Passing Information from both the in- and out-neighbors and edges is important for node classification. In a digraph, however , messages from the out-neighbors and edges are usually ignored in traditional GNNs such as GCN [ 25 ], GraphSAGE [ 26 ], GIN [ 27 ], etc. In addition, edge attributes are pr esent in some real-world data, and they can be significantly more informative than node attributes. Therefore, we build our simple MVGNN ( M ulti- V iew G raph N eural N etwork) model within the Dir -GNN framework [ 11 ]. As illustrated in Figure 2 , the model considers messages from both in-neighbors and out-neighbors. Our MVGNN differs fr om Dir-GNN by (1) parameter sharing between message aggr egation functions for in- and out-neighbors, and (2) the message combination function for the two types of messages. Details are pr ovided below . Figure 2. V isual illustration of two-way message passing in MVGNN. At the ( l + 1 ) -th MVGNN layer , aggregation of messages m ( l + 1 ) v in and m ( l + 1 ) v ou t can be expressed by the following: m ( l + 1 ) v in = ∑ w k ∈ N in ( v ) M ( l + 1 ) in  [ h ( l ) w ∥ e vw ]  , (1) m ( l + 1 ) v ou t = ∑ w k ∈ N ou t ( v ) M ( l + 1 ) o ut  [ h ( l ) w ∥ e vw ]  , (2) where M ( l ) in and M ( l ) o ut are fully connected layers with ReLU activation, N in ( v ) and N o ut ( v ) denote the in- and out-neighbors of node v , e vw denotes the features of an edge between node v and w , and ∥ denotes concatenation. The node embeddings are updated with these messages by the equation below: h ( l + 1 ) v = U ( l + 1 )  h ( l ) v , C ( l + 1 )  m ( l + 1 ) v in , m ( l + 1 ) v ou t  , (3) where the vertex update function U ( l + 1 ) is a fully connected layer with ReLU activation. For the message combination function C ( l + 1 ) , we experiment with 2 different methods. For brevity and clarity , superscript is omitted, i.e., C ( l + 1 ) , m ( l + 1 ) v in , and m ( l + 1 ) v ou t will become C , m v in , and m v ou t respectively . Combine by W eighted Sum. T o differ entiate the importance of messages from in- neighbors and out-neighbors, similarly to [ 11 ], we propose a variant MVGNN- add , which multiplies weight scalars with m v in and m v ou t : C  m ( l + 1 ) v in , m ( l + 1 ) v ou t  = α l + 1 m ( l + 1 ) v in + ( 1 − α l + 1 ) m ( l + 1 ) v ou t , (4) where α l + 1 ∈ R is a learnable scalar (instead of a hyperparameter as in [ 11 ]). AI 2025 , 6 , 69 6 of 21 Combine by Concatenation. T o better model the element-wise interaction between messages from these two types of neighbors, MVGNN- cat is proposed (instead of using weighted sum as in [ 11 ]). The two types of messages are concatenated and then passed to a linear layer F ( l + 1 ) : C  m ( l + 1 ) v in , m ( l + 1 ) v ou t  = F ( l + 1 )  [ m ( l + 1 ) v in ∥ m ( l + 1 ) v ou t ]  . (5) T o boost model efficiency , we share parameters for aggr egation maps, message combi- nation maps, and vertex update functions for both types of neighbors in our experiments. In other words, the primary power of distinguishing between in- and out-messages lies in the learnable parameter α , or the linear layer F . In experiments, this proposed model remains competitive. In short, at a high level, the ( l + 1 ) -th MVGNN layer can be expressed as follows: h ( l + 1 ) v = MVGNNLayer ( l + 1 )  { h ( l ) u , h ( l ) w , e ( l ) vu , e ( l ) vw | u ∈ N in ( v ) , w ∈ N o ut ( v ) }  , (6) where MVGNNLayer = { MVGNNLayer-add , and MVGNNLayer-cat } , e ( l ) vu and e ( l ) vw are edge embeddings. Since there are no edge updates, e ( l ) vu = e vu and e ( l ) vw = e vw . “MVGNNLayer-add” denotes message combination by weighted sum defined in Equation ( 4 ), while “MVGNNLayer -cat” denotes message combination by concatenation followed by a linear layer . T o prevent over-smoothing neighbors’ information, similar to [ 28 ], personalized PageRank-based aggregation mechanism of GNN [ 29 ] is adopted to obtain the final aggre- gated embedding z v of node v : z v = L − 1 ∑ l = 1 β l h ( l ) v + 1 − L − 1 ∑ l = 1 β l ! × h ( L ) v , (7) where β l ∈ R is a learnable parameter . Subsequently , a fully connected layer maps each z v to a prediction vector y v . 3.3. Line Graph V iew Whether a transaction is illicit also depends on the previous transactions due to the significance of the money flow . T o facilitate capturing this information, we leverage line graphs to perform edge feature propagation. T o the best of our knowledge, few studies have leveraged line graphs in anti-money laundering. One example is LaundroGraph [ 30 ] which repr esents edges as nodes and implements a GNN on the line graph. The original transaction graph G is transformed into a line graph G ′ = L ( G ) , in which each node t ( v , w ) ∈ G ′ corresponds to edge e vw ∈ G (transaction from w to v ). In the line graph G ′ , a directed edge e ′ t ( v , w ) t ( u , s ) exists if u = w (i.e., the payee u of transaction t ( u , s ) is the payer w of transaction t ( v , w ) ). Figure 3 and Algorithm 1 show how our pr oposed LineMVGNN leverages the line graph view and propagates edge featur es. In the line graph G ′ , edge feature pr opagation is performed with G ′ before each message passing in G . LineMVGNN uses separate MVGNN layers for the original graph G and the line graph G ′ , respectively , as defined in Equation ( 6 ). T o seamlessly involve the line graph view in every message passing in the original graph G , inspired by the cross-stitch networks [ 31 ], we update the edge featur es with G ′ before each message propagation in G as shown from line 5 to 7 in Algorithm 1 . Residual connection [ 32 ] is adopted for edge embedding updates, as shown in line 6 . Note that a dummy edge feature [ 1 ] can be used for models requiring edge features, which are usually absent in G ′ . W ith the use of line graphs, edge features can be effectively AI 2025 , 6 , 69 7 of 21 propagated. In other words, information about a particular transaction can be pr opagated to the next transactions, capturing money flow . Figure 3. V isual illustration of LineMVGNN. Algorithm 1 LineMVGNN Input : Graph G = ( V , E ) ; input node features { h ( 0 ) v , ∀ v ∈ V } ; input edge features { e ( 0 ) vw , ∀ v ∈ V , w ∈ N o ut ( v ) } ; model depth L Parameter : { MVGNNLayer ( l ) G , MVGNNLayer ( l ) G ′ } : ∀ l ∈ L Output : V ector z v , ∀ v ∈ V 1: G ′ ← L ( G ) 2: Initialize in G : h ( 0 ) v ← x v . 3: Initialize in G ′ : t ( 0 ) v , w ← e ( 0 ) vw . 4: for l = 0 to L do 5: W ith t ( l ) ( v , w ) in G ′ , apply MVGNNLayer ( l + 1 ) G ′ and obtain t ( l + 1 ) ( v , w ) by Equation ( 6 ). 6: e ( l ) vw ← t ( l + 1 ) ( v , w ) 7: W ith e ( l ) vw in G , apply MVGNNLayer ( l + 1 ) G and obtain h ( l + 1 ) v by Equation ( 6 ). 8: end for 9: z v ← Equation ( 7 ) 10: return z v The LineMVGNN model is further divided into 2 variants: LineMVGNN- add if using weighted sum for combining messages as defined in Equation ( 4 ), or LineMVGNN- cat which concatenates message vectors followed by a linear layer as defined in Equation ( 5 ). 3.4. Computational Complexity The overall complexity of this pr opagation scheme is dominated by the edge pr opa- gation on the line graph G ′ , which has a complexity of O ( L | E | 2 ) for L model layers, and the line graph construction, which has a complexity of O ( | E | 2 ) in the worst case, such as in a complete graph. The propagation scheme can be optimized by avoiding explicit line graph construction. Edge propagation can be performed dir ectly on the original graph G , as shown in Algorithm 2 . In this refined LineMVGNN, edges in G are tr eated as first-class entities. For each edge e vw ∀ ( v , w ) ∈ E , messages ar e aggregated fr om neighboring edges that shar e a common node. W ith a sparse graph, this reduces the asymptotic computational complexity to O ( L | E | ) , which is the same with GCN [ 25 ]. AI 2025 , 6 , 69 8 of 21 Algorithm 2 Refined LineMVGNN (W ithout Explicit Line Graph Construction) Input : Graph G = ( V , E ) ; input node features { h ( 0 ) v , ∀ v ∈ V } ; input edge features { e vw , ∀ v ∈ V , w ∈ N o ut ( v ) } ; model depth L Parameter : { MVGNNLayer ( l ) node , MVGNNLayer ( l ) edge } : ∀ l ∈ L Output : V ector z v , ∀ v ∈ V 1: h ( 0 ) v ← x v . 2: for l = 0 to L do 3: # Edge Featur e Propagation 4: Scan for N in ( e vw ) and N o ut ( e vw ) . 5: e l vw ← apply MVGNNLayer ( l ) edge by Equation ( 6 ), treating e vw as the first-class entities. 6: # Node Featur e Propagation 7: Scan for N in ( v ) and N o ut ( v ) . 8: h ( l + 1 ) v ← apply MVGNNLayer ( l + 1 ) node by Equation ( 6 ). 9: end for 10: z v ← Equation ( 7 ) 11: return z v 4. Experiments 4.1. Datasets T wo datasets are used to evaluate LineMVGNN in the classification of benign nodes against illicit counterparts. 4.1.1. Ethereum (ETH) Datasets Ethereum Phishing T ransaction Network data is a real public transaction graph dataset available on Kaggle. Each node repr esents an address, and each edge r epresents a transac- tion. The nodes ar e labeled as either phishing or not. Each edge contains two attributes: the balance and the timestamp of the transaction. T o address the node class imbalance and huge graph size, we adopt graph sampling strategies similar to [ 33 , 34 ], creating two data subsets with different subgraph sizes, namely ETH-Small (12,484 nodes and 762,443 edges ) and ETH-Large (30,757 nodes and 1,298,315 edges). Since no node attributes are pr ovided in the original dataset, we further derive variant data subsets by adding structural node features (SNFs): in-degrees and out-degrees, yielding a total of four data subsets: ETH-Small (w/ SNF), ETH-Large (w/ SNF), ETH-Small (w/o SNF), and ETH-Large (w/o SNF). W e adopt a semi-supervised transductive learning setting to classify nodes as illicit (fraud) nodes or benign, and treat all non-central nodes as unlabeled. Nodes of each extracted subgraph are randomly split into training, validation, and test sets at a ratio of 60%:20%:20%. 4.1.2. Financial Payment T ransaction (FPT) Dataset Provided by our industry partner , this transaction dataset contains e-wallet payment transaction data from January 2022. After data preprocessing, a transaction graph is con- structed for each day , where accounts and transactions ar e repr esented by nodes and edges, respectively . No SNFs are added. As we assume that the real data contain no anomalous money laundering patterns, synthetic anomalies are injected into the graphs, following the injection strategy from [ 35 ]. The proportion of illicit (money laundering) nodes constitutes roughly one-third of the total node count. On average, there are 1,048,512 nodes and 1,092,895 edges in the graphs for each day . A supervised transductive learning setting is used to classify nodes as either illicit (money laundering) or benign. The transaction graphs for the whole month are chr onologically split into training, validation, and test sets at a ratio of 60%:20%:20%. For more details, please r efer to Appendix A . AI 2025 , 6 , 69 9 of 21 4.2. Compared Methods and Evaluation Metrics The following baseline GNNs are compared with our proposed methods: (1) Non- digraph GNNs include GCN [ 25 ], GraphSAGE, [ 26 ], MPNN [ 14 ], GIN [ 27 ], PNA [ 36 ], and EGA T [ 37 ]; (2) Digraph GNNs includes DiGCN [ 7 ], MagNet [ 6 ], SigMaNet [ 5 ], FaberNet [ 4 ], and Dir-GCN and Dir -GA T [ 11 ]. Since jumping knowledge [ 38 ] by concatenation (“cat”) and by max-pooling (“max”) are applied in FaberNet, Dir-GCN, and Dir-GA T , we build the corresponding variant models, namely FaberNet (cat), FaberNet (cat), Dir-GCN (cat), Dir -GCN (max), Dir -GA T (max), and Dir-GA T (cat). For models that do not naturally support edge features, we concatenate them with node features. Since some datasets are very imbalanced, the F1 score for the illicit class is used for model performance evaluation, which is similar to what is used in real-world scenarios. 4.3. Results T able 1 summarizes the F1 scores of the illicit class acr oss five datasets for all models. Our LineMVGNN model, especially the variant model LineMVGNN- cat , achieves state-of- the-art results on nearly all datasets. T able 1. F1 scores of the illicit class. For each dataset, the highest F1 score is bold and underlined , and the 1st runner-up is underlined. “OOM” indicates out of memory . Category Methods ETH-Small ETH-Large FPT w/ SNF w/o SNF w/ SNF w/o SNF w/o SNF Non-Digraph GNNs GCN 0.8770 0.8998 0.9068 0.9072 0.8817 GraphSAGE 0.8752 0.6705 0.8984 0.6705 0.8802 MPNN 0.7857 0.8912 0.8854 0.9087 OOM GIN 0.9055 0.8954 0.9117 0.8950 0.8802 PNA 0.9352 0.9105 0.9130 0.9249 OOM EGA T 0.8916 0.6705 0.9195 0.6705 OOM Digraph GNNs DiGCN 0.8192 0.8055 0.8650 0.8290 OOM MagNet 0.9009 0.9012 0.9330 0.9354 0.9616 SigMaNet 0.8072 0.8319 0.8018 0.8300 0.5033 FaberNet (cat) 0.9352 0.9393 0.9476 0.9451 0.9934 FaberNet (max) 0.9336 0.9376 0.9381 0.9460 0.9945 Dir-GCN (cat) 0.9240 0.8987 0.9168 0.9188 0.6402 Dir-GCN (max) 0.8577 0.9000 0.8598 0.9207 0.6402 Dir-GA T (cat) 0.8831 0.6705 0.8769 0.6705 0.9768 Dir-GA T (max) 0.7958 0.6705 0.8515 0.6705 0.9908 Our Digraph GNNs MVGNN-add 0.9231 0.9333 0.9300 0.9365 0.9821 MVGNN-cat 0.9331 0.9301 0.9439 0.9394 0.9858 LineMVGNN-add 0.9362 0.9407 0.9598 0.9048 0.9905 LineMVGNN-cat 0.9441 0.9455 0.9394 0.9565 0.9954 Compared to non-digraph GNN baselines, LineMVGNN beats the competing methods by an average of 9.68% across all datasets. For each dataset, the improvement is at least 0.89%, 3.50%, 4.03%, 3.16%, and 11.37% on the ETH-Small (w/ SNF), ETH-Small (w/o SNF), ETH-Large (w/ SNF), ETH-Large (w/o SNF), and FPT datasets, respectively . Compared to digraph GNN baselines, LineMVGNN improves the prediction illicit F1 scores by an average of 10.79% across all datasets. For each dataset, the increment is at least 0.89%, 0.62%, 1.22%, 1.05%, and 0.09%, respectively . It is noteworthy that our LineMVGNN model achieves over 99% in the FPT dataset without SNFs. This shows the effectiveness of leveraging both the line graph view and the r everse view for node classification tasks (illicit account detection). In addition, even with shared parameters among GNN aggregation maps for in-neighbors and those for out-neighbors, the performance of our MVGNN variant models matches with other competing digraph GNN methods. This confirms that high model efficiency and expr essiveness were achieved. AI 2025 , 6 , 69 10 of 21 4.4. Discussion 4.4.1. Effect of Dif ferent V iews As shown in T able 2 , regardless of the presence of structural node features (SNFs), the performance in the illicit node detection drops when we remove the line graph view for LineMVGNN-cat and LineMVGNN-add. A more significant dr op is caused when we remove the r eversed view (i.e., only aggregating messages from in-neighbors). This reflects the contributions of each view in the illicit node classification tasks. T able 2. Illicit F1 in the ablation experiments. “TWMP” and “LGV” stand for “two-way message passing” and “line graph view”, respectively . Method Components ETH-Small ETH-Large FPT w/ SNF w/o SNF w/ SNF w/o SNF w/o SNF LineMVGNN-cat TWMP + LGV 0.9441 0.9455 0.9394 0.9565 0.9954 - LGV 0.9331 0.9301 0.9439 0.9394 0.9858 - TWMP 0.9009 0.8922 0.9042 0.9031 0.8188 LineMVGNN-add TWMP + LGV 0.9362 0.9407 0.9598 0.9048 0.9905 - LGV 0.9231 0.9333 0.9300 0.9365 0.9821 - TWMP 0.9009 0.8922 0.9042 0.9031 0.8188 4.4.2. Effect of SNF As shown in T able 2 , regar dless of the presence of dif ferent views, our LineMVGNN- cat model performs robustly when SNF (including node in-degrees and out-degrees) is masked in the ETH-Small and ETH-Large datasets. In addition, when LineMVGNN-cat is compared with LineMVGNN-add, message combination by concatenation is superior to the weighted sum method. W ithout SNF , the performance of LineMVGNN-add dr ops a little in various scenarios over Eth-Small and Eth-Large. Also, the performance of our LineMVGNN-cat model is less sensitive to the absence of SNF than LineMVGNN-add. This reflects the superiority of message combination by concatenation following a linear transformation over message combination by weighted sum. 4.4.3. Effect of Parameter Sharing Our LineMVGNN variants extend from our MVGNN model, which is designed within the Dir-GNN framework due to its versatility . A key distinction between the MVGNN and Dir-GNN models lies in their parameter sharing approach. For Dir-GNN models, two independent sets of learnable parameters for M ( l ) in and M ( l ) o ut are used, theoretically enhancing expressiveness but doubling the number of parameters for each Dir-GNN layer . In contrast, our MVGNN models employ parameter sharing, cr eating a lighter yet effective model. Our empirical experiments on r eal-world payment transaction datasets indicate that separating the two sets of learnable parameters may not always be necessary . As shown in Figure 4 , the illicit class F1 scores of our MVGNN variants consistently outperform those of the Dir-GNN variants. On average, the F1 scor es of our MVGNN-add surpasses those of the other competing methods by +7.01%, +21.48%, +6.22%, and +20.75% on the ETH-Small (w/ SNF), ETH-Small (w/o SNF), ETH-Lar ge (w/ SNF), ETH-Large (w/o SNF), and FPT datasets, respectively . Meanwhile, our MVGNN-cat outperforms the others by +8.17%, +21.07%, +7.81%, and +21.12% on these five datasets, respectively . AI 2025 , 6 , 69 11 of 21 ETH-Small (w/ SNF) ETH-Small (w/o SNF) ETH-Large (w/ SNF) ETH-Large (w/o SNF) 0.4 0.6 0.8 1 Illicit Class F1 Score Dir-GA T (max) Dir-GA T (cat) Dir-GCN (max) Dir-GCN (cat) MVGNN-add MVGNN-cat Figure 4. Illicit class F1 score of Dir -GNN family and MVGNN family models across datasets. 4.4.4. Effect of Learning Rate Learning rate can influence the training dynamics and final performance of deep learn- ing models, such as LineMVGNN-add and LineMVGNN-cat. W e conducted experiments with dif ferent learning rates chosen from 0.1, 0.01, 0.001 while using default values for other hyperparameters on the ETH-Small (w/ or w/o SNF), ETH-Lar ge (w/ or w/o SNF), and FPT datasets. The results ar e summarized in Figures 5 – 7 . For the ETH-Small dataset (Figure 5 ) and the ETH-Large dataset (Figure 6 ), in general, the models achieve the highest F1 score for the illicit class at a learning rate of 0.01 regar dless of the presence of SNF . A large learning rate of 0.1 leads to an unstable suboptimal perfor - mance, likely due to overshooting the optimal weights during gradient descent. Conversely , a small learning rate of 0.001 results in slower convergence and slightly lower performance in general. The models might have got trapped in local minima during training. For the FPT dataset (Figure 7 ), the model performs best with a small learning rate of 0.001. This dataset is more complex, with more attributes and a larger number of nodes and edges. A smaller learning rate allows for more stable training, reducing the risk of overshooting the optimal solution, leading to better generalization. A large learning rate of 0.1 results in poor performance due to the instability of training, while a learning rate of 0.01 performs well but not as effectively as 0.001. Comparing LineMVGNN-cat and LineMVGNN-add, although both variants exhibit similar behavior under the variation of learning rates, LineMVGNN-add is more r obust to the variation of learning rate, especially when the transaction graph data is more com- plex. As shown in Figure 7 for the FPT dataset, although both variant models score over 0.99 for the illicit class F1, LineMVGNN-add’s performance degrades less noticeably than LineMVGNN-cat when the learning rate increases from 0.001 to 0.1. This reflects that LineMVGNN-cat, which combines messages from in- and out-neighbors via concatenation followed by a linear transformation, requires more careful tuning to ensure stable con- vergence. The linear transformation layer in LineMVGNN-cat introduces mor e learnable parameters, making the model mor e susceptible to oscillations or diver gence if the learning rate is too high. AI 2025 , 6 , 69 12 of 21 0.001 0.01 0.1 0 0.2 0.4 0.6 0.8 1 Learning Rate Illicit Class F1 Score ETH-Small (w/ or w/o SNF) LineMVGNN- add , w/ SNF LineMVGNN- cat , w/ SNF LineMVGNN- add , w/o SNF LineMVGNN- cat , w/o SNF Figure 5. Illicit class F1 against learning rate for ETH-Small (w/ or w/o SNF). 0.001 0.01 0.1 0 0.2 0.4 0.6 0.8 1 Learning Rate Illicit Class F1 Score ETH-Large (w/ or w/o SNF) LineMVGNN- add , w/ SNF LineMVGNN- cat , w/ SNF LineMVGNN- add , w/o SNF LineMVGNN- cat , w/o SNF Figure 6. Illicit class F1 against learning rate for ETH-Large (w/ or w/o SNF). 0.001 0.01 0.1 0 0.2 0.4 0.6 0.8 1 Learning Rate Illicit Class F1 Score FPT dataset LineMVGNN- add LineMVGNN- cat Figure 7. Illicit class F1 against learning rate for FPT . 4.4.5. Effect of Embedding Size The embedding size determines the dimensionality of the node and edge r epresenta- tions learned by the models. Generally , a larger embedding size can capture more complex AI 2025 , 6 , 69 13 of 21 patterns and nuances in the data. T o explore the impact of embedding size on the perfor- mance of our LineMVGNN variants, we conduct experiments with differ ent embedding sizes on the FPT dataset. As shown in Figure 8 , we vary embedding sizes of 16, 32, and 64. The models achieve the highest F1 scor e for the illicit class with an embedding size of 64. An embedding size of 32 results in a significant drop in the F1 score, likely due to insufficient capacity to capture the complex r elationships in the transaction graph. On the other hand, doubling the embedding size fr om 32 to 64 pr ovides little improvement: + 0.1711% and + 0.0704% for LineMVGNN-cat and LineMVGNN-add, respectively . The results suggest that an embedding size of 32 or 64 provides a good balance between model capacity and computa- tional efficiency for the FPT dataset, while avoiding the pitfalls of overfitting and excessive computational cost associated with larger embedding sizes. Comparing LineMVGNN-cat and LineMVGNN-add, LineMVGNN-cat, which lever - ages concatenation for message combination, is less sensitive to the increase in embedding size when the size doubles from 16. In contrast, LineMVGNN-add, which uses a weighted sum for message aggregation, is slightly less robustness. Its performance drops more notice- ably with smaller embedding sizes. This suggests that the weighted sum mechanism may struggle to capture sufficient information with lower-dimensional repr esentations. This differ ence highlights the advantage of LineMVGNN-cat’s concatenation-based approach, which provides gr eater flexibility in combining features and is less sensitive to embedding size variations. Overall, both models benefit from an embedding size of 64, but LineMVGNN-cat demonstrates superior stability and performance across differ ent embedding sizes, par- ticularly in more complex datasets like FPT . W e did not experiment on the ETH-Small and ETH-Large datasets due to the lack of attributes in the datasets. There are only three attributes in the datasets, which ar e transaction timestamps and amounts, and a label to indicate all fraud nodes. Given the limited feature set, the embedding size was set to match the dimensionality of the available attributes, ensuring that the model could effectively capture the sparse information pr esent in the ETH datasets. 16 32 64 0.95 0.96 0.97 0.98 0.99 1 Embedding Size Illicit Class F1 Score LineMVGNN- add LineMVGNN- cat Figure 8. Illicit class F1 against embedding size for FPT dataset (w/ SNF). 4.4.6. Qualitative Discussion LineMVGNN improves over existing methods primarily through its ability to ef fec- tively propagate and leverage transaction-level information, which is crucial for AML de- tection. Unlike traditional GNNs that focus solely on node-level interactions, LineMVGNN introduces a line graph view that explicitly models edge (transaction) adjacencies. This allows the model to capture the flow of money between transactions, which is essential for AI 2025 , 6 , 69 14 of 21 identifying suspicious patterns such as temporary r epositories of funds or cyclic transac- tions. By propagating edge featur es before updating node featur es, LineMVGNN ensures that transaction-level details are pr eserved and utilized in the learning process. Addition- ally , the two-way message passing mechanism in LineMVGNN (and in MVGNN), which considers both in- and out-neighbors, enhances the model’s ability to capture the direc- tional flow of funds, further improving its detection capabilities. These features collectively enable LineMVGNN to outperform other competing methods, as demonstrated by its superior performance on the ETH-Small (w/ or w/o SNF), ETH-Lar ge (w/ or w/o SNF), and FPT datasets. 5. Limitations and Future W ork 5.1. Scalability While the asymptotic computational complexity of our refined LineMVGNN model has the same asymptotic complexity O ( L | E | ) as GCN [ 25 ], the refined LineMVGNN may have a higher practical runtime due to the additional overhead of edge propagation. Processing the entir e graph in a single batch may not be feasible for extremely lar ge graphs due to memory constraints. T o alleviate this, graph sampling techniques can be leveraged, yielding smaller subgraphs for minibatch training. 5.2. Adversarial Robustness In financial applications such as fraud detection, malicious actors may attempt to manipulate the graph to evade detection. Our current setup does not apply adversarial training for the model. In our future work, we can generate perturbed versions of the graphs during training and optimizing the model to perform well on both clean and perturbed data. On the other hand, befor e applying the model, graph purification can be integrated into the model, detecting and r emoving adversarial edges (transactions) or nodes (accounts). These techniques can enhance the model’s resilience to adversarial perturbations. 5.3. Regulatory Considerations When deploying neural network models in highly regulated domains such as finance, the transparency and traceability of decisions ar e paramount. The proposed LineMVGNN model inherently provides explainability by explicitly modeling and propagating edge- level information (e.g., transactions) and aggr egating it to the node level (e.g., accounts). This design allows the model to pr opagate and compare transactions (edge information) related to the same account (node). This highlights the contributions of individual edges (transactions) to the final node representations, making its decision-making process in- terpretable. For further improvement, explainable artificial intelligence (XAI) techniques can be integrated into the model, such as attention mechanisms and post hoc explainabil- ity methods. 6. Conclusions Existing GNNs for digraphs face differ ent challenges such as ignoring multi- dimensional edge features, the lack of model interpretability , and suboptimal model effi- ciency . T o address these issues, we intr oduce a lighter -weight yet ef fective model, MVGNN, which shares parameters in the aggregation maps for payment and receipt transactions. This effectively and efficiently leverages the local original view and the local reversed view . Extended from MVGNN, we propose LineMVGNN which leverages line graph transformation for the enhanced propagation of transaction information. LineMVGNN surpasses SOT A methods in detecting money laundering activities and other financial frauds in real-world datasets. AI 2025 , 6 , 69 15 of 21 Author Contributions: Data curation, C.-H.P . and J.-H.C.; Funding acquisition, C.C.; Investigation, C.-H.P . and J.-H.C.; Methodology , C.-H.P .; Project administration, C.C.; Software, C.-H.P . and J.-H.C.; Supervision, J.K. and C.C.; W riting—original draft, C.-H.P .; W riting—review and editing, C.-H.P . and J.K. All authors have read and agr eed to the published version of the manuscript. Funding: This research was funded by the Innovation and T echnology Fund of the Hong Kong Special Administrative Region. The article processing char ges were funded by Logistics and Supply Chain MultiT ech R&D Centre. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data A vailability Statement: The Ethereum Phishing T ransaction Network (ETH) dataset is available at https://www .kaggle.com/datasets/xblock/ethereum- phishing- transaction- network (accessed on 1 January 2025). The Financial Payment T ransaction (FPT) Dataset is not readily available because of privacy regulations. Acknowledgments: W e gratefully acknowledge the Logistics and Supply Chain MultiT ech R&D Centre for providing computational resour ces and supporting the article processing charges for this publication. Conflicts of Interest: The authors declare no conflicts of inter est. Abbreviations The following abbreviations ar e used in this manuscript: SOT A State-of-the-Art AML Anti-Money Laundering GNN Graph Neural Network ETH Ethereum FPT Financial Payment T ransaction SNF Structural Node Featur e MPNN Message Passing Neural Network Dir-GNN Directed Graph Neural Network MVGNN Multi-V iew Graph Neural Network LineMVGNN Line-Graph-Assisted Multi-V iew Graph Neural Network GCN Graph Convolutional Network GraphSAGE Graph SAmple and aggreGatE GIN Graph Isomorphism Network PNA Principal Neighborhood Aggr egation EGA T Graph Attention Network with Edge Features DiGCN Digraph Inception Convolutional Networks Dir-GCN Directed Graph Convolutional Network Dir-GA T Directed Graph Attention Network MagNet Digraph GNN Based on the Magnetic Laplacian SigMaNet Digraph GNN Based on the Sign-Magnetic Laplacian FaberNet Spectral Digraph GNN Using Faber Polynomials OOM Out of Memory TWMP T wo-W ay Message Passing LGV Line Graph V iew eq equation w/ with w/o without AI 2025 , 6 , 69 16 of 21 Appendix A. FPT Dataset There are a total of 43 attributes in the dataset. Fourteen relevant attributes are extracted to construct a transaction graph for each day: currency , transaction type, business service, payment category purpose, status, reject reason code, r eturn reason code, outwar d input source, inward delivery channel, real-time counterparty verification, settlement amount, settlement time, credit participant account type, and debit participant account type. T able A1 shows detailed statistics of each graph of each day in our FPT dataset. T able A1. Graph statistics of the FPT dataset. Set Day Number of Nodes Number of Edges T rain 1 1,116,969 1,160,635 2 1,013,391 1,039,540 3 1,259,733 1,294,309 4 1,175,766 1,208,983 5 1,165,737 1,217,110 6 1,101,062 1,141,048 7 1,137,598 1,185,835 8 911,029 955,756 9 924,847 976,963 10 1,117,958 1,167,333 11 997,538 1,037,538 12 1,036,556 1,094,068 13 970,965 1,008,168 14 976,630 1,012,067 15 888,321 920,889 16 875,318 925,757 17 1,029,538 1,070,001 18 975,762 1,012,953 19 1,024,562 1,077,209 V alidation 20 1,024,570 1,061,908 21 982,044 1,025,592 22 843,878 879,405 23 848,317 902,103 24 1,044,676 1,094,133 25 1,057,020 1,097,454 T est 26 1,117,969 1,176,023 27 1,173,160 1,221,627 28 1,265,930 1,314,902 29 1,022,037 1,064,851 30 1,001,363 1,060,849 31 1,423,624 1,474,741 As we assume that the real data contain no anomalous money laundering patterns, synthetic anomalies are injected into the graphs. Following the anomaly injection strategy from [ 35 ], we embed directed paths, cycles (rings), cliques, and multipartite structures as shown in Figure A1 . The sizes of paths and cycles are randomly chosen from 10 to 20, while the size of cliques ranges fr om 5 to 10. The source, intermediate, and destination layer in a directed multipartite network have fixed sizes of 5, 3, and 1 respectively . The steps of anomaly synthesis for each transaction graph of a day are as follows: 1. Randomly select a pattern from path, cycle, clique, multipartite graph. 2. Generate the pattern with n nodes (and e edges). 3. Randomly select e rows of transaction data from the FPT dataset. 4. Assign each row of transaction attribute values to each edge and the corr esponding end nodes. (T o simulate a flow of money in paths and cycles, the selected rows of transaction data are sorted and assigned, such that for each node the transaction AI 2025 , 6 , 69 17 of 21 timestamp of incoming edges is earlier than that of the outgoing edges except one edge in each cycle pattern. Similarly , in multipartite graphs, the selected transaction data are sorted and assigned such that transaction timestamps in all edges in the first layer are earlier than those in the second layer . Also, in each path and cycle, transaction amounts of edges within a given anomaly are set by randomly choosing from one out of e r ows of selected transaction data). 5. Insert the anomaly into the transaction graph. 6. Repeat the steps above until a desired number of synthetic nodes has been reached. Figure A1. Embedded money laundering patterns following [ 35 ]. Appendix B. Compared Methods • GCN [ 25 ] leverages spectral graph convolutions to capture neighbor hood information and perform various tasks, such as node classification. Since it does not naturally support multi-dimensional edge features, we concatenate in-node features with in- edge features during message passing. • GraphSAGE [ 26 ] utilizes neighborhood sampling and aggregation for inductive learning on large graphs. W e choose the pooling aggregator and full neighbor sampling as the baseline model setting. Since it does not naturally support multi- dimensional edge features, we concatenate in-node features with in-edge features during message passing. • MPNN [ 14 ] is a framework for processing graph structured data. It enables the ex- change of messages between nodes iteratively , allowing for information aggregation and updates. Specifically , it proposes the message function to be a matrix multiplica- tion between the source node embeddings and a matrix, which is mapped by the edge feature vectors with a neural network. • GIN [ 27 ] is designed to achieve maximum discriminative power among WL-test equivalent graphs. It uses sum aggregation and MLPs to process node featur es and neighborhood information. Since it does not naturally support multi-dimensional edge features, we concatenate in-node features with in-edge featur es during message passing. Although [ 39 ] extends GIN by summing up node features and edge features, AI 2025 , 6 , 69 18 of 21 we find it inappropriate for our datasets because of (1) the considerable context differ ence between node and edge features and (2) the dif ference in featur e sizes. • PNA [ 36 ] enhances GNNs by employing multiple aggregators and degree scalars. For aggregators, we picked mean, max, min, and sum; for degr ee scalars, amplification, attenuation, and identity are used. • EGA T [ 37 ] extends graph attention networks, GA T , by incorporating edge features into the attention mechanism. The unnormalized attention score is computed with a concatenated vector of node and edge features. In this work, we use thr ee attention heads by default. • DiGCN [ 7 ] extends graph convolutional networks to digraphs. It utilizes digraph convolution and kth-or der proximity to achieve larger receptive fields and learn multi- scale features in digraphs. As suggested in the paper , we compute the approximate di- graph Laplacian, which alters node connections, during data prepr ocessing because of considerable computation time. Since it does not naturally support multi-dimensional edge features and performs edge manipulation (such as adding/removing edges), we aggregate all features from in-edges by summation and update node featur es by concatenating with the aggregated edge featur es. • MagNet [ 6 ] is a spectral GNN for digraphs that utilizes a complex Hermitian matrix called the magnetic Laplacian to encode both undirected structure and directional information. W e set the phase parameter q = [ 0, 0.25 ] to be learnable and initialize it as 0.125. Unless otherwise specified, other model parameters are set to default values from PyT or ch Geometric Signed Directed (version 0.22.0) [ 40 ]. Since it does not natu- rally support multi-dimensional edge features and performs edge manipulation (such as adding/removing edges), we aggr egate all features from in-edges by summation and update node features by concatenating with the aggr egated edge features. • SigMaNet [ 5 ] is a generalized graph convolutional network that unifies the treatment of undirected and directed graphs with arbitrary edge weights. It introduces the Sign- Magnetic Laplacian which extends spectral GCN theory to graphs with positive and negative weights. Since it does not naturally support multi-dimensional edge features and performs edge manipulation (such as adding/removing edges), we aggregate all features fr om in-edges by summation and update node features by concatenating with the aggregated edge featur es. • FaberNet [ 4 ] leverages Faber Polynomials and advanced tools from complex analysis to extend spectral convolutional networks to digraphs. It achieves superior results in heterophilic node classification. Unless specified, default parameters in that paper are used. W e experimented with two differ ent jumping knowledge options (“cat” and “max”), producing variant models FaberNet (cat) and FaberNet (cat), respectively . W e use real FaberNets because [ 4 ] proves that the expressive power of r eal FaberNets is higher than complex ones given the same number of real parameters. Since it does not naturally support multi-dimensional edge features, we concatenate in-node featur es with in-edge features during message passing. • Dir-GCN and Dir -GA T [ 11 ] are instance models under the proposed Dir -GNN frame- work for digraph learning. It extends message passing neural networks by performing separate message aggregations fr om in- and out-neighbors. W e experimented on the base models, GCN and GA T , respectively , with two different jumping knowledge options (“max” and “cat”) with learnable combination coefficient α , producing four variant models, namely Dir-GCN (cat), Dir-GCN (max), Dir-GA T (max), and Dir-GA T (cat). For details about jumping knowledge, readers can r efer to [ 38 ]. AI 2025 , 6 , 69 19 of 21 Appendix C. Implementation Details PyT orch [ 41 ], Deep Graph Library (DGL) [ 42 ], and PyT orch Geometric (PyG) [ 43 ] ar e used for implementing all algorithms on a single NVIDIA GV100GL [T esla V100 SXM3 32 GB] GPU. For all GNN models, the depth is 2 by default. W e train the models with the Adam optimizer [ 44 ]. W e use cosine annealing with warm restarts at epoch 10, 20, 40, and so on [ 45 ]. Cross-entr opy loss is used. The initial learning rate, l r , is optimized by grid search, wher e l r ∈ { 0.1, 0.01, 0.001 } . 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