## 5 Results
### Experiments with Graph Combinations
The reason for this is that there is more information related to the graph structure (more extended period), and also, there are more features that help to distinguish between fraud and expected behavior. At the same \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Graph** & **Vertex** & **Edges** & **Users** & **Entities** & **Edge-to-Vertex Ratio** \\ \hline Cards & 648,022 & 392,950 & 266,164 & 381,858 & 60.63 \\ \hline Devices & 988,107 & 691,611 & 318,736 & 669,371 & 69.99 \\ \hline Bank Accounts & 252,482 & 165,604 & 103,381 & 149,101 & 65.59 \\ \hline Cards–Devices & 1,404,593 & 1,084,561 & 353,364 & 1,051,229 & 77.21 \\ \hline Cards–Bank Accounts & 805,111 & 558,554 & 274,152 & 530,959 & 69.37 \\ \hline Devices–Bank Accounts & 1,138,913 & 857,215 & 320,441 & 818,472 & 75.26 \\ \hline Cards–Devices–Bank Accounts & 1,554,229 & 1,250,165 & 353,899 & 1,200,330 & 80.43 \\ \hline IPs–Devices–Bank Accounts & 1,285,705 & 1,120,085 & 320,577 & 965,128 & 87.11 \\ \hline IPs–Cards–Devices & 1,551,262 & 1,347,354 & 353,377 & 1,197,885 & 86.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Characteristics of the extended graphs. The data span from March 2021 to February 2024. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Graph** & **F\({}_{1}\)** & **AUC** \\ \hline Cards & 0.3131 \(\pm\) 0.2106 & 0.4578 \(\pm\) 0.0678 \\ \hline Devices & 0.1763 \(\pm\) 0.0743 & 0.6755 \(\pm\) 0.0525 \\ \hline Bank Accounts & 0.1508 \(\pm\) 0.0720 & 0.6285 \(\pm\) 0.0142 \\ \hline Cards + Devices & 0.4213 \(\pm\) 0.1341 & 0.6727 \(\pm\) 0.0552 \\ \hline Cards + Bank Accounts & 0.3136 \(\pm\) 0.2212 & 0.5330 \(\pm\) 0.0737 \\ \hline Devices + Bank Accounts & 0.4121 \(\pm\) 0.1437 & 0.7014 \(\pm\) 0.0249 \\ \hline Cards + Devices + Bank Accounts & 0.4523 \(\pm\) 0.0710 & 0.7255 \(\pm\) 0.0196 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of the experiments performed on each graph. Each graph is a bipartite graph between users and cards, users and devices, and users and bank accounts. On the other hand, the AUC scores show more favorable results. A notable difference in the values for AUC can be observed when additional edges are included in the analysis, particularly when comparing graphs with single-edge types to those with two- or three-edge types. In summary, it has been found that including more features and structural and temporal context enhances fraud detection, according to the values of the metrics: \(F_{1}\) and AUC. Additionally, there appears to be no direct correlation between metric results and the percentage of fraudulent observations, suggesting that graph behavior information is more critical than the raw fraud data itself (as was shown in Tables 3 and 5). When comparing the results with the additional event (IP address) to the same events of Table 5, it can be noticed that the aggregation of this event also helps to improve the metric results. This concurs with the previous outcomes, which show that adding more interaction events helps improve the metrics. For this work, there are only results of those new graph combinations because adding more events also has an impact on the execution time of the algorithm. The improvement in the metrics due to the addition of more entities (for example, the cards-devices-bank accounts (Table 5) or IPs-cards-devices graphs (Table 8)) also means that more fraud events are captured in the detection. This can be appreciated from the results in Table 9, where the recall value (also known as true positive rate) tends to improve when adding more structural information. This makes sense from an empirical context because adding more data (IPs, devices, cards, bank accounts, etc.) has been noticed to help differentiate between fraud patterns and standard patterns. As shown in Tables 5 and 8, adding structural information correlates with improving the metrics. This concurs with the empirical context of fraud detection, where fraudulent events are expected to exhibit different patterns from regular events by nonfraudulent users.