Partitioning Message Passing for Graph Fraud Detection
Authors: Wei Zhuo
Zemin Liu
Bryan Hooi
Bingsheng He
Guang Tan
Rizal Fathony
Jia Chen
Venue: ICLR 2024 poster (2024)
Keywords: Graph Neural Networks
Fraud Detection
Primary area: learning on graphs and other geometries & topologies
TL;DR: Applyiing GNNs on fraud detection by distinguishing neighbors during message passing.

Abstract: Label imbalance and homophily-heterophily mixture are the fundamental problems encountered when applying Graph Neural Networks (GNNs) to Graph Fraud Detection (GFD) tasks. Existing GNN-based GFD models are designed to augment graph structure to accommodate the inductive bias of GNNs towards homophily, by excluding heterophilic neighbors during message passing. In our work, we argue that the key to applying GNNs for GFD is not to exclude but to {\em distinguish} neighbors with different labels. Grounded in this perspective, we introduce Partitioning Message Passing (PMP), an intuitive yet effective message passing paradigm expressly crafted for GFD. Specifically, in the neighbor aggregation stage of PMP, neighbors with different classes are aggregated with distinct node-specific aggregation functions. By this means, the center node can adaptively adjust the information aggregated from its heterophilic and homophilic neighbors, thus avoiding the model gradient being dominated by benign nodes which occupy the majority of the population. We theoretically establish a connection between the spatial formulation of PMP and spectral analysis to characterize that PMP operates an adaptive node-specific spectral graph filter, which demonstrates the capability of PMP to handle heterophily-homophily mixed graphs. Extensive experimental results show that PMP can significantly boost the performance on GFD tasks.

Published as a conference paper at ICLR 2024
PARTITIONING MESSAGE PASSING FOR GRAPH FRAUD DETECTION
Wei Zhuo', Zemin Liu', Bryan Hooi', Bingsheng H Guang Tan'; Rizal Fathony Jia Chen 'Shenzhen Campus of Sun Yat-sen University, "National University of Singapore, GGrabTaxi Holdings Pte. Ltd.
zhuow5êma 112. wtt.aa....ssuueleao.a. sysu.edu.co. en, zeminliugnus, edu. sg, [bhooi, hebs } @comp. nus. edu. sg, tanguangemai i www.sueddueeo (rizal. fathony, jia. (hweneegrab....
ABSTRACT
Label imbalance and momophiiy--hterophhie mixture are the fundamental problems encountered when applying Graph Neural Networks (GNNs) to Graph Fraud De- tection (GFD) tasks. Existing GNN-based GFD models are designed augment graph structure to accommodate the inductive bias of GNNs towards homophily, by excluding heterophilic neighbors during message passing. In our work, we argue that the key to applying GNNs for GFD is not to exclude but to distin- guish neighbors with different labels. Grounded in this perspective, we introduce Partitioning Message Passing (PMP), an intuitive yet effective message passing paradigm expressly crafted for GFD. Specifically, in the neighbor aggregation stage of PMP, neighbors with different classes are aggregated with distinct node- specific aggregation functions. By this means, the center node can adaptively adjust the information aggregated from its heterophilic and homophilic neighbors, thus avoiding the model gradient being dominated by benign nodes which occupy the majority of the population. We theoretically establish connection between the spatial formulation of PMP and spectral analysis to characterize that PMP operates adaptive node-specific spectral graph filter, which demonstrates the capability of PMP to handle results show that PMP can significantly boost the performance on GFD tasks. Our code is available at aatatts:///ggo //tpt/gbg.ttt///// com/ com/Xtra ra computinng/a0
mixed graphs. Extensive experimental
INTRODUCTION
With the explosive growth of online information, fraudulent activities have significantly increased in financial networks (Ngai et al. 2011: [in et al., 2021), social media (Deng al. (022), review networks (Rayana & Akoglu 2015). 1, and academic networks (Cho et al.) (201). making the detection of such activities an area of paramount importance. To fully exploit the rich graph structures contained in fraud graphs, recent studies have increasingly adopted Graph Neural Networks (GNNs) (Wu et al. 2020 to Graph Fraud Detection (GFD). Applying message passing GNNs Gilmer et al., 2017) in GFD encounters two significant challenges: label imbalance (Liu al. 2023) and mixture of heterophily and homophily (Gao et al. (023a). Network attackers often employ sophisticated tactics to mimic regular network patterns, by strategi- cally injecting limited number of fraud nodes in the main contexts of the target graph to hide their fraudulent activities. The label imbalance problem causes the GNN to primarily capture the patterns and characteristics of benign nodes, compromising their ability to accurately identify fraudulent ones. Additionally, the heterophily-homophiy mixture violates the homophily (Zhu et al. (2020) inductive bias of GNNs, as fraud nodes are often strategically placed within benign communities to exhibit heterophily, while the context around benign nodes exhibits homophily. Therefore, it is pivotal to develop GNNs that can navigate these challenges adeptly.
To alleviate these issues, several methods have been proposed from a spatial perspective to diminish the impact of heterophilic neighbors during the aggregation process. These strategies commonly involve utilizing trainable approach or predetermining a mechanism to resample neighbors (Doul
et al. 2020 Liu et al. 2021c 2020) or reweight edges (Wang et al. 2019 Cui et al. 2020) Shi et al.
"Corresponding authors.

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Published as a conference paper at ICLR 2024
vi
h22
Comb (B-1)
(h. (1-1) 12-1)
h Comb (-1)
(
()
(-11)
カ-11
um
(a) Generic message passing.
(b) Partitioning message passing.
Comparison between generic message passing GNN and partitioning message passing GNN. Red, blue and grey mean fraud, benign and unlabeled nodes, respectively.
Figure 1:
2022 Liu al., 2021a). From spectral view, GHRN (Gao et al.) 2023a) shows using high-pass filter helps identify heterophilic neighbors. pruning inter-class edges to highlight the high frequencies of the graph. BWGNN (Tang et al., 2022) reveals the 'right-shift' phenomenon of spectral energy and suggests Beta wavelet-based band-pas, aand-pass graph filter.
However, from the spatial perspective, augmenting the graph structure to exclude heterophilic neighbors during message passing to accommodate the inductive bias of GNNs is non-trivial for two reasons. Firstly, semi-supervised GFD usually makes pruning or reweighting of unlabeled neighbors based on predicted logits unreliable (Gao et al. 2023c). In other words, the edge weight or pruning probability depends on the similarity of the predicted representations between nodes, therefore any prediction eπrоr may accumulate and impact the final result. Secondly, these methods face scalability issues on large-scale graphs. On the other hand, spectral- based models can keep the graph fixed and learn the spectral graph filter beyond low-pass (Bo et al., [02T He al. 202 Tang al. 2022 Gao et al.) 2023c), which demonstrate efficacy in addressing heterophily. However, these models usually struggle with label imbalance due shared parameters and require the entire graph input, hindering the mini-batch training.
In this paper, we argue that complicated trainable or predetermined strategies for excluding her- erophilic neighbors are unnecessary. Instead, the key of applying GNNs on fraud graphs is to distinguish neighbors during the message passing process, rather than exclusion. A powerful model should inherently have the capacity to adaptively modulate the information derived from both ho- mophilic and heterophilic neighbors.
Based on this insight, we propose an intuitive yet effective message passing paradigm named Partitioning Message Passing (PMP). the neighbor aggregation stage PMP, we employ distinct aggregation functions to independently handle homophilic and heterophilic neighbors. It ensures that parameter sharing confined nodes within the same class, thereby preventing the gradients from being dominated by the majority class nodes. For neighbors with unknown classes, we configure their aggregation function to be a flexible composite of the aggregation functions used for labeled neighbors, where the combination weight is derived from adaptable scalar function unique to the center node. Besides, since the fraud graph is homophily- heterophily mixed, the amount of information that different nodes obtain from homophilic neighbors and heterophilic neighbors should also be adaptively adjusted according to the node. To achieve it, we treat the parameter matrices of aggregation functions as the output of weight generators with respect to the center node. By this means, each node can adaptively adjust the influence from different classes of neighbors without additional parameters. Moreover, our theoretical analysis for PMP bridges the gap between the spatial form of the model and the spectral explanation, which proves that PMP can be interpreted as a mode-specift spectral convolution, i.e., each node has its own spectral graph filter. shows that our model is more suitable for graphs with homophilic and heterophilic mixtures because the uniform graph filter can not balance both aspects. Extensive experiments on four benchmark datasets demonstrate the effectiveness of PMP.
2
BACKGROUND AND MOTIVATION
2.1 BACKGROUND
Notations. A multi-rclational attributed fraud graph denoted as 9 = (V, |= X, )), where (" represents the node set with N nodes, F,, represents the edge set under the
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r-relation, and R is the total number of relations. Thus, G can be seen combination of separate single-relational graphs, each of which is characterized by an adjacency matrix A, E RN*N derived from its corresponding edge set E,, i... if an edge Cij E,, Ar, = 1; otherwise A,, = 0. X E Nxd is node feature matrix, wherein the i-th row 2; is the feature vector of vg. 3' represents the set of labels, where each node U; is assigned a binary label yi E V i.e., yi = denotes fraud node, and 0 denotes benign node. Due the label imbalance nature in fraud graphs, the quantity of benign nodes (the majority class) substantially outweighs that of the fraud ones (the minority class). For a node v and its neighbor Vy. if ys- Vd Uy is homophilic neighbor; otherwise, it is heterophilic.
Graph Neural Networks. In a typical GNN, the I-th message passing layer for a node is operates by iteratively aggregating data from its immediate neighbors, and then combining the aggregated messages with its own node representation (Gilmer et al., 2017):
(1) h = Comb ( (folr (h. (n
ell11) age
((n)))))e)N())))
(I)
where (.) is the 1-th layer hidden representation of vs Comb(.) is a combination function, fsei(()
is function on the center node, fagg()) 1) is permutation invariant neighbor aggregation function, and ((s)) is the neighbor set of 0z. Saza() typically encompasses two essential components:
feature transformation and message fusion. Existing GNNs have primarily focused on elaborating
on message fusion functions based on Laplacian (Kipf & Welling 2017: Defferrard et al.) 2016))
diffusion (Gasteiger et al. 2019a b Chien et al. 2021), attention (Veličković et al.J 2018 or
combination of multiple aggregators (Corso et al. 2020). However, these methods adopt simple linear feature transformations by employing weight matrices shared across all neighbors. This can lead to a learning process that is biased by the potential label imbalance. The backpropagation of gradients, which aims to minimize the overall loss, tends to be dominated by the majority class neighbors. This limits the ability of the model to learn from the minority class effectively. If the center node belongs to a minority class, the graph embodies heterophily. Such interplay of heterophily and label imbalance in fraud graphs provides unique challenge for GNNs.
Related Work. (1) Label Imbalanced Learning on Graphs. GNNs are known to be sensitive to label imbalance (Liu et al. 2023). Currently, several studies have proposed different strategies to address this challenge, such as by implementing adversarial constraints (Shi et al., 2020), generating minority instances using techniques like GAN (Qu et al. (222)) or SMOTE (Zhao et al. 2021). and modifying the degree of imbalance compensation (Song al. 2022). Despite their effectiveness, their success heavily relies on augmenting graph structure to balance the label distribution, which can hardly be adaptive to fraud detection tasks on large-scale graphs. (2) Graph Heterophily Learning. Heterophily has emerged as a significant concern for GNNs. This issue was initially highlighted (Zhu et al. 2020 Platonov et al.. 2023 Hamilton et al.) (017) proves to be an effective technique for learning on heterophilic graphs. Given that nodes sharing the same class are distantly placed within heterophilic graphs, several approaches aim to extend the local neighbors to non-local ones by integrating multiple layers (Chien et al. 202
by Pei et al. (2020). Separating ego- and
Abu-EE-Haija et al. 2019), and identifying potential neighbors through attention mechanisms (Liu
similarity measures (Zhuo & Tan 2022 Jin et al., (02TT) Spectral- based methods (Luan et al. 202T) overcome this challenge by introducing additional graph filters and mixture strategy, which aims to adaptively integrate information by emphasizing certain frequencies.
al. 2021b] Yang et al. 2002bb
This approach shares a similar goal with our work. We discuss in detail how our model relates to this method in Appendix C (3) GNN-based Fraud Detection. GNNs have been leveraged to detect fraudulent activities in financial services (Rao et al., 2022); Xu et al.) 202T Lin et al.) 2021 Chen
al. 2024), telecommunications (Nabeel et al. (022)) social networks (Deng et al., 2022) ) and
healthcare (Cui et al.] 2020). CARE-GNN (Dou et al. 2020) employs label-aware similarity measure to identify informative neighbors and leverages reinforcement learning to selectively integrate similar neighbors. PC-GNN (Liu et al. 2021c) utilizes label-balanced samplers for sub- sub-graph training. Spectral-based methods (Tang et al. 2022 Gao et al. 2023c) conduct spectral analysis on fraud graphs, designing band-pass
high-pass graph filters specifically tailored to detect fraud nodes. GAGA (Wang et al.] (002)) groups neighbors by augmenting neighbor features with their labels explicitly and treating unlabeled neighbors new class.
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2.2 MOTIVATION ANALYSIS
To probe the reasons why message passing based GNNs experience shortcomings in fraud detection tasks, our investigation is conducted through analysis of the mutual influence between nodes resulting from the message passing process. Inspired by (Xu et al.] 2018 /Zhang et al.) 2021). the influence of node by on the center node V can be quantified by measuring how alterations in the input feature of by affect the representation of 2 after k iterations of message passing. For any U
and its neighbor v E (us), given the message passing form H A "H W where H X. considering the li-th feature of X, the influence of vy on the final representation of o; is defined as:
HH
I(k)es =
= A W.
jh VWE(L.-.dd)
Since the gradient is independent of the feature dimension h (Zhang al. 2021), ), the final result omits the h. In fraud graph characterized by imbalanced label distribution, let m represent the number of benign neighbors (v;) of node Us and n represents the number of fraud neighbors Nir(vs). In this context, m > n. Since we specifically study the class imbalance problem, to rule out potential interference from other variables, we assume the graph to be regular. Consequently, all non-zero off-diagonal entries in A & are assumed to be equal, and this common value is denoted by 7. The total influence of benign neighbors on the center node Vn is (o,,(,))-rr = EjENs(()) A w = m W, where we use y to scale the influence score. Similarly, for fraud neighbors we have (e = nW. Then, the total influence from neighbors is given by (m + n)W ING.(1,))-o; which tends to over- amplify the influence from the majority class neighbors (benign) while neglecting that from the minority class neighbors (fraudulent). Such an imbalance in influence can skew the message passing process, causing it to be insufficiently responsive to the nuances of the minority class nodes. As a result, the capacity of the network to capture critical features from the minority class neighbors diminishes, potentially undermining its effectiveness in fraud detection especially when the graph exhibits heterophily.
To address the aforementioned problems, many existing GNN-based fraud detection models adopt strategies such as re-weighting neighbors from different classes using predefined indicators (Liu
al. 2021ch or learnable attention values (Wang al. 2019 Liu al. 2021a Shi et al.) 2022 Cui
et al. 2020). Additionally, they often incorporate a learnable sampler to selectively focus on potential neighbors belonging to the same class as the center node (Dou et al.) 2020 Liu al.) 2020). These works typically utilize additional modules to augment the graph structure, aiming to homogenize contextual information in graphs. The essential goal of these methods is to modify the graph to exclude heterophilic neighbors (i.e., neighbors that belong to different classes than the center node), thereby tailoring the graph topology for more effective processing by GNNs in fraud detection tasks. However, augmenting the graph structure often leads to high time and memory complexity, limiting scalability to large graphs. In our work, we argue that adapting the graph to fit GNNs is both costly and unnecessary, but keeping the graph fixed and modifying the GNN model to fit the graph can be more efficient and effective.
Upon observing the formation of the total influence (m n)) W, we find that both fraud and benign neighbors are weighted equally with W, which leads to benign neighbors dominating the gradient of W during the backpropagation. naturally motivates us to ask whether handling neighbors of two distinct classes with separate weight matrices might allow for adaptive adjustment in their influence on the center node, i.c., m W, + nW,, Such an approach may effectively mitigate issues related to label imbalance and heterophily, without doing any operations on the graph itself.
3 PARTITIONING MESSAGE PASSING
Our preliminary analysis in Section 2.2 reveals a relationship between parameter sharing within GNNs and biases in the learning process. In this section, we formally present our method, Partitioning Message Passing (PMP), which is simple, intuitive, yet powerful approach tailored for the fraud detection task. The basic idea of PMP is to utilize the label information to partition the message passing process, enabling the model to distinguish neighbors according to their classes by learning different weights for each class during message passing, thereby enhancing its ability to adaptively adjust the influence propagated from class-imbalanced neighboring nodes. As shown in Fig. Tb the
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I-th message passing iteration of PMP for node is described as follows:
h = (Imbl
(11)
f0011) (nf--)) JE (((-1) (h los E @Nu(vi)),
(3)
bb 10-11) (n5-11u 1 E Nuo(ms) fm 1 (hg 1 & Noe(os (Na(o))
(l2 (.), and (.) correspond to the handling of neighbors categorized as fraud, benign, and unlabeled, respectively. These functions are parameterized by the weight matrices w! w and w IRdxd" aligning with the aforementioned categories. We simply set the message be
where fü (1), Jbc
(ll-1) (b) 11 E NN((v.) = E,,EN(()))) bj w fusion of aggregation functions to sum: Jfr
Handling unlabeled neighbors as a weighted combination of fraud and benign labels. For unlabeled neighbors, face the challenge of determining appropriate transformation that ac- knowledges the uncertainty of these connections. Treating Jun for unlabeled neighbors independently from those for fraud and benign neighbors is not suitable, it would mean forcing additional, definite label on these neighbors. For binary classification fraud detection task, unlabeled neighbors share characteristies with both benign and fraud categories, while separate, definite label might make it harder to model the continuous spectrum between clear-cut benign and fraud behavior and can not leverage the knowledge captured by fir and fbe. Instead, PMP aims treat unlabeled neighbors in a way that reflects their uncertain and mixed nature. Thus we define wi()) as a weighted combination of w!l and w
w a. (1) ( + (1-
where ( E (0.1)) is scalar to modulate the class tendency of unlabeled neighbors associated with the center node Vi. In other words, large at means that the model treats unlabeled neighbors more similarly to fraud nodes, indicating tendency or suspicion toward fraud for those particular connections, and vice versa. The parameter a thus serves continuous dial that allows the model to smoothly interpolate between treating unlabeled neighbors as either benign or fraud nodes, by dynamically adjusting a;
(t) based on characteristies of each center node Uy. Since a. should be
node-specific and adaptive, we define
as a function of v; (1) (h; (111), where $ Rd - IR single-layered MLP with sigmoid activation shared across all nodes.
Root-specific weight matrix generation. In Eq. (3) the influence from the majority and minority class neighbors on U; can be adaptively adjusted by Whe and Wtr respectively. Following the analysis in Section 2.2 the total influence from labeled neighbors on is is m Wbe + n2 n ft. where Whe and Wi, are shared across all center nodes when performing message passing. Further, in various contexts across the graph, nodes often exhibit unique characteristics and patterns of interactions. These differences can arise from distinct node attributes or their positional contexts within the graph structure. Given this variability, a uniform approach to regulating the influence from different classes of neighbors might oversimplify these interactions. Instead, a more adaptive and node-specific method would better capture the intricate dynamics between node and its diverse neighborhoods. However, directly equipping every center node with individualized weight matrices, w
fr and (! is intractable. To circumvent this, introduce learnable weight generators that model w(( and w(()) be as functions of the center node. Importantly, while these generators produce node-specific weight matrices, their underlying parameters are shared across all center nodes. Such design strikes balance between adaptability and model compactness, eliminating the risk of an exponential surge in parameter count. Specifically, for center node U; the (1 + 1)-th layer weight matrices of its aggregation functions are defined as:
w the (b())
w (1) be.t
(w
(5)
where Va, The Rdd ddx two learnable weight generators defined as (x((x) MLP (d)ag(()) and The (z) = MLP M.Pbbeddaa (diag(z)), and they are both implemented as single-layered MLPs; diag(r)) is the diagonal matrix of the input vector. Each node e; receive distinct ((!)
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Spacra
023 0J5 OCE 0U5
Figure 2: (a) A Barabási-Alber graph with 500 nodes, with 10% nodes are fraud. Features of benign nodes (depicted in blue) follow Gaussian distribution, A (1, 1), while those of fraud nodes (shown in red) are drawn from (5, 1). (b) Spectral convolution filters of PMP for random node v..
and W be tailored to its unique features, while the generation process MLPp: and MLP remains consistent and is governed by shared parameters across all nodes.
To summarize, PMP works Algorithm of Appendix A under the r-th relation, utilizing mini-batch training. For Rerelltionnl graphs, we perform PMP propagation for each relation separately, resulting in R representations for each node. We then add a concatenation pooling followed by MLP to integrate R node representations for each node to produce the final node representation.
4 THEORETICAL INSIGHTS
As mentioned in Section 2.1 graphs with fraud usually exhibit a seterophily-homophhee mixture, as attackers would like sparsely inject limited number of fraud nodes into benign communities to camouflage their activities and spread influence. For regions exhibiting homophily, where the center node and its neighbors predominantly share the same label, the desired GNN should act loss-pass filter to smooth the feature representations in the locality. Conversely, in heterophilic regions, where the center node's 's label diverges from most of its neighbors, the GNN must adaptively shift its spectral response to capture such contrasting label information. Some work (Yang et al. [202aa] Wang & Zhang, 2022) show that assigning each feature dimension a separate spectral filter improves the performance of GNNs. Differently, our model achieves adaptivity at the node level, with each node being assigned a separate spectral filter. Specifically, we present the following theorem.
Theorem 1. Consider an undirected graph 9, let UAUT represent the eigendecomposition of the symmetric normalized Laplacian L = I- D 1-11/-D 1/2 1/2, where U is the matrix of eigenvectors the diagonal matrix of eigenvalues and 0 > S... 5... An S Given two sets of d' graph signals X W and X XWin be, PMP scheme described in Eq. (3) operates adaptive graph filter, with the node-specific spectral convolution for node v on X &Wr and X XWW. be is given as:
and A diag((A))i=11..>))
H(v.) - (L)X + (L)
(A)U XW, U (A)U XW
(6)
where the spectral convolution filters are diagonal matrices defined as:
By E (p(())) (n)())
ey E Nyr(ss))
(A) (111))))
(A)(i.1
1- (1-as)(1 (1-1,))
ug E Nie( otherwise
otherwise
(7)
where (1s and (u: respectively the fraud neighbors and benign neighbors of training set. The i-th row of the matrix H(v.), iee. Hi,;)i,, is the representation of U.-
0s in the
We provide a proof in Appendix B The inherent mixed samophily-heterophhin characteristics of fraud graphs underline the importance of utilizing node-specific adaptive filters. As different center nodes locate in diverse contexts within the graph, they each reflect distinct degrees of homophily heterophily. This variability implies that one- siee-size-fin fits--fi----a -its-fil-llll-- approach, using universal graph filter across all nodes (Defferrard et al.) 2016 Kipf & Welling 2017| Tang et al. (022). is not only suboptimal but could lead inaccuracies in fraud detection. PMP ensures that the filters are tailored

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Table 1: Experiment Results on Yelp and Amazon (40% training ratio).
Yelp
Amazon
Method
AUC
F1-Macro
G-Mean
AUC.
F1-Macro
G-Mean
GCN
59.83
56.20
43.65+2.s
83.69
64.86+6.4
17.18
GAT
57.15
48.79
16.59
81.0 31.02
66.75
GraphSAGE
89.38
75. =5.s
73.07
88.26
83,46+1.4
GPRGNN
82.85-ao
63.19
75.62++1.4
93. 73.72+aas
80.66=1az
85,56+273
FAGCN
74.23
61.18
67.37
95.00+aa1
87.29
79.62
Care-GNN
76.19
63.32gaw
67.91
90.671149
86.39
70.521421
PC-GNN
79.87
63.00
71.6021.0
15.861831
89.561877
90.30ga44
H2-FDetector
89,48-125
74.38+242
79.15+227
96,03+aus
86.91 +
91.74++047
BWGNN
90.54.0.9
76.96=a.9
77.12
97.42
91.72
90.01 +:.16
GHRN
90.57
77.54::1e
74.21
97.07
92.36
90.58
GDN
90.34-0.0
76.05
=0.84
97.09+0.6
90.78 92.03
90.68
PMP
93.97-015
81.96=a56
83.92=1.4
97.57+
91.85
Table Experiment Results on T-Finance and T-Social (40% training ratio). OOM: out of memory: OOT: out of time (running time > 1 day).
T-Finance
T-Social
Method
AUC
F1-Macro
G-Mean
AUC
F1-Macro
G-Mean
GCN
93.3 =0.75
86.86sa7
80.43aax
76.00=247
46.85
71.19
GAT
92.77=1.12
62.19
77.46s220
63.45+136
$7.22=us1
69.53+2.1
GraphSAGE
95.05
90.37
85.35
94.19
74.97
72.07a39
GPRGNN
54.82-a.7
56.25
32,96
82.79
49.23
50.03
FAGCN
OOM
OOM
OOM
OOM
OOM
OOM
Care-GNN
93.79
82.59
84.581126
51.87
63.602am
PC-GNN
92.09
55.81=034
$1.96+042
88.98=am
44.13+1..
75.07=0.0
H2-FDetector
94.99
74.21
85.91
OOT
OOT
OOT
BWGNN
95. 25.8
88.6620.2
85. 16.116
94.72
84.062219
81.51
GHRN
95.77=0.0
87.92=075
81. 31.65+043
90.60+1.v
68.28
63 43.44+2211
GDN
95.53=c.74
88.75+223
87. 877
88.06=
56.89+1.1
30.33+357
PMP
97.10:
91.90:0.50
88.51
99.62
95.27
92.97
the unique neighborhood characteristics of each node, enhancing both sensitivity and specificity in identifying fraud patterns. Fig. shows an example of PMP on BA(500) graph. We can find that (A) and (A) are two complementary filters that cover all frequencies the spectral domain.
5
EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Datasets and Baselines. We evaluate our approach using five datasets tailored for GFD:
Yelp (Rayana& & Akoglu 2013), Amazon (McAuley Leskovee, 2013). T-Finance, T-Social (Tang et al. 2022). Besides, have evaluated our model on large-scale real graph from industry partner, Grab. The statistics of these datasets are provided in Table Appendix [ We compare our model against 11 state-of-the-art approaches, including generic GNNs: GCN (Kipf & Welling (017).
GraphSAGE (Hamilton et al. (017)). ) and GAT (Veličković et al.) 2018); beyond homophily GNNs:
GPRGNN (Chien al.] 202 and FAGCN (Bo al. [021]; fraud detection tailored GNNs: Care-
GNN (Dou et al. 22000) PC-GNN (Liu et al. 2021c), ,) H2-FDetector (Shi et al. 2022). BWGNN (Tang
al. 2022). GHRN (Gao al. 2023c), and GDN Gao al. 2023b).
Metries and Implementation Details. We employ three commonly used metrics for imbalanced classification in deep learning evaluations: AUC, F1-Macro and G-Mean. We provide detailed explanation of each metric in Appendix D.3 Following (Tang et al.) 2022), we adopt the data splitting ratios of 40%:20%:40% for training, validation, and test set in the supervised scenario. In the semi--un supervised scenario, the data splitting ratio 1%:10%:89%. For consistency in our evaluations, each model underwent 10 trials with varied random seeds. We present the average performance
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GCN PHP
GCN
2500
600
BAVGNN
1200
BWIGNN
Numbbr of Nodbo
PMP
1000
400
800
1500
200
400
500
200
-40
-20
20
100
200
400
600
800
1000
2000
4000
6000
8000 10000
lb
1-
4-4
(a) Yelp
(b) Amazon
(c) T-Finance
Figure 4: Influence distribution.
and standard deviation for each model as benchmarks for comparison. We provide the detailed hyperparameter tuning strategies of baselines and hyperparameter setting of PMP in Appendix D.4
5.2 PERFORMANCE COMPARISON
95
For public datasets, results derived from the supervised setting can be found in Tables 1 and 2 while those from the semi-supervised setting are detailed in Tables 6 and of Appendix E.1 For the Grab dataset, results are shown in Appendix The results demonstrate that PMP con- sistently surpasses baseline performances across almost all datasets and metrics. One explanation for its enhanced performance over generie weights of the aggregation function are uniformly applied across all neighbors-is that PMP encodes class-specific discriminative information into the model parameters. This is achieved by distinguishing neighbors of distinct classes during the message passing phase. Besides, the adaptive modulation between Wtr and Wee allows nodes to judiciously calibrate the information flow from distinct classes of neighbors. By segregating the processing of neighbors based on their labels, our model can adaptively emphasize and give importance to rare patterns, which is vital in imbalanced and heterophilic scenarios. Among generic GNNs, GraphS exhibits better performance due to its separation between ego- and neighbor embeddings, which is beneficial when learning under heterophily, and our model also inherits such design.
90
85
CARE. GANN
PC-GNN
80
BWGNN
GraphSAGE
75
GHRN
GDN
70
2000
the learnable
Testing
Figure 3: AUC vs. Social
testing time on T-
In comparison to the six GNNs tailored for GFD, our model also demonstrates markedly superior performance. For instance, on Yelp, our model shows improvements of 3.4% AUC, 4.42% in F1-Macro, and 3.08% in G-Mean. Interestingly, our evaluation reveals that GraphSAGE, despite its fundamental design, serves as a potent baseline, even surpassing some models specifically crafted for GFD. Many of these specialized baselines incorporate intricate preprocessing steps rooted in feature engineering (e.g., GHRN) or employ learnable edge reweighting/sampling techniques (e.g., H2-FDetector) for graph augmentation. Such findings suggest that elaborate manipulations tailor the graph structure for the model might be superftuous. Instead, adapting the model to align better with the inherent graph characteristics could prove be more effective strategy. Our model presented in Eq. (3) can be seen as a variant of GraphSAGE. Notably, relative to GraphSAGE. our model exhibits substantial performance enhancements, underscoring the efficacy of the proposed partitioning message passing strategy. In Fig. 3 we show that our model achieves the optimal trade-off between inference speed and effectiveness comparison to baselines. PMP follows the iterative aggregation framework of GNNs, thus the number of layers and the dimension of hidden layers are two core parameters that determine the receptive field and representation power of the model. Thus, we analyze the sensitivity of the model to these two parameters in Appendix D.5
5.3 How DOES PMP SOLVE THE PROBLEMS OF HETEROPHILY AND LABEL. IMBALANCE?
To answer this question, we investigate the influence of neighbors with different labels on the center fraud nodes. Given a center fraud node U, we measure the influence of its fraud neighbors on the final representation of v as (1;) = Σn ax,' where Z, = H.(1) is the output representation of the last layer. Correspondingly, the influence of benign neighbors is captured by
az,
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Table 3: Ablation study
Mothod
Yelp
Amazon
T-Finance
AUC
F F-IMMacro
AUC
FI-Macro
AUC
F1-Macro
GraphSAGE
89.38
75.46
93.16
88.26
95.05
90.37
+partition
89.64
95.92
91.43
++
+tadaptive combination
93.51
79.93
97.61
91.29
96,86
91.31
+++ruot- weights (full PMP)
93.97
81.96
97.57
97.10
91.90
10(e)) - - En xx," Then, the larger - 1o())). the larger influence of homophilic neighbors on the center fraud node, which indicates that the model is robust, effectively mitigating the issues of label imbalance and heterophily. In Fig. 4 we show the distribution of 1 Io with respect to all fraud nodes. It is evident that, in comparison to GCN and BWGNN, the representations learned through PMP exhibit a higher T To for greater number of fraud nodes. This metric underscores PMP's efficacy in enhancing the impact of homophilic (fraud) neighbors on the center fraud node, despite them being in the minority, the center node, thereby showcasing its superior ability to resist the challenges posed by label imbalance and heterophily.
a2.
5.4 ABLATION STUDY
As shown in Eq. (3), the main difference between PMP and GraphSAGE rests in their aggregation methods: while GraphSAGE employs uniform aggregation with shared feature transformations for all neighbors, PMP distinctly partitions message passing, applying varied feature transformations contingent upon neighbor labels. The results in Section (.2 [5.2 show our model consistently outperforms GraphSAGE across all datasets and evaluation metrics, which demonstrates that such simple design can significantly enhance GNN performance in GFD.
Additionally, we delve deeper to validate two pivotal components of our model: the adaptive blending
of unlabeled neighbors as a weighted fusion of labeled data, as shown in Eq. (4). and the root-specific weight matrices in Eq. (5) Table 3 shows the results, where employ GraphSAGE as the benehmark model, given its conceptual proximity to our proposal, to methodically illustrate the incremental benefits introduced by our design choices. "+partition" denotes adopting distinct weight matrices for labeled fraud benign neighbors, while for unlabeled neighbors, a separate, independent weight matrix employed. Subsequently, "++adaptive combination" denotes that the weight matrix for unlabeled neighbors is treated as an adaptive combination of the weights of labeled neighbors as introduced in Eq. ( *++++root-speci"" weights" implies that the weight matrices for fraud and benign neighbors are dynamically generated as functions of the center node as presented in Eq. (5).
From our evaluations, it evident that GraphSAGE lags behind in performance across all metrics. This underscores the inherent limitation of uniformly aggregating information from all neighbors during the message passing process. Such an approach, while general and versatile, might miss out on capturing the relationships and contextual dependencies intrinsic to GFD. The "+partition" contributes to major improvements, which affirms that the core feature of our model, distinguishing between different classes of neighbors during message passing, stands as pivotal mechanism for enhancing GNNs' efficacy in GFD. Besides, the subsequent extensions, "adaptive combination" and "root-specific weights"". also contribute positively to the overall performance. Specifically, the "++adaptive combination", , building upon the "*partition", brings incremental improvements across all datasets. The +ootttt-peeciffee weights", further extending the preceding component, enhances performance on most metrics in most datasets. Combined, these two components contribute to a cumulative improvement of more than 1% over the "+partition" baseline.
CONCLUSIONS
This work presents simple yet effective GNN framework, PMP, for fraud detection task. We propose that the key of applying GNNs on fraud detection to distinguish neighbors with different labels during message passing. With this insight, neighbor aggregation is partitioned based on labels using distinct node-specific aggregation functions. In this way, the center node can adaptively adjust the information propagated from its homophilic and heterophilic neighbors, thus avoiding model overfitting the majority class nodes (i.e., benign), and alleviating the problem of heterophily. The theoretical analysis of PMP demonstrates that PMP learns an adaptive spectral filter for each node separately. Extensive experiments show that our model achieves new state-of-the-a art results on several public GFD datasets, verifying the power of partitioning message passing for GFD.
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ACKNOWLEDGMENTS
This research supported by the National Research Foundation, Singapore under its Al Singapore Programme (AISG Award No: AISG2-TC-2021-000), Shenzhen Baisc Research Fund under grant JCYI20020100112213355
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A ALGORITHMIC DETAILS
Algorithm 1 PMP forward propatation
Input: Fraud graph 9 = (V, E,, X, Y));; Depth L; Batch size B;
Output: Logits Z IR
for I {1..... . do
2: for each batch S G of size B do
3:
for v2 E batch do
h t Sseff (h. ( (
((111)
4:
aj (1-1) $(h; (1-1)
5:
( + (n, )); be.i
(oo
un, = a; ((-"))((-)) + (1
6:
fr
(111)
" Generate weight matrices for
ai
be,
a, (1)
( hg
1a-""yy eNr((s)) + she,1)
hh 1o Nho(v:))
7:
[1) 1111// & Noe U Ni((s)) // pll"), sll and _- are parame-
terized by w (1-1) w and wil-")) respectively.
h 1 + +a
(!)
8:
9: end for
end for
11: end for
H (L)((H()) Nx1
13: Z, < Sigmoid (H)
L E, (ye log Zz) + (1 yi) v;) Z.)) Cross-entropy loss
Time Complexity The time complexity of an L-layer PMP propagation is O + L|E(d). For the adaptive combination of unlabeled neighbors, a single-layer MLP is employed, resulting in complexity of ONd)) Additionally, generating the root-specific weight matrix carries a time complexity of ((Nd). Thus, the aggregate time complexity of the PMP can be summarized as OLLdd+ (L)E) NNd)).
B PROOF OF THEOREM 1
Proof. Assuming the node indices fixed, let F, B be diagonal label mask matrices, which mask benign nodes and fraud nodes respectively with in the main diagonal elements. Specifically, for node v; labeled 1 (fraud) in the training set, assign Fa 1, otherwise O. Similarly, for training benign nodes, B otherwise O. Then the mask matrix of unlabeled nodes is thus F B. For the sake of simplicity, we analyze the first layer of PMP as example and omit superscripts. Specifically, the feature transformation step of Eq. 3 can be reformulated as:
FXW, f + BXW &o (I F B (o: W + (1 Wbe)
(8)
= -F + aII aFF a,B) XW fr + (B + (1 a,) I- (1- a. (1 (1-a, B1
Let K(v,) F + a,I a.F B, Eq. g can be rewritten
K K( + K(w:)) XW
by E
where K(n))ii =
0
by Nbo(t))
Oz otherwise
To facilitate the spectral analysis, we align the definition of the graph convolution with GCNs (Kipf & Welling 2017), using the normalized adjacency matrix, D D-1/2AD-1/2 = I L. Note that although our model utilizes summation the convolution method, this doesn't 't alter its inherent
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spectral properties (Dong al. 2021: Zhu al. 2021 Since the matrix K(-) is node-specific for V/ induces a separate message passing matrix form:
H (1)) =I - (1; XW + (I- L)(I K(o))
(10)
-A)) XW, (A) XWb.
where the node-specifer convolution filters (A) and (A) can be derived Eq. 7 RNx'd representation matrix induced by the convolution kernel functions associated with U; thus its i-th row H( gives the PMP representation for For another node V/, it yields corresponding PMP representation matrix H(v;) whose j-th row is the representation of U3- Besides, we can find that the spectral filters depend on the node indices and the label distribution. For the frequency A with its index j satisfying vj & Nir(us) U Nue then its response g' g²()) is node- adaptive given that a, is inherently function of v Since in real- weal-world networks, where most nodes remain unlabeled, we can conclude that PMP is an adaptive node-specific spectral convolution. M
H(:))
C RELATION OF PMP TO ACM
The basic idea of Adaptive Channel Mixture (ACM) (Luan et al., 2021) to attribute negative weights to heterophilic neighbors through spectral synthesis of low ww-pass and high-pass filters. This technique fundamentally differs from our PMP. It is because the use of negative weights in ACM is indicative of removing (or subtracting) the heterophilic components from the homophilic neighbors. This can be interpreted as a form of information "forgetting" where heterophilic features de-emphasized in favor of homophilic features. It is different from PMP which preserves and utilizes all available information. From the perspective of methodology. ACM is rooted in spectral graph theory, while PMP is spatially oriented. This difference is not just theoretical but has practical implications. The spectral nature of ACM inherently influences its scalability, particularly the context of large graphs. Typically, spectral- graph as input, which can pose challenges for mini- batch training and thereby limit scalability. In contrast, our PMP model, with its spatial-based framework, inherently supports mini- training and is thus more adaptable to large-scale graphs. Furthermore, considering the application of ACM in GFD tasks, its formulation, with removed nonlinearity for simplicity's sake, can be
models, including ACM, require the entire
H
expressed as (1-1 oFL
H H
((O) (1)
9F
H be
w where FL and Fu distinct
H
H are both shared spectral filters. We can observe that the trainable weight matrix W and (((l)
across all nodes. In contrast, our PMP model, when represented in matrix form, can be rewritten
H W1
as H PMP
F
H W2
where F is the normalized adjacency matrix. Unlike
be
H (aW, +
W2)
ACM, trainable weight matrices in PMP, Wi and W2, are specifically applied to nodes with different labels, where W1 and W2 capture the information of fraud nodes and benign nodes respectively, reflecting a more label-aware and adaptive approach to message passing in the context GFD.
D EXPERIMENTAL DETAILS
D.1 DATASETS
Table summarizes the dataset statistics, including the number of nodes, edges, and relations, the proportion of fraud nodes, feature dimension, and the homophily score. The imbalance-aware homophily score (Lim et al. 2021 is defined as:
C-1
IC N
i
(11)
C C-1
R=
where [+] = max(-; 0). C, representing the number of classes, is set to for GFD. C₁ denotes the set of nodes in class k. with k = corresponding benign nodes and l to fraud nodes. Ma is the
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class-wise homophily metric:
En,ec, (vxe))]
(12)
Eu,ec, W(x))
where (1)) is the neighbors of with label k. n E 10 where ń 1 corresponds to fully homophilic graph, while îj < 0.5 indicates a highly heterophilic graph.
We compare the fraud detection performance of different baselines on YelpChi, Amazon, T-Finance, and T-Social. Among them, 1) YelpChi comprises both filtered (spam) and recommended (legitimate) reviews of hotels and restaurants, as collected by Yelp.com. This dataset has three types of relations: R-U-R, representing reviews by the same user, R-S-R, for reviews under the same product receiving identical star ratings; and R-T-R, which groups reviews for the same product posted within the same month. 2) The Amazon dataset collects reviews from the musical instruments category on Amazon. also has three relations, where the U-P-U relationship links users who have reviewed the same product; the U-S-U relationship connects users who assigned identical ratings within a week; the U-V-U relationship bridges users showcasing the top 5% of mutual review text similarities measured by TF-IDF metrics. 3) I--Finance is single-relational fraud graph, capturing anomalous accounts within transaction networks. In this dataset, nodes represent anonymized accounts, with attributes including registration days, login activities, and interaction frequencies. Edges mean the existence of transaction records between two accounts. Nodes are annotated as anomalies by human experts they exhibit behaviors characteristic of fraud, money laundering, or online gambling. 4) T-Social is designed to identify anomalous accounts within social networks. two nodes are interconnected if they sustain a friendship for over three months. 5) The industrial graph includes the transaction of month in leading super app. Due to the anonymity and privacy policy, we exclude the exact details of the graph but roughly the graph includes over 1 million nodes and over 10 million edges.
Table 4: Summary of dataset statistics.
Datasct
# Nodes
Edges
Relations
Frauds (%)
Features
YelpChi
45,954
3,8466979
14.53%
32
0.0538
Amazon
11,944
4.398.392
6.87%
25
0,0512
T-Finance
39,357
21,222,543
4.58%
10
0.4363
Social
5,781,065
73,105.508
3.01%
10
0.1003
Industrial graph
~IM
~10M
0.54%
17
D.2 LABEL DISTRIBUTION
In Fig.
5 show the label distributions of the labeled neighborhoods of training nodes. We use W represent the number of fraud neighbors relative benign neighbors for a central node.
N
indicates that the number of labeled fraud neighbors is fewer than benign neighbors.
Specifically, if Wi=- 0.5, it indicates that the central node has far fewer fraud neighbors compared [NseT to benign ones, denoting a highly imbalanced neighborhood. The height of each bar means the number of center training nodes under a certain 'Nue!" Our results show that commonly used datasets such as Yelp, Amazon, and T- F-Finance suffer from significant label imbalance in the neighborhoods, because the neighborhoods' label distribution is long-tail, i.e., the number of fraud neighbors is far less than benign ones. It requires the model can enhance the influence from the minority class neighbors on the center nodes during neighbor aggregation. Combining with the empirical results in Section 5.3 our PMP method effectively achieves this, enhancing the influence from minority class neighbors more efficiently than traditional GNN approaches.
[
D.3 METRICS
AUC is the area under the ROC curve, and it provides aggregate measure of performance across all possible classification thresholds, reflecting the model's 's ability to distinguish between positive and negative classes. F1-Macro computes the Fl score for each class independently and then takes the average. Lastly, the G-Mean, or geometric mean, calculates the square root of the product of the sensitivity and specificitty, offering insight into the balance between true positive rate and
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Published as a conference paper at ICLR 2024
5000
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0.
0.0
0.4
0.6
0.8
[A6]
[A6-]
[X6-]
(a) Yelp
(b) Amazon
(c) T-Finance
Figure 5: Label distributions of the labeled neighborhoods of the training nodes.
true negative rate performance. Higher values of these metrics signify superior performance of the methods.
D.4 HYPERPARAMETER SETTINGS
To mitigate bias, we individually fine-tune the hyperparameters of each model for every benchmark and report the best performance the validation set. For each model, explored the following searching ranges for general hyperparameters: learning rate Ir E {0.01, 0.005, 0.001), weight decay wd {0, 5e 5, 1e 4), dropout do {0, 0. 0.3, 0.4, 0.5, 0.6, 0.7, 0.8), hidden dimension E (32, 64, 128, 256, 512]. For spatial-hased models, hatch size is an important hyperparameter that highly depends on the graph size. Specifically, for Yelp, Amazon, and T-Finance, batch size bs E {64, 128, 256, 512, 10243. For the large-scale graph T-Social, bs E {227, 22 2 For model- specific hyperparameters, we also carefully calibrate parameters in accordance with varying datasets and training sizes. Here we provide the optimal hyperparameters of PMP in Table 5
Table 5: Hyperparameter settings for PMP.
Dataset
fr
wd
Yelp 0.01
512
256
Amazon
0.01
0
0.6
128
128
T-Finance
0.01
256
64
T.Sccial
0.001
0
2
128
D.5 PARAMETER STUDY
We investigate the sensitivity in relation the key hyperparameters in model: the number of layers L and the hidden dimension d'. For each dataset, we vary L over the range [1, 3), and d' over [32, 64, 128, 256, 512]. We employed a grid search methodology to test combinations of hyperparameters. As illustrated in Fig. 6 PMP consistently performs optimally with single layer, and more layers bring about a continuous decrease effect. This phenomenon can be attributed the issue of oversmoothing. We find that all these datasets have a high average degree. Specifically, the average degree of Yelp, Amazon and T-Finance are 167, 740, and 1078, respectively, and we further investigate the number of nodes within the first two hops of neighborhoods. Eliminating duplication, the average number of nodes in the first two-hop neighborhoods across all nodes stands at 1229 for Yelp, 11338 for Amazon, and 24480 for T-Finance. The remarkable density of these benchmark datasets implics that even a two-layer GNN might aggregate an overwhelming quantum information from the global, thereby obfuscating the essential local features. Additionally, Furthermore, findings reveal that the optimal value of d' varies across datasets. This calls for meticulous tuning to ascertain peak performance. A potential rationale for this sensitivity lies in the substantial variability in the dimensions of input node features across different datasets.
D.6 COMPUTING RESOURCES
For all experiments, we single NVIDIA A100 GPU with 80GB GPU memory.
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Published as a conference paper at ICLR 2024
97
03.27
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93.10
93.97
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92
96
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91.99
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92.47
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32
64
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128
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512
92
128
2.56
512
0
(a) Yelp
(b) Amazon
(c) T-Finance
Figure 6: Sensitivity study of different hyperparameter combinations.
E MORE EXPERIMENTAL RESULTS
E.
SEMI-SUPERVISED FRAUD DETECTION ON PUBLIC DATASETS
In Table Table 7 show the performance of fraud detection under semi-supervised setting
(1% training ratio).
Table 6: Experiment Results Yelp and Amazon (1% training ratio).
Yelp
Amazon
Method
AUC
FI FI-Macro
G-Mean
AUC
FI-Macro
G-Mean
GCN
54.06147
12.48
46.53
82.85
6
58.122147
GAT
50.95x1.x
50.27sam
26.83
13.45
60.84:2m
42.24
GraphSAGE
82.59
64.21emu
57,09
87.50
77.88=23
70.00+a2s
GPRGNN
$4.19
46.07
8.29
80.77
$1.14:2m
19.93
FAGCN
70.92.a 46727
5.902
92.07
84.82
80.13
Care-GNN
73.96aa
61.25
63.87
30.30
69 69.24sxzz
78.17
PC-GNN
75.35
55.05
68.05
91.731a9
87.5810
80.3511
H2-FDetector
74.19aam2
57.42
67. 21
83. 23.26
67. 60smm
55 s5.25
BWGNN
79.31.107
66.12.4a
88.37
86.50.494
GHRN
76.76sa37
64.30
61.73
90.27
89.16:am
83.68
GDN
72.31
$8.28121
13.07
86.38
26c20
83.19
PMP
83,94
68.60
72.39
91.22:044
87.72+1as
83,77
Table 7: Experiment Results on T-Finance and T-Social (1% training ratio)
T-Finance
T-Social
Method
AUC
FI-Macro
G-Mean
AUC
F1-Macro
G-Mean
GCN
74.37
57.51
72.58
59.61
g4
GAT
83,384as7
67.18aus
19.71
68.55e14
47.03
71.26
GraphSAGE
92.53
85.09
82.86
88.89
49.24
22.76 22.76
GPRGNN
89.57
80.47
69.34
74.30
49.24
2.72
FAGEN
0OM
0OM
00M
OOM
OOM
0OM
Care-GNN
90.73gm
26.25
34.38
68.32gi
13.31
63.75
PC-GNN
92.29
76.17
81.93
84.66e12
49.38
57,06+2ss
H2-FDetector
94,37
83.44-255
79.32
OOT
OOT
OOT
BWGNN
89.54
85.17
78.99
86.40.uu
77.35
72.17
GHRN
89.18
73.20
70.08
87.28
65.37
57.93
GDN
93.32
87.26
:3.04
87.37
16.48
29.34
PMP
93.78
88.89..43
84.91
97.14.am
84.58
83.161as1
E.2 EXPERIMENTAL. RESULTS ON GRAB
In Table 3 and Table we show the fraud detection performance on the Grab dataset under supervised and semi-supervised settings.
E.3 COMPARISON WITH R-GCN
As an extension of the comparative analysis presented in Fig. 4 we include R-GCN due to its ability to handle multiple types of relationships within graph. R-GCN (Schlichtkrull al. 2018) extends
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Published as a conference paper at ICLR 2024
Table 8: Experiment Results on Grab (40% training ratio)
Grab
Method
AUC
F1-Macro
G-Mean
Care-GNN
99.58
60.8
98.31
PC-GNN
98.97 66.32
98.08+0:4
H2-FDetector
OOT
OOT
OOT
BWGNN
29.792mm
80.63204t
99.322m01
GHRN
99.71
76.65
99.1 1 2001
GDN
99.73=004
80.83
99.64:00:
PMP
99.82
82.42
99.631008
Table 9: Experiment Results on Grab (1% training ratio)
Grab
Method
AUC
Fl F1-Macro
G-Mean
Care-GNN
99.56
71.73+o22
98.65+am
PC-GNN
96.35+024
71.51+233
98.98+a15
H2-FDetector
OOT
OOT
OOT
BWGNN
98.78aa12
81.53
99.40
99.53
69
98 98.42:.23
GDN
99.72
79.33
99.27
PMP
99.72:009
79.73+0m
99.54
the GCN model to account for different types of relations in the graph. Fig. iillustrate the influence distributions for the multi-relational Yelp and Amazon datasets. evident that accounting for multiple relationships results in a rightward (positive) shift in the influence distributions. Despite the relational capabilities of R-GCN, our model demonstrates superior performance on both datasets.
R-GCN
600
R-GCN BWGNN
Number of Nodes
5000
PMP
BWGNN
Number of Nodes
500
PMP
4000
400
3000
300
2000
200
1000
100
50
100
150
200
250
200
400
600
800
1000
/
-
lb
(a) Yelp
(b) Amazon
Figure 7: Influence distribution.
19


Reviews:
[{"review_id": "QlUncxLps7", "reviewer": "Reviewer_69ev", "summary": "This paper describes a variant of GNN trained for graph fraud detection tasks.\nApparently, previously existing GNN's efficacy suffer due to label imbalance (a common occurrence with fraud data). \nThe authors' solution to this problem was in label-aware partitioning of aggregated contributions during message passing stage. Instead of aggregating contribution from all nodes from a given root node's neighborhood the proposed method would aggregate label-aware contributions separately which would include separate weight matrices for benign nodes and for fraud-related nodes.\nThe algorithm's theoretical examination shows that it independently learns an adaptive spectral filter for each node in the graph. The model also proved to outperform existing state-of-art solutions on publicly available datasets.", "questions": "On page 4, equation 4: Shouldn't a small \\alpha^{(l)}_{i} indicate that the model should treat unlabeled neighbors more similarly to benign nodes (instead of fraud nodes as written in the paper)?\n\nOn Figure 4 (page 8): Influence distribution: why GCN (and not R-GCN) was chosen as one of the methods for comparison? My understanding is that GCN cannot distinguish between diffenent relations and both Yelp and Amazon datasets include multiple relationship types. May be that is why GCN did better at T-Finance (with only one relation type)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper is very well motivated and clearly written. The idea seems to be simple enough yet previous researchers' work focused on augmenting graph structure and label-augmented features and have not augmented message passing process only.\n\nI found the visualization of the difference between the influence of fraud nodes in neighborhood of a given node and influence of benign nodes from the same neighborhoods to be very persuasive when using this metric for comparing minority class contribution with various GNN models.", "weaknesses": "I did not find any glaring weaknesses. I would just mention that it would be easier for a reader to follow if the same notation would not be used for different purposes.\nh (page 15) - class homophili metric\nh_{i}^{(l)} (page 3) - l-th layer hidden representation of v_i", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oYlUll4x3b", "reviewer": "Reviewer_8onS", "summary": "In this paper, the authors address fundamental challenges faced in applying Graph Neural Networks (GNNs) to Graph Fraud Detection (GFD) tasks, namely, label imbalance and the mixture of homophily-heterophily. Existing GNN-based GFD models modify graph structures to accommodate GNNs' homophilic bias by excluding heterophilic neighbors during message passing. However, the authors propose a novel perspective: instead of excluding, they advocate for distinguishing neighbors with different labels. They introduce a method called Partitioning Message Passing (PMP), a message passing paradigm tailored for GFD. In PMP, neighbors with different classes are aggregated using distinct node-specific aggregation functions. This approach allows the central node to adaptively adjust the information gathered from both heterophilic and homophilic neighbors. By doing so, PMP prevents the model gradient from being dominated by benign nodes, which constitute the majority of the population. The authors establish a theoretical connection between the spatial formulation of PMP and spectral analysis, characterizing PMP as an adaptive node-specific spectral graph filter. This demonstrates PMP's ability to handle graphs with mixed heterophily and homophily. Extensive experiments validate the effectiveness of PMP, showing significant performance improvements in Graph Fraud Detection tasks. PMP's innovative approach of distinguishing rather than excluding neighbors with different labels showcases its potential in enhancing the capabilities of GNNs for fraud detection on graphs.", "questions": "- The strategy of partitioning message passing is very similar to GNNs with adaptive channel mixing used in [1] although [1] is from the spectral perspective. It will be interesting to discuss the differences and relationships between your proposed method and [1].\n\n- The discussion on the relationships between heterophily and imbalance is not detailed. A detailed empirical and/or theoretical analysis of relationships between heterophily and imbalance on graphs should be conducted to better understand the problem.\n\n[1] Is Heterophily A Real Nightmare For Graph Neural Networks on Performing Node Classification? 2021", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The solution to adaptively learn from heterophilous and homophilous nodes for fraud detection is interesting and the theoretical analysis is sound.\n\n- The paper is generally well-written and almost clear everywhere.\n\n- Experiments conducted on datasets with different sizes show the effectiveness and efficiency of the proposed method in graph fraud detection.", "weaknesses": "- The relationships between heterophily and imbalance (which is specific in fraud detection) are not clear. This is important to understand the problem.\n\n- The relationships between the proposed method and some previous spectral GNNs, e.g., [1], have not been discussed. A lack of discussions about differences may limit the novelty of the proposed method.\n\n- A minor issue: repeated references 2nd and 3rd articles.\n\n[1] Is Heterophily A Real Nightmare For Graph Neural Networks on Performing Node Classification? 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bHaxHFYUBA", "reviewer": "Reviewer_XqA8", "summary": "This work addresses the challenges of label imbalance and the complex interplay between homophily and heterophily in Graph Neural Networks for Graph Fraud Detection. It introduces a novel approach called Partitioning Message Passing (PMP). In this method, neighboring nodes of different classes are processed using distinct, node-specific aggregation functions. Furthermore, the central node has the ability to adaptively fine-tune the information it gathers from both its heterophilic and homophilic neighbors. Empirical results reveal that the PMP method outperforms other competitive algorithms across a range of datasets, including Yelp, Amazon, and T-Finance, while maintaining an optimal balance between performance and computational time.", "questions": "1. On Page 2, in the penultimate line of \"where each node v_i is assigned a binary label y_i∈Y\", please confirm whether it is \"Y\" or \"fi\"?\n2. In the description of Eq.(4) on page 4, it is mentioned that \"In other words, a small α_i^((l)) means that the model treats unlabeled neighbors more similarly to fraud nodes\". In conjunction with Eq.(4), shouldn't it be the case that a smaller α_i^((l)) makes unlabeled neighbors more biased to benign nodes?\n3. Throughout the paper, sections 2, 3, and 7 are more similar to one part of the content, related work, could they be synthesized into one section?\n4. When introducing PMP, the article mentions the use of one layer of MLP in the generation of α_i^((l)). Still, it does not note how many layers of MLP are used in the subsequent \"Root-specific weight matrices generation.\n5. In the experimental results for the T-Social dataset, the AUC value is much higher than that of the comparative algorithm, close to 100%, is it possible to analyze the reason for this excellent result?\n6. In the last sentence of Section 6.4, \"wherein each of these designs brings more than a 1% improvement across most metrics\", does that mean that each approach delivers at least a 1% improvement over GraphSAGE, or does it mean that it provides a 1% improvement over the previous pivotal components? Please be as precise as possible.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The PMP method is well-designed, offering a straightforward solution that is easy to understand and implement.\n2. The article employs a comprehensive analytical framework to validate the effectiveness of the PMP method, adding to its credibility.", "weaknesses": "1. The primary innovation in the PMP method lies in the use of different weighting matrices for aggregating nodes of various classes. While effective, this focus may be perceived as lacking in breadth in terms of overall innovativeness.\n2. The article would benefit from a more meticulous attention to the use of symbols and language. Ensuring consistent and clear terminology and notation would contribute to the paper's readability and accessibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CZAynB3brp", "reviewer": "Reviewer_wN2Y", "summary": "This paper discusses the challenges in applying Graph Neural Networks (GNNs) to Graph Fraud Detection (GFD) tasks, specifically the issues of label imbalance and the mixture of homophily and heterophily. While existing GNN-based GFD models typically exclude heterophilic neighbors during message passing, the authors argue for distinguishing neighbors with different labels instead of exclusion. They introduce a new approach called Partitioning Message Passing (PMP), which adapts the information aggregated from heterophilic and homophilic neighbors, preventing the model gradient from being dominated by benign nodes. Theoretical connections and extensive experiments demonstrate that PMP significantly enhances GFD task performance, effectively addressing these challenges.", "questions": "Please refer to the weakness.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. This paper maintains a high level of self-containment and coherence, as the authors support the claims made within the manuscript through detailed explanations and experimental validation.\n2. This paper addresses the issues of label imbalance and the mixture of homophily-heterophily by introducing a new approach that assigns distinct parameter matrices to neighbors from different classes.\n3. The comparison algorithms used in experiments are state-of-the-art.", "weaknesses": "1. There are some reservations regarding the novelty of this paper for the following reasons:\n\n     a) The issues of label imbalance and homophily-heterophily have been effectively addressed in the past. Notably, Tang et al. [1] offered a theoretical explanation for these concerns using spectral graph analysis.\n     \n     b) Building upon the findings of [1], this paper makes incremental enhancements in the field of fraud detection while questioning the complexity of earlier algorithms. Nonetheless, this paper does not provide comprehensive comparison of time and space complexities with previous works.\n\n2. The approach of treating fraud detection as a binary classification problem has inherent limitations. Real-world scenarios often involve anomalies that cannot be neatly classified into a single class and that do not adhere to clustering assumptions. Given the multitude of existing papers that have already presented effective solutions to address the challenges of imbalance and homophily-heterophily by binary classification algorithms, it raises questions about the need for publishing similar articles at the cutting-edge conference ICLR.\n\n3. An analysis about the labeled neighborhoods’ label distributions of central nodes in the training dataset is necessary. This is crucial as the proposed method relies heavily on the labels of neighboring nodes. Furthermore, it is important to emphasize the ratio of normal nodes to anomaly nodes in the training data.\n\n4. Concerns have been raised about the correctness of equation (7).\n\n[1] Tang, Jianheng, et al. Rethinking graph neural networks for anomaly detection. ICML, 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)