# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 2 Materials and Methods
### Experimental Design  
The analysis of our multiplex criminal network was conducted using the Python module unnet.multinet (available at: [[https://bitbucket.org/uuinfolab/py_multinet/src/master/](https://bitbucket.org/uuinfolab/py_multinet/src/master/)]([https://bitbucket.org/uuinfolab/py_multinet/src/master/](https://bitbucket.org/uuinfolab/py_multinet/src/master/)), accessed on 10 April 2022) created by [PERSON], [PERSON] and [PERSON] to analyze multiplex social networks. This library is the Python version of the multinet library for the analysis of multilayer networks released by the same authors for the R framework [48]. Algorithm 1 shows the design of our experiments. ```
% Initialization;  set three undirected and weighted graphs \(G=(V,E)\); % Generation of the multiplex network;  add each graph \(G\) as layer \(l\) of a multiplex network \(M=(\mathcal{A},\mathcal{L},V,E)\); % Descriptive measures; for each layer \(l\in\mathcal{L}\)do  compute \(N\), \(L\), \(dir\)., \(|cc|\), \(lcc\), \(\delta\), \(\langle C\rangle\), \(\langle d\rangle\) and \(d_{\text{max}}\);  end for % Actor measures; for each actor \(a\in\mathcal{A}\)do  compute \(k(a,\mathcal{L})\);  end for set the 20 high degree actors \(K_{\mathcal{A}}\); for each actor \(a\in K_{\mathcal{A}}\)do  compute \(\sigma_{k}\) and \(n(a,\mathcal{L})\); for each layer \(l\in\mathcal{L}\)do  compute \(xn(a,l)\), \(r(a,l)\) and \(xr(a,l)\);  end for  end for % Layer measures; for each pair of layers \((l,l^{\prime})\in\mathcal{L}\)do  compute \(J_{\mathcal{A}}(l,l^{\prime})\), \(J_{E}(l,l^{\prime})\), \(P_{r}(l,l^{\prime})\) and \(D_{J}(l,l^{\prime})\);  end for
```  
**Algorithm 1** Multilayer approach. We started with the creation of the three simple graphs Meetings (\(G_{1}\)), Phone Calls (\(G_{2}\)) and Crimes (\(G_{3}\)) described in Section 2.2, and we added them, respectively, as the first layer (\(l_{1}\)), second layer (\(l_{2}\)) and third layer (\(l_{3}\)) of a multiplex network \(M\). Figure 1: Multiple layers representing different kinds of relationships, i.e., meetings (violet), phone calls (blue), and crimes (light blue), among suspects in the Montagna operation. We applied to each layer \(l\) some of the most traditional SNA measures such as the 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 (see Section 2.1.1). Then, we studied the actors in the network (see Section 2.1.2). We computed the highest degree actors on the whole multiplex network focusing on the 20 most central actors. We also compared the degree values of each actor in each specific layer computing the standard deviation of the degree. It can be a useful function to estimate what actors possess similar or different degrees on the various layers. Given the set of the 20 most central actors \(K_{\mathcal{A}}\), we computed the neighborhood for these actors considering the whole network with all the three layers (\(\mathcal{L}\)) and the exclusive neighborhood on each layer \(l\). The neighborhood is not evaluated on each layer because on a single layer, neighborhood and degree have the same value. The exclusive neighborhood is calculated on a layer because it allows knowing if a layer is important to maintain the actor connectivity. We also calculated relevance and exclusive relevance for the actors in \(K_{\mathcal{A}}\) on each layer \(l\). Relevance and exclusive relevance allow studying the relation between the multilayer network and the actors identifying those who are highly connected on a given layer or actors that are connected uniquely through a layer. At the end, we compared layers using four different approaches (see Section 2.1.3). We computed the overlapping of actors and edges using the Jaccard correlation coefficient. Consequently, we determined the correlation between the degrees using the Pearson correlation coefficient to know if actors with a high degree on one layer had a similar behavior on the other layers.