The greatest length or breadth of a full grown inhabitant of Flatland may be estimated at about eleven of your inches. Twelve inches may be regarded as a maximum.
– The Square
In this chapter we present a set of measures that can be used to compute quantitative descriptions of multilayer networks. Some of these measures have counterparts in classical SNA, whereas others focus on the different layers or on their interplay and do not have specific equivalents among traditional measures.
We organize the chapter in to two main sections. Actor measures are used to describe the characteristics of actors with respect to their connections on the different layers. Some are extended versions of existing SNA measures, for example, degree, betweenness, and clustering coefficient, whereas others are specific to multilayer networks and can be used to quantify the relevance of one or more layers for an actor. Layer measures, alternatively, focus on the relationships between layers, for example, their similarity.
Along with the description of the measures, we also show their application to our running example, represented in Figure 1.2. In addition, we apply a selection of these measures to a real multilayer network to better illustrate their goals and consequences. The real multilayer network we are going to use for our analysis is the AUCS network, described in Section 2.3 and containing five types of relationships among the employees of a university department: Facebook friendship, having lunch together, being coauthors of published research papers, collaborating at work, and spending leisure time together.
The aim of this chapter is not only to introduce metrics to describemultilayer social networks in the same way we used to describe single-layer networks but also to stress how and why multilayer networks present a unique set of features and problems requiring specific approaches to be addressed.
Four Main Approaches
When we study multilayer networks, the additional complexity introduced by relations existing between the layers can be handled in different ways, depending on the interpretation of the network data and on the goals of the analysis. On a general level, and with the due level of abstraction, we can identify four different approaches.