## 5 Results
### Experiments with Graph Combinations  
It is of utmost importance to note that Algorithm 1 computes embedding from raw information, i.e., precomputed embeddings are not used during the training of the algorithm, in comparison to [19]. The entire process is evaluated using four types of vertices (users, cards, devices, and bank accounts), resulting in seven independent sets of bipartite graphs constructed from the data provided by Moneypool. The first three sets (as detailed in Table 3) consist solely of independent connections; one set links users through shared cards, another through shared devices, and the third through shared bank accounts. The remaining four sets (also listed in Table 3) combine these graphs, such as cards combined with devices, cards with devices and bank accounts, devices with bank accounts, and finally, cards with devices and bank accounts. No previous works use combinations of events in an ETG approach to serve as input for a TGN-type algorithm. Figure 2 illustrates how the temporal graph is configured, while Figure 3 presents a subgraph (sample) of the cards-devices-bank account graph for a schematic visualization. From Table 3, entities represent a subset of vertices. The term is a standardization used in the online payment platform to refer to vertices that are not users, and this helps clarify the difference between users, cards, devices, etc. Figure 3: Sample subgraph of the cards–devices–bank accounts graph. Gray vertices represent users, blue vertices are devices, green vertices are cards, and purple vertices are bank accounts. In this sample, it can be observed that users share devices (blue vertices) and bank accounts (purple vertices) more prominently than they do cards (green vertices). In addition, some users (gray vertices) act as bridges (long edges) between some connected components of the graph. Each edge has a corresponding timestamp (not shown in this image). This is not a general behavior; the figure is only intended as an example. Figure 2: An example schema of the temporal graph with timestamps. The graph includes four distinct types of vertices: users (brown), devices (blue), cards (green), and bank accounts (orange). This particular graph represents the card–device–bank account registration, where each edge has an associated timestamp. The main objective of independently testing the MPTGNN algorithm on seven different graphs is to determine if the aggregation of structural information impacted the values for \(F_{1}\) and AUC metrics. \(F_{1}\) is a harmonic mean between precision and recall, two additional metrics related to true positives (observations that are predicted as fraud and indeed are fraud) and false positives (observations that are predicted as fraud but are not fraud). Because of these considerations, the \(F_{1}\) metric is commonly used in imbalanced class problems. On the other hand, AUC stands for the Area Under the Curve and is the area under the ROC curve; the ROC curve is a graphical representation between the true positives and false positives at each classification threshold. Tables 3 and 4 detail the characteristics of the seven graph sets, while Table 5 presents the results from the fraudulent user classification task. The values shown are the averages from 10 different runs. Given the class imbalance, class weighting was applied to the cross-entropy loss function. An extended dataset (also provided by Moneypool) was used for completeness, consisting of the same events and additional IP address registration events. Also, the period is extended from March 2021 to February 2024, i.e., seven extra months of information. The extended data contain 118 features, and the characteristics can be observed in Tables 6 and 7. For these extended graphs, the percentage of fraud decreased (compared to Table 4) due to the expected behavior of the data provided by the digital platform. Despite that, the metric results improved significantly (see Tables 8 and 9).