## 5 Results
### Characteristics of the Data and the Algorithm  
The events considered in this work encompass various interactions, including card registration, device registration, IP registration, official identification registration, and bank account registration. After constructing the ETG, it was divided into training, validation, and test sets using a straightforward chronological split (70%, 15%, 15%). Specifically, the training set comprises the first 70% of the data in chronological order, followed by the validation set with the next 15%, and the test set with the remaining 15% of the data. The algorithm samples data in each iteration during the training process to enhance efficiency, as suggested in [12]. In this case, data sampling focuses on the \(n\) neighboring vertices with temporal interactions occurring before an event. The MPTGNN algorithm incorporates information from these neighboring vertices, and for this study, the 20 neighboring vertices with the most recent interactions were selected. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Graph** & **Normal** & **Fraud** &
\begin{tabular}{c} **Fraud** \\ **Percentage** \\ \end{tabular} \\ \hline Cards & 314,894 & 3347 & 1.05 \\ \hline Devices & 505,797 & 2424 & 0.48 \\ \hline Bank Accounts & 123,574 & 1199 & 0.96 \\ \hline Cards–Devices & 820,691 & 5771 & 0.7 \\ \hline Cards–Bank Accounts & 438,468 & 4546 & 1.02 \\ \hline Devices–Bank Accounts & 629,371 & 3623 & 0.57 \\ \hline Cards–Devices–Bank Accounts & 944,265 & 6970 & 0.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of normal and fraud events present in each graph. The numerical values in the “Normal” and “Fraud” columns indicate the corresponding number of events. The “Fraud Percentage” indicates the percentage of fraudulent events in the graph. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Graph** & **Vertex** & **Edges** & **Users** & **Entities** & **Edge-to-Vertex Ratio** \\ \hline Cards & 533,761 & 318,241 & 224,258 & 309,503 & 59.62 \\ \hline Devices & 752,973 & 508,221 & 261,612 & 491,361 & 67.49 \\ \hline Bank Accounts & 196,343 & 124,773 & 82,122 & 114,221 & 63.54 \\ \hline Cards–Devices & 1,100,761 & 826,462 & 299,897 & 800,864 & 75.08 \\ \hline Cards–Bank Accounts & 655,785 & 443,014 & 232,061 & 423,724 & 67.55 \\ \hline Devices–Bank Accounts & 869,315 & 632,994 & 263,733 & 605,582 & 72.81 \\ \hline Cards-Devices–Bank Accounts & 1,215,588 & 951,235 & 300,503 & 915,085 & 78.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the graphs utilized in this study. The numerical values represent the counting of vertices or edges within each graph. Each graph is a bipartite graph between users and entities (cards, devices, bank accounts, etc. ), i.e., entities are a subset of the vertices. This term is a standardization used in the online payment platform to refer to vertices that are not users. The edge-to-vertex ratio represents an intuitive density measure of edges over vertices. As outlined in the previous section and illustrated by the MPTGNN Algorithm 1, four key modules are central to the MPTGNN: the message function, message aggregator, memory updater, and embedding module. The message function gathers and processes information on the memory states of the vertices involved in an interaction event (which creates an edge between two vertices). This is accomplished using a Multi-Layer Perceptron (MLP--a fully connected neural network with at least three layers and a non-linear activation function). In this setup, the memory states serve as inputs to the MLP, and the output is utilized in the subsequent steps, as shown in lines 4 and 5 of Algorithm 1. Following this, the message aggregator combines the information from the message function associated with the same vertex. Typically, this combination can be achieved using a learnable function, but only the most recent message from a given vertex is used for simplicity.