# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 2 Materials and Methods
### Multilayer Networks
#### 2.1.1 Descriptive Measures  
One of the main approaches to study multilayer networks consists in applying typical measures of the traditional SNA to each layer separately and then comparing these results [2]. This kind of approach can be useful for producing an initial overview of the data before applying truly multilayer measures.  
Different layers have in fact their specific characteristics in terms of number of nodes or edges, edge directionality, graph density, clustering coefficient, components, or network diameter when observed one at a time.  
**Definition 4** (Directionality).: _Edge directionality is a key property that defines the type of graph together with edge weight (i.e., a numerical value attached to each edge). If all edges are bidirectional, then the graph is called undirected. If edges have directionality, then the graph is called directed._  
**Definition 5** (Density).: _Graph density is a measure of how many edges between nodes exist compared to how many edges between actors are possible. It is defined as:_  
\[\delta=\frac{L}{N(N-1)}. \tag{1}\]  
**Definition 6** (Diameter).: _Graph diameter \(d_{\max}\)[41] is the maximum shortest path in the graph, where the shortest path (or distance) \(d_{v_{i}v_{j}}\) between nodes \(v_{i}\) and \(v_{j}\) is the path with the fewest number of edges. A path is a sequence of nodes such that each node is connected to the next node along the path by an edge._  
**Definition 7** (Average path length).: _The average path length \(\langle d\rangle\)[41] is the average distance between all pairs of nodes in the graph. It is defined as:_  
\[\langle d\rangle=\sum_{v_{i},v_{j}\in V}\frac{d_{v_{i}v_{j}}}{N(N-1)}. \tag{2}\]  
**Definition 8** (Connected component).: _A connected component cc [41] is a subset of nodes in a graph, so that there is a path between any two nodes that belongs to the component, but one cannot add any more nodes to it that would have the same property._  
**Definition 9** (Largest connected component).: _The largest or giant connected component lcc can be found typically in real undirected graphs, and it contains most of the nodes in the graph. The rest of the graph usually is divided into a large number of small components disconnected from the others._  
**Definition 10** (Degree).: _The degree \(k\)[36] is a key property of each node in a graph and represents the number of connections it has to other nodes. It is defined as:_  
\[k(v_{i})=|\{v_{j}:(v_{i},v_{j})\in E\vee(v_{j},v_{i})\in E\wedge v_{i}\
eq v_{j} \}|. \tag{3}\]  
**Definition 11** (Degree distribution).: _The degree distribution \(p_{k}\)[41] provides the probability that a randomly selected node \(v_{i}\) in the graph has degree \(k\). For a graph with \(N\) nodes, the degree distribution is a normalized histogram given by:_  
\[p_{k}=\frac{N_{k}}{N}, \tag{4}\]  
_where \(N_{k}\) is the number of nodes with degree equal to \(k\)._  
**Definition 12** (Average clustering coefficient).: _The average clustering coefficient \(\langle\mathrm{C}\rangle\)[41] is the probability that two neighbors of a randomly selected node link to each other. It is defined as:_  
\[\langle\mathrm{C}\rangle=\frac{1}{N}\sum_{1}^{N}\mathrm{C}(v_{i}), \tag{5}\]  
_where \(C(v_{i})\) is the clustering coefficient of a node \(v_{i}\) which captures the degree to which the neighbors of \(v_{i}\) link to each other. It is defined as:_  
\[C(v_{i})=\frac{2L_{i}}{k(v_{i})(k(v_{i})-1)}, \tag{6}\]  
_where \(L_{i}\) represents the number of edges between the \(k\) neighbors of the node \(v_{i}\)._