# The Whole Is Greater than the Sum of the Parts: A Multilayer Approach on Criminal Networks +
## 1 Introduction  
### Contextualization  
A long time before the development of social network analysis (SNA), people dealt with multiple social networks. Even if this has always been done with no effort, it is not a trivial activity that can be overlooked. One of the possible views of the problem is represented by the connections between individuals through multiple types of relational ties [1; 2; 3; 4]. The world cannot be defined considering different kinds of relationships as ontologically equivalent and without taking into consideration the interactions among different kinds of connections from which invisible relationships emerge. For this reason, we cannot look only at a single kind of relationship within a single social network [1; 2; 3; 4]. A single-layer perspective has been used to study the social interactions in most cases. These interactions have been measured for a long time through the simple graph, which is one of the most powerful tools from SNA. A simple graph is defined as a set of nodes with edges between them. There are no edges that connect a node to itself [2]. Nodes are oftencalled actors, which usually represent individuals or organizations. Edges are also known as connections or links, and they usually represent relationships between individuals such as friendships. The type of graph is defined by edge weight and directionality. If there exists a numerical value (i.e., a weight) attached to each edge, then the graph is weighted. If all edges are bidirectional, then the graph is called undirected. If edges have directionality, then the graph is called directed. In practical contexts, the most applied set of SNA tools is probably represented by the family of centrality measures. Centrality is an intrinsically relational concept. In fact, an actor needs to have relations to be important [2]. An actor is important if it is connected to other important nodes or to a large number of different nodes. An actor can also be central if his absence breaks the network into many isolated components. The position of the actor within a network can give him power in terms of the control over the information flowing through the network. The number of interactions of each actor is quantified by the most traditional centrality measure, which is the degree. Social networks are usually treated as monodimensional objects, but they can at least have three different dimensions: (i) a structural dimension, (ii) a compositional dimension and (iii) an affiliation dimension. The full complexity of a social structure can be understood only through these dimensions [1]. The first one corresponds to the simple graph made by actors and their relationships. The second one describes the actors and their personal information. The third one indicates the belonging of these actors to the same organization or family. Multiple relationships can also be considered according to an alternative conceptual approach as a set of layers (i.e., connected levels) [2; 3; 4]. Each simple graph with nodes and edges can be organized into multiple layers. Each layer can represent different types of edges, nodes, online social networks, communities, social contexts, and so on. In such a multilayer network, nodes in the different layers refer to a global set of actors, and edges can also connect nodes in the same layer (i.e., intralayer edges) or in different layers (i.e., interlayer edges). Analyzing multiple layers, we can obtain information that is not present in each single layer and would not be achieved considering layers independently of each other. Many relevant works on multilayer networks have been reviewed and discussed by [PERSON] et al. [5], who also presented a general framework for these kinds of networks [6; 7; 8]. This framework introduced the cumulative constraints on the general model [9], explaining the different types of networks: monoplex or single layer [6], multiplex [10; 11; 12], interdependent, and networks of networks.