# Many-Objective Optimization for Anomaly Detection on Multi-Layer Complex Interaction Networks
## 3 Method
### Network Centrality  
To find the most influential node in a network, we can utilize the notion of network centrality. This is formalized using different network centrality metrics. In everyday life, for example, we perceive a person or some people to be very important in a community or in an organization if in some way those have a certain influence on others or can generate important decision for a community. Identifying important person or organizations can then be considered as identifying key players in a community.  
Figure 1: Overview on the procedural steps of the proposed approach: first, centralities and densities are estimated before many-objective optimization is applied in order to obtain the Pareto Front, i.e., the set of less important nodes. Finally, these are assessed in a human-in-the-loop approach.  
Figure 2: Depiction sketch of a multi-layer network consisting of ten nodes, with two types of different links. The different sets of links in this network are illustrated by different colors (of the respective links/edges).  
Network centrality methods enable the identification of such key players or most influential nodes in a social setting [56; 57]. They include, e.g., the following centrality measures:  
* closeness centrality, measuring the distance from a specific node to all other nodes [58];
* betweenness centrality, taking into account the number of shortest paths past the certain nodes in the network [59];
* eigenvector centrality, considering the number of links from other nodes, their importance, and to how many these nodes themselves point to, c.f., [60]; and
* degree centrality, centering on the number of connected peers for a node [61; 62].  
For our proposed approach, we consider eigenvector centrality which is computed for each layer of the network. We applied eigenvector centrality, since this precisely corresponds to our intuition for estimating the notion of connections to important nodes and/or parts of the network, which is relevant for anomaly detection. However, our method can potentially also be adapted to other centrality measures, e.g., see those above.  
The eigenvector centrality can be defined as follows: For a given graph \(G:=(V,E)\) with \(|V|\) denoting the number of nodes, let \(A=(a_{v,t})\), \(v\in\{1,...,|V|\}\), \(t\in\{1,...,|V|\}\) be the adjacency matrix, i.e., \(a_{v,t}=1\) if vertex \(v\) is linked to vertex \(t\), and \(a_{v,t}=0\) otherwise.  
The relative centrality score of vertex \(v\) is defined as  
\[x_{v}=\frac{1}{\lambda}\sum_{t\in M(v)}x_{t}=\frac{1}{\lambda}\sum_{t=1}^{|V| }a_{t,v}x_{t}\,,\]  
where \(M(v)\) is a set of the neighbors of \(v\) and \(\lambda\) is a constant.  
Furthermore, the eigenvector centrality in vector notation can be written as follows: \(Ax=\lambda x\). It is clearly defined that \(x\) is an eigenvector of \(A\). Since there could be many eigenvectors of \(A\), by convention, the eigenvector that corresponds to the largest eigenvalue is considered. There are two important factors that influence the eigenvector centrality of each node: (1) the number of neighbors that point to the node and (2) the weight of neighbors that point to the node.  
Furthermore, there is also a possibility that nodes with a larger number of neighbors have a lower eigenvector centrality compared to nodes with fewer neighbors. This situation can occur, for example, when the neighbors of the less connected node have higher weights.