# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 2 Materials and Methods
### Multilayer Networks
#### 2.1.2 Actor Measures  
Actors in multilayer networks can be described by a specific set of metrics [2]. Some of them are direct extensions of their single-layer counterparts and measure actors based on their local edges and relationships. Other measures have no counterpart for single-layer networks. They characterize the relevance of a specific layer or set of layers within the context of the connectivity of the actors. **Definition 13** (Multilayer actor degree). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the degree [2] of an actor \(a\in\mathcal{A}\) on a set of layers \(L\subseteq\mathcal{L}\) is the number of his connections on all these layers. It is defined as:_  
\[k(a,L)=|\{\{(a,l),(a^{\prime},l^{\prime})\}\in E:l,l^{\prime}\in L\}|. \tag{7}\]  
When \(L=\mathcal{L}\), the degree of the actor is within the whole multilayer network, whereas when the set of layers contains only one layer, the traditional degree is computed as shown in Equation (3) for the actor in that layer. **Definition 14** (Multilayer actor degree deviation). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the degree deviation [2] of an actor \(a\in\mathcal{A}\) on a set of layers \(L\subseteq\mathcal{L}\) is defined as the standard deviation of the degree of a over the layers in \(L\):_  
\[\sigma_{k}=\sqrt{\frac{\sum\limits_{l\in L}\left(k(a,\{l\})-\frac{k(a,L)}{|L|} \right)^{2}}{|L|}}. \tag{8}\]  
The degree deviation indicates the presence of an actor in the multilayer network and quantifies to what extent actors have similar or different degrees on the different layers. **Definition 15** (Multilayer actor neighborhood). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the neighborhood [2] of an actor \(a\in\mathcal{A}\) on a set of layers \(L\subseteq\mathcal{L}\) is the number of the neighbors \(ns\) of \(a\), i.e., those distinct actors that are connected to \(a\) on a specific layer or set of layers. It is defined as:_  
\[n(a,L)=|\{a^{\prime}\in\mathcal{A}:\{(a,l),(a^{\prime},l^{\prime})\}\in E\wedge l,l^{\prime}\in L\}|=|ns(a,L)|. \tag{9}\]  
When computed on a single layer network, degree and neighborhood coincide. However, the degree of an actor cannot be defined anymore as the number of adjacent actors in a multilayer network where actors can be connected to different individuals depending on the layer. **Definition 16** (Multilayer actor exclusive neighborhood). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the exclusive neighborhood [2] of an actor \(a\in\mathcal{A}\) on a set of layers \(L\subseteq\mathcal{L}\) counts the neighbors that are adjacent to \(a\) only on the input layers \(L\). It is defined as:_  
\[xn(a,L)=|ns(a,L)\setminus ns(a,\mathcal{L}\setminus L)|, \tag{10}\]  
_where the symbol \(\setminus\) indicates the set difference operation._  
The exclusive neighborhood considers the actors that are connected exclusively on a specific layer or set of layers, and it is used to explore the role of specific layers for specific actors. If an actor has a high exclusive neighborhood on a layer, this means that this layer is important to maintain the actor connectivity. In fact, if the layer was removed, the actor's neighbors would also disappear. **Definition 17** (Multilayer actor relevance). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the relevance [2] of an actor \(a\in\mathcal{A}\) on a set of layers \(L\subseteq\mathcal{L}\) is the ratio between the neighbors of \(a\) on the specific set of layers \(L\) and the total number of his neighbors. It is defined as:_  
\[r(a,L)=\frac{n(a,L)}{n(a,\mathcal{L})}. \tag{11}\]  
The relevance describes the specific signature of each actor, i.e., to be present on the different layers. **Definition 18** (Multilayer actor exclusive relevance).