# Many-Objective Optimization for Anomaly Detection on Multi-Layer Complex Interaction Networks
## 4 Results
### Case Study III: Real-World Multi-Layer Network--Kapferer Tailor Shop
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Layer** & \(\mu\) & \(\sigma\) & \(D\) \\ \hline Layer 1 & 0.3854688 & 0.2125424 & 0.4264507 \\ \hline Layer 2 & 0.4899419 & 0.2459141 & 0.6018893 \\ \hline Layer 3 & 0.260266 & 0.2709372 & 0.1470985 \\ \hline Layer 4 & 0.2506608 & 0.2362966 & 0.1983806 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical characterization: Kapferer Tailor Shop multiplex network. Figure 7: Pareto Front chart for the Kapferer multiplex network and the different layers (\(L1-L4\)): it depicts the Pareto Front (PF) based on minimization of the nodes’ centrality on the right side of the picture; the PF for the minimization is shown in the lower part of the vertical PF line. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No** & **L1** & **L2** & **L3** & **L4** & **Node** & **Label** & **Level** \\ \hline
1 & 0.397101 & 0.076899 & 0.227271 & 4.36E-19 & 6 & **[PERSON]** & 1 \\ \hline
2 & 0.17709 & 0.255465 & 1.57E-17 & 0.075394 & 8 & **[PERSON]** & 1 \\ \hline
3 & 0.059998 & 0.341019 & 1.57E-17 & 0.255959 & 10 & **[PERSON]** & 1 \\ \hline
4 & 0.297021 & 0.373386 & 1.57E-17 & 0.034534 & 18 & **[PERSON]** & 1 \\ \hline
5 & 0.013455 & 0.084826 & 0.028562 & 0.160698 & 20 & **[PERSON]** & 1 \\ \hline
6 & 0.022149 & 0.259987 & 0.006686 & 0.078514 & 22 & **[PERSON]** & 1 \\ \hline
7 & 0.154452 & 0.480842 & 1.57E-17 & 2.47E-18 & 26 & **[PERSON]** & 1 \\ \hline
8 & 0.143018 & 0.172989 & 0.020164 & 2.47E-18 & 27 & **[PERSON]** & 1 \\ \hline
9 & 0.481191 & 0.104547 & 0.228507 & 0.050905 & 4 & **SEAMS** & 2 \\ \hline
10 & 0.141911 & 0.352355 & 0.122665 & 0.147189 & 17 & **ENOCH** & 2 \\ \hline
11 & 0.237573 & 0.803971 & 0.105287 & 0.141186 & 25 & **MESHAK** & 2 \\ \hline
12 & 0.235435 & 0.23692 & 0.158304 & 0.012102 & 23 & **BEN** & 2 \\ \hline
13 & 0.372248 & 0.28798 & 0.045572 & 0.081653 & 35 & **CHRISTIAN** & 2 \\ \hline
14 & 0.446693 & 0.310456 & 0.060973 & 0.077575 & 38 & **CHILUFA** & 2 \\ \hline
15 & 0.250912 & 0.241652 & 0.07816 & 1.14E-17 & 39 & **MABANGE** & 2 \\ \hline
16 & 0.201821 & 0.607807 & 0.23093 & 0.353406 & 1 & **KAMWEFU** & 3 \\ \hline
17 & 0.241237 & 0.655692 & 0.203435 & 0.623016 & 5 & **CHIPATA** & 3 \\ \hline
18 & 0.276116 & 0.307887 & 0.302346 & 0.038746 & 15 & **NYIRENDA** & 3 \\ \hline
19 & 0.307472 & 0.335643 & 0.180225 & 0.252708 & 28 & **MPUNDU** & 3 \\ \hline
20 & 0.351199 & 0.284229 & 0.344007 & 0.224243 & 37 & **ANGEL** & 3 \\ \hline
21 & 0.417402 & 0.403791 & 0.084483 & 3.21E-17 & 33 & **CHOBE** & 3 \\ \hline
22 & 0.40163 & 0.475965 & 0.604055 & 0.304077 & 9 & **CHILWA** & 4 \\ \hline
23 & 0.318112 & 0.617183 & 0.282275 & 0.255397 & 2 & **NKUMBULA** & 4 \\ \hline
24 & 0.373917 & 0.571605 & 0.240467 & 0.491226 & 14 & **[PERSON]** & 4 \\ \hline
25 & 0.430833 & 0.481783 & 0.172491 & 0.237627 & 29 & **[PERSON]** & 4 \\ \hline
26 & 0.438643 & 0.721808 & 0.09152 & 0.188956 & 21 & **[PERSON]** & 4 \\ \hline
27 & 0.495024 & 0.405638 & 0.093042 & 0.025297 & 31 & **[PERSON]** & 4 \\ \hline
28 & 0.467563 & 0.418168 & 0.129541 & 0.092453 & 36 & **[PERSON]** & 4 \\ \hline
29 & 0.637761 & 0.658291 & 0.77509 & 0.411734 & 12 & **[PERSON]** & 5 \\ \hline
30 & 0.333699 & 0.666146 & 0.432999 & 0.649743 & 7 & **[PERSON]** & 5 \\ \hline
31 & 0.508462 & 0.881521 & 0.277381 & 0.506618 & 24 & **[PERSON]** & 5 \\ \hline
32 & 0.510248 & 0.730396 & 0.252753 & 0.268102 & 30 & **[PERSON]** & 5 \\ \hline
33 & 0.64177 & 0.543767 & 0.299718 & 0.303266 & 32 & **[PERSON]** & 5 \\ \hline
34 & 0.666847 & 0.649858 & 0.134423 & 0.455691 & 34 & **[PERSON]** & 5 \\ \hline
35 & 0.677078 & 0.792611 & 1 & 1 & **11** & **[PERSON]** & 6 \\ \hline
36 & 0.514091 & 0.810266 & 0.433572 & 0.390044 & 13 & **[PERSON]** & 6 \\ \hline
37 & 1 & 0.877987 & 0.92856 & 0.431482 & 16 & **[PERSON]** & 2 \\ \hline
38 & 0.585384 & 0.826399 & 0.900669 & 0.744139 & 3 & **ABRAHAM** & 7 \\ \hline
39 & 0.806727 & 1 & 0.674242 & 0.412089 & 19 & **MUKUBWA** & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Pareto Front and network centralities of the Kapferer Tailor Shop multiplex network.