# Many-Objective Optimization for Anomaly Detection on Multi-Layer Complex Interaction Networks
## 4 Results
### Case Study I: Synthetic Multi-Layer Networks  
In our first case study for evaluating our approach, we focus on synthetically generated networks. These are generated via a random graph according to the Erdos and Renyi model. In this complete graph model, each edge has a probability of \(1-p\) to be removed from the network. By varying the probability and number of nodes, we can generate different networks and layers, respectively. We generated networks with a specific number of nodes (\(n=100\)) and link probabilities ranging from \(p=0.03\) to \(p=0.3\); for the number of layers, we chose \(l=5\) layers. We denote the layers with \(g_{1}\) to \(g_{5}\). Table 2 provides an overview on the respective network statistics. Figure 3 shows the applied synthetic network based on Erdos-Renyi graph generation. Each layer corresponds to one synthetic network with different link probabilities \(p\in\{0.03,0.05,0.20,0.25,0.30\}\). We performed a round of experiments, and take this specific instantiation as an example to demonstrate the methodological approach. Other instantiations, e.g., varying the number of layers, showed similar results. For our example case, as a result of the estimation of eigenvector centrality and the application of many-objective optimization for synthetics networks, we detected 38 nodes in the Pareto Front, c.f., Table 3. Applying our selection methodology described in Section 3, we started with those nodes in the Pareto Front in order to identify candidate nodes as sets of anomalous nodes. Therefore, starting with these 38 nodes, we observe a set of nodes with very high ACE score values or some nodes that are very close to these and show a large deviation from the respective centrality mean \(\mu\). In particular, we took those nodes with very high ACE score in specific consideration, where we observe them being contained in the layers that have high density \(D\). In our case, these are the layers \(g_{3}\), \(g_{4}\), and \(g_{5}\). As a result, we can identify six nodes (35, 39, 48, 68, and node 99), which were then categorized as anomalous nodes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Type** & **Number of Nodes** & **Number of Layers** & **Number of Edges** \\ \hline
**Synthetic** & 100 & 5 & 8158 \\ \hline
**CS-Aarhus** & 61 & 5 & 620 \\ \hline
**Kapferer-Tailor** & 39 & 4 & 552 \\ \hline
**Social Media** & 6407 & 3 & 74,862 \\ \hline
**Trade Network** & 207 & 11 & 148,727 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary: statistical overview on the applied networks, describing the overall number of nodes, the number of layers, and the overall number of edges of the respective multiplex networks. As can be observed in Tables 3 and 4 and Figure 3, in this way, we selected the nodes with the highest ACE score that are also in the Pareto Front, focusing on the denser layers, while the low-density layers are neglected since these also feature quite low ACE scores. Intuitively, this makes sense since a node with a low connectivity and/or high eigenvector centrality on a layer with higher density then gets a higher chance to be proposed as an anomalous node. It is important to note that layers \(g_{1}\) and \(g_{2}\) feature some unconnected nodes, which can then be indicated as anomaly candidates for these layers based on a connectivity criterion as well. Such a (simple) criterion can of course be implemented complementing the Pareto-front-based selection. However, in our many-objective optimization approach, we consider all layers at the same time to first find initial candidates on the Pareto Front by minimizing eigenvector centrality, which are then refined using further criteria such as density and connectivity, which are formalized in the novel ACE score. Figure 3: Synthetic network generated by an Erdős Rényi random graph generation strategy. Each layer consists of 100 nodes. For the different layers (1–5), layer 1 top left, layer 2 top right, etc.