Understanding network formation and evolution is crucial for a wide variety of network tasks. Whether used to support theories or for the practical tasks of planning for future size and structure, understanding the possible implications of interventions or extrapolating from a smaller data set to a larger one, they all require the ability to create and evolve networks according to a specified model. Any network model is typically evaluated on its ability to reproduce factors of interest for the network in question. This means that a first necessary step to be done to select the specific model we need is to answer the following rather difficult questions:What kinds of characteristics are really important for us? What factors should be incorporated into the model we decide to use? Obviously, these relevant factors change according to both the domain we are in and our final goals.

After a long tradition of social network research, we now have a general understanding of what elements should be taken into account when we think of a model able to create a network showing some level of similarity with the networks we observe in the real world. At minimum these factors include degree(s), the degree-degree correlations (assortativity and disassortativity), and the clustering coefficient. The present chapter first provides a quick overview of how these elements have been implemented in the currently available models for network formation, and then it presents the state of the research about the extension to multilayer networks.

General Properties for Social Network Formation

Degree Distribution

The first consideration for the formation of social networks is the degree distribution, or how likely or common it is for people to have a given number of connections. As we may expect, it is far more likely for someone to have a small number of connections than it is to have a large number of connections. Besides this commonsense expectation, many years of analysis of social networks have revealed that unlike, for example, height, there is no “typical” number of connections: the degree distribution of social networks is almost invariably fat-tailed (Amaral et al., 2000; Adamic et al., 2001; Clauset et al., 2009).