# Many-Objective Optimization for Anomaly Detection on Multi-Layer Complex Interaction Networks
## 5 Conclusions  
In this paper, we proposed a novel approach for centrality-based anomaly detection on multi-layer networks using many-objective optimization. In summary, the presented approach identifies anomaly candidates based on a new method to obtain a Pareto Front of potential anomalous nodes via minimizing eigenvector centrality. Given these candidates, we applied a statistical network metrics criterion--formalized in the novel ACE score--which balances the connectivity and eigenvector centrality of a node in relation to density relative to the containing layer. For our evaluation, we conducted experiments using synthetic as well as real-world multi-layer network data. Specifically, we utilized a synthetic dataset generated by an Erdos Renyi random graph generation approach. Furthermore, we applied four different real-world multiplex networks capturing different (social) relations. From our results and the analysis in detecting anomalous nodes in the synthetic network generated by an Erdos Renyi generator, we could clearly identify the anomalous nodes given the different layer structure on the synthetic data. Regarding the real-world networks, such as the Aarhus, Kapferer, trade, and social media networks, we could also apply our proposed approach well, identifying some special cases such as singular anomalies and anomalies supported by layers with different densities. In this respect, our results \begin{table}
\begin{tabular}{c c c c} \hline \hline
**NO** & **L1:Youtube** & **L2:FriendFeed** & **L3:Twitter** & **Node** & **Level** \\ \hline
16 & 0 & 3.41E-19 & 4.85E-05 & 1095 & 3 \\ \hline
17 & 0 & 3.43E-19 & 3.20E-06 & 1915 & 3 \\ \hline
18 & 0 & 3.40E-19 & 0.0003068 & 2543 & 3 \\ \hline
19 & 0 & 3.42E-19 & 2.96E-05 & 3089 & 3 \\ \hline
20 & 0 & 4.29E-19 & 3.00E-19 & 1302 & 3 \\ \hline
21 & 0 & 9.03E-19 & 0 & 2307 & 3 \\ \hline
22 & 0 & 4.08E-19 & 6.50E-19 & 149 & 3 \\ \hline
23 & 0 & 3.85E-19 & 6.53E-19 & 2973 & 3 \\ \hline
24 & 0 & 3.47E-19 & 6.93E-19 & 6173 & 3 \\ \hline
25 & 0 & 3.46E-19 & 0.0005896 & 796 & 4 \\ \hline
26 & 0 & 4.29E-19 & 4.05E-19 & 164 & 4 \\ \hline
27 & 0 & 4.60E-19 & 3.97E-19 & 3392 & 4 \\ \hline
28 & 0 & 1.19E-18 & 0 & 2922 & 4 \\ \hline
29 & 0 & 4.15E-19 & 6.52E-19 & 2226 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**ACE Score with High Density Network Only L2 and L3** \\ \hline \hline
**L2:FriendFeed** & \multicolumn{2}{c}{**L3:Twitter**} \\ \hline ACE Score & Node & ACE Score & Node \\ \hline
0.000130404 & 143 & 0.00029655 & 270 \\ \hline
0.0001304 & 1101 & 0.000296548 & 1101 \\ \hline
0.0001304 & 1102 & 0.000296548 & 1102 \\ \hline
0.0001304 & 2019 & 0.000297 & 2551 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: ACE score of multiplex network centrality for the social media network, only considering high-density layers L2 and L3. were quite clear in identifying the sets of anomalous nodes. Finally, by measuring the mean \(\mu\), standard deviation \(\sigma\), and edge-density \(D\), from the population of nodes centrality in each layer, the ACE score of each layer can be directly calculated. Then, the nodes with the highest ACE scores, which imply potentially anomaly nodes, can be easily identified and assessed. Overall, our results thus indicate that we can identify anomalous nodes quite well based on the criteria of connectivity and importance (as estimated by the eigenvector centrality), always relative to the density of the containing network/layer. Additionally, a specific advantage of the proposed method is its interpretability and explainability, referring to the network structure and topological features, which can be intuitively assessed in a human-centered approach, c.f. [70]. In future work, we aim to extend the analysis towards further real-world complex networks in order to capture and investigate further real-world phenomena about potential anomalies, e.g., in feature-rich networks [4] as well as taking temporal relations into account [4; 41].