# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 2 Materials and Methods
### Multilayer Networks
#### 2.1.3 Layer Measures  
Actor measures such as the relevance can be used to know the role of a layer (or a set of layers) with respect to its actors. In order to know the relationship between different layers (or sets of layers), some measures of layer similarity need to be introduced. Layer similarity can be studied from two different perspectives: the actor-centered perspective and layer-centered perspective [2]. The first one describes the differences between different layers as a sign of different behaviors of the actors, strategically selecting what kinds of connections they want to establish on every layer. The second one describes the differences between layers in terms of interlayer influences, and it can be investigated by applying existing methods to compute correlation and similarities to multilayer networks. [PERSON] et al. [42] developed the idea of layer correlation as a multilayer network version of the classical Jaccard correlation coefficient. **Definition 19** (Jaccard correlation coefficient). : _Given two finite sample sets \(A\) and \(B\), the Jaccard correlation coefficient [43] is defined as the size of the intersection divided by the size of the union of the sample sets:_  
\[J(A,B)=\frac{|A\cap B|}{|A\cup B|}. \tag{13}\]  
**Definition 20** (Jaccard layer correlation coefficient). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the Jaccard layer correlation coefficient [42] computes the ratio of pairs of actors connected on a set of layers \(L\subseteq\mathcal{L}\) and the total number of pairs of actors connected in at least one layer in \(L\). It is defined as:_  
\[J(L)=\frac{\left|\bigcap_{l\in L}P_{l}\right|}{|\bigcup_{l\in L}P_{l}|}, \tag{14}\]  
_where \(P_{l}\) denotes the set of pairs of actors connected in each layer \(l\in L\)._  
The Jaccard coefficient can be used to measure the overlapping of the actors, i.e., the presence of common actors between pair of layers in a multilayer network. This coefficient can also be used to measure the overlapping of the edges to know if the common actors between two layers behave in a similar way. This measure can show edges that actually exist in each pair of layers and actors that are highly connected on a layer but not on the other layer. If actors who are connected by an edge in one layer are also connected in the other layer, the value of edge overlapping will be equal to 0. On the contrary, this value will be 1 if the actors are connected in both layers or in none of the layers. [PERSON] et al. [8] introduced an actor-centered approach trying to quantify the similarity between the degree of actors across various layers. In this case, it is possible to use the Pearson correlation coefficient. **Definition 21** (Pearson correlation coefficient). : _Given two random variables \(a\) and \(b\), the Pearson correlation coefficient [44] is defined as:_  
\[P_{r}=\frac{cov(a,b)}{\sigma_{a}\sigma_{b}}, \tag{15}\]  
_where \(cov(a,b)\) is the covariance of \(a\) and \(b\) and \(\sigma_{a}\sigma_{b}\) is the product of their standard deviations._  
**Definition 22** (Pearson interlayer correlation coefficient). : _Given a multilayer network \(M=(\mathcal{A},\mathcal{L},V,E)\), the Pearson correlation coefficient [12] is defined as:_  
\[P_{r}(l,l^{\prime})=\frac{\langle k(a,l)k(a,l^{\prime})\rangle-\langle k(a,l) \rangle\langle k(a^{\prime},l^{\prime})\rangle}{\sigma_{k(a,l)}\sigma_{k(a^{ \prime},l^{\prime})}}, \tag{16}\]  
_where \(k(a,l)\) and \(k(a^{\prime},l^{\prime})\) are the degrees of actor \(a\), respectively, at layer \(l\) and layer \(l^{\prime}\), and \(\sigma_{k(a,l)}\sigma_{k(a^{\prime},l^{\prime})}\) is the product of their standard deviations._  
For each pair (actor, layer), the Pearson correlation does not depend on the connected actors but only on the number of incident edges on an actor. The difference between the degree distributions in different layers can instead been quantified using the Jeffreys dissimilarity function [45]. **Definition 23** (Jeffreys dissimilarity function).