# Many-Objective Optimization for Anomaly Detection on Multi-Layer Complex Interaction Networks
## 4 Results
### Case Study II: Aarhus Real-World Multi-Layer Network
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**No** & **L1** & & **L2** & & **L3** & & **L4** & & **L5** \\ \hline  & ACE score & Node & ACE score & Node & ACE score & Node & ACE score & Node \\ \hline
1 & 0.003489 & 7 & 0.029332 & 22 & 0.39361 & **68** & 0.520676 & 40 & 0.576491 & **35** \\ \hline
2 & 0.003489 & 12 & 0.029332 & 70 & 0.37581 & 66 & 0.398917 & **39** & 0.556781 & **48** \\ \hline
3 & 0.003489 & 15 & 0.028154 & 57 & 0.367554 & **99** & 0.38098 & 60 & 0.512226 & 49 \\ \hline
4 & 0.003489 & 19 & 0.026904 & 41 & 0.36481 & 77 & 0.373612 & 85 & 0.49261 & 71 \\ \hline
5 & 0.003489 & 20 & 0.025192 & 49 & 0.363811 & 71 & 0.357277 & 27 & 0.364353 & 75 \\ \hline
6 & 0.003489 & 21 & 0.024367 & 96 & 0.362636 & 78 & 0.339156 & 3 & 0.332967 & 14 \\ \hline
7 & 0.003489 & 22 & 0.024012 & 21 & 0.354883 & 33 & 0.290516 & 93 & 0.327641 & 64 \\ \hline
8 & 0.003489 & 26 & 0.023848 & 50 & 0.300521 & 8 & 0.278613 & 78 & 0.323315 & 43 \\ \hline
9 & 0.003489 & 33 & 0.023426 & 87 & 0.298178 & 39 & 0.270124 & 47 & 0.303674 & 30 \\ \hline
10 & 0.003489 & 35 & 0.023066 & 78 & 0.297861 & 32 & 0.26853 & 18 & 0.301767 & 98 \\ \hline
11 & 0.003489 & 36 & 0.022582 & 14 & 0.249337 & 50 & 0.265734 & 30 & 0.287986 & 66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ACE score of the synthetic/artificial network with 100 nodes consisting of five layers. From those nodes in the Pareto Front shown in Table 6, we assessed those with the highest ACE score in each layer--these are summarized in Table 7. As we can see in the table, in layer 1, the highest score is achieved by node 60, while in layer 2, there are nodes 1 and 14; in layer 3, there are nodes 4, 8, 12, and 14; in layer 4, there are nodes 1, 14, and 60; and finally, in layer 5, there is node 1. Since layer 3 exhibits a very low density \(D\), we did not consider this layer as a high indicator in consideration in order to find anomalous nodes. Additionally, as can be seen from the scores and the different layers, we can actually focus on those nodes that occur in more than two layers as interesting candidates, which also supports our intuitions. Therefore, in this way, we can actually collect \"evidence\" in this network towards designating nodes as anomalous from several layers of the network. As a consequence, we can identify four nodes out of six nodes in the Pareto Front to be selected as candidate anomalous nodes in our semiautomatic human-centered process, as we have discussed above. As an outcome, the designated final set of anomalous nodes is given by nodes 1, 12, 14, and 60. In the social context, this then indicates, for example, that node 1, node 12, node 14, and node 60 are not very active in the interaction with their colleague in terms of special interactions (layer1 = Facebook, layer 2 = Leisure, layer 3 = Work, layer 4 = Co-authorship, and layer 5 = Lunch): node 60 and node 12 are very rare in interactions with their colleagues on Facebook, and node 1 and node 14 are not very active in communication with their colleague related to leisure and co-authorship. Additionally, node 1 shows little interaction in leisure and co-authorship and is less active in interacting with his colleagues regarding lunch. The analysis process described above provides for a human-centered interpretable and explainable approach since simple network metrics  
Figure 5: Visualization of the Aarhus network, i.e., layer 5. Here, we observe clearly how the nodes connect to others with many connections, few connections, or even without a single connection. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Layer** & \(\mu\) & \(\sigma\) & \(D\) \\ \hline Layer 1 & 0.257986 & 0.291311 & 0.1054645 \\ \hline Layer 2 & 0.273081 & 0.331148 & 0.06775956 \\ \hline Layer 3 & 0.073182 & 0.240734 & 0.01147541 \\ \hline Layer 4 & 0.176249 & 0.22261 & 0.04808743 \\ \hline Layer 5 & 0.266831 & 0.20674 & 0.1060109 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical characterization: Aarhus multiplex network.