# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 3 Results  
Table 1 shows our preliminary analysis of the multiplex network considering each layer as an independent graph. The third layer is the smallest one and presents different characteristics especially in terms of number of connected components, size of the largest connected component and graph density. The first two layers seem to be more similar with the exception of the average clustering. The results obtained applying the actor measures on the Montagna multiplex network are shown in Figure 2 and more in detail in Tables 2-4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Layer** & \(\mathbf{N}\) & \(\mathbf{L}\) & **Dir.** & \(|\mathbf{cc}|\) & \(\mathbf{lcc}\) & \(\delta\) & \(\langle\mathbf{C}\rangle\) & \(\langle d\rangle\) & \(\mathbf{d_{\text{max}}}\) \\ \hline Meetings (\(l_{1}\)) & 101 & 256 & False & 5 & 92 & 0.050693 & 0.417647 & 3.308887 & 7 \\ Phone Calls (\(l_{2}\)) & 100 & 124 & False & 5 & 89 & 0.025051 & 0.072687 & 3.378192 & 7 \\ Crimes (\(l_{3}\)) & 25 & 74 & False & 2 & 23 & 0.246667 & 0.779626 & 2.407115 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive measures for the flattened Montagna multiplex network and its three layers: number of nodes, number of edges, directionality, number of connected components, size of the largest connected component, density, clustering coefficient, average path length and diameter. Table 2 contains the degree of the 20 most central actors in the whole network and in each single layer. It also shows the standard deviation of the degree. Our actors are specific individuals involved in criminal activities or members of Mafia families which were at the center of the Montagna operation. We reconstructed the actor roles reading court documents of the anti-mafia operation. These roles have been analyzed in our previous work [36] making particular reference to the structure of a Mafia family which is characterized by typical figures such as boss, underboss, consigliere, messaggero, caporegimes, soldiers and associates. Considering the layered structure, the actors we identified as the most central are 18, 47 and 27. These actors are respectively a caporegime of the [PERSON] family, a deputy caporegime and a caporegime of the [PERSON] family. Therefore, they are effectively important. Caporegimes manage their crew of soldiers (i.e., average type criminals) within a Mafia family in a specific geographical location. These actors have similar degrees on the  
Figure 2: Degree \(k(a,\mathcal{L})\) and neighborhood \(n(a,\mathcal{L})\) computed on all the three layers \(\mathcal{L}\) compared with degree \(k(a,l)\), exclusive neighborhood \(xn(a,l)\), relevance \(r(a,l)\) and exclusive relevance \(xr(a,l)\) computed on each of the three layers Meetings (\(l_{1}\)), Phone Calls (\(l_{2}\)) and Crimes (\(l_{3}\)) for each actor \(a\) of the 20 highest-degree actors. first two layers and different degrees on the third one. The Crimes layer does not include some of the most central actors such as 68, 12, 22, 11, 43 and 25. Actor 22 is a pharmacist which can be an important figure because chemical or pharmacological knowledge is required during a process of drug synthesis. Actor 11 is a criminal activity coordinator in Messina which is central to know the connection among the [PERSON] and [PERSON] families with other criminal organizations sited in Messina. Node 43 is the messaggero who is a key figure in a Mafia family who functions as a connection between families. He limits the boss exposure, reducing the necessity to meet publicly. Actor 29 seems to have equal importance for all the three layers as confirmed by a low degree deviation. He is an entrepreneur. Compared with the two-layer multiplex network described in [36], adding the third layer brings to light the importance of the entrepreneurs such as actors 54, 64 and 63, who did not appear as prominent figures from the analysis of the first two layers.