# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 3 Results
Table 5 shows the Jaccard correlation coefficient computed among pairs of layers in the Montagna multiplex network to verify the presence of the same actors on different layers. If two layers do not share any actor, the coefficient will be equal to 0. If two layers share the same actors, the coefficient will be equal to 1. In our case, we can observe the strongest overlapping between the actors in the Meetings and Phone Calls layers, which is slightly lower between the Meetings and Crimes layers, and very low between the Phone Calls and Crimes layers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a\) & _n(a,_L_) & _xn(a,l_1)_ & _xn(a,l_2)_ & _xn(a,l_3)_ \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Neighborhood computed on 20 highest-degree actors of the whole Montagna multiplex network and exclusive neighborhood computed on its single layers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(a\) & \(r(a,I_{1})\) & \(xr(a,I_{1})\) & \(r(a,I_{2})\) & \(xr(a,I_{2})\) & \(r(a,I_{3})\) & \(xr(a,I_{3})\) \\ \hline
18 & 0.545455 & 0.318182 & 0.568182 & 0.340909 & 0.181818 & 0.068182 \\
47 & 0.655172 & 0.275862 & 0.724138 & 0.344828 & 0.137931 & 0.00000 \\
27 & 0.666667 & 0.416667 & 0.458333 & 0.208333 & 0.166667 & 0.125000 \\
29 & 0.650000 & 0.350000 & 0.450000 & 0.100000 & 0.400000 & 0.200000 \\
61 & 0.166667 & 0.083333 & 0.708333 & 0.541667 & 0.333333 & 0.208333 \\
68 & 0.789474 & 0.473684 & 0.526316 & 0.210526 & N/A & N/A \\
54 & 0.352941 & 0.058824 & 0.294118 & 0.058824 & 0.764706 & 0.588235 \\
45 & 0.705882 & 0.411765 & 0.352941 & 0.117647 & 0.352941 & 0.176471 \\
51 & 0.733333 & 0.33333 & 0.266667 & 0.000000 & 0.466667 & 0.266667 \\
48 & 0.857143 & 0.428571 & 0.071429 & 0.000000 & 0.571429 & 0.142857 \\
64 & 0.461538 & 0.230769 & 0.153846 & 0.000000 & 0.769231 & 0.461538 \\
12 & 1.000000 & 0.937500 & 0.062500 & 0.000000 & N/A & N/A \\
22 & 0.933333 & 0.866667 & 0.133333 & 0.066667 & N/A & N/A \\
11 & 0.800000 & 0.733333 & 0.266667 & 0.200000 & N/A & N/A \\
75 & 0.307692 & 0.230769 & 0.615385 & 0.538462 & 0.230769 & 0.076923 \\
43 & 0.818182 & 0.545455 & 0.454545 & 0.181818 & N/A & N/A \\
36 & 0.615385 & 0.538462 & 0.307692 & 0.230769 & 0.153846 & 0.153846 \\
25 & 1.000000 & 0.923077 & 0.076923 & 0.000000 & N/A & N/A \\
63 & 0.44444 & 0.111111 & 0.111111 & 0.000000 & 0.888889 & 0.444444 \\
50 & 0.888889 & 0.555556 & 0.111111 & 0.000000 & 0.444444 & 0.111111 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relevance and exclusive relevance of the 20 highest-degree actors on each layer of the Montagna multiplex network. Figure 3: Jaccard overlapping of the actors \(J_{A}\), Jaccard overlapping of the edges \(J_{E}\), Pearson correlation between actor degrees \(P_{r}\) and Jeffreys dissimilarity between degree distributions \(D_{J}\). The overlapping between edges in a pair of layers can be checked to know if actors are connected to the same other actors on different layers. In this case, if no actors who are connected on a layer are also connected on the other layer, the Jaccard coefficient will be equal to 0. If all pairs of actors are connected in none of layers or in both of them, the Jaccard coefficient will be equal to 1. Table 6 shows that there is no overlapping of the edges in our Montagna multiplex network. This means that in most cases, the actors of our network who represent suspected criminals have different connections among them. When there is an overlapping between the actors in the layer of a multiyer network, it can be interesting to verify if actors have a high or low degree on all the layers in which they are present. In other words, we want to know if these actors behave in a similar way. The Pearson correlation allows us to know this information. Table 7 shows the Pearson correlation coefficient computed among the degree of the actors in pairs of layers of the Montagna multiplex network. The Pearson correlation is in the interval \([-1,1]\). It is equal to \(-1\) when actors with high degree values in one layer have low degree values in the other and vice versa.