## Thesis: 1.2 Longitudinal networks

Another limitation of traditional methods used to study networks is related to the way in which networks are collected. As described above, datasets are often collected through surveys or interviews at a single point in time. Consequently, most network measures have been developed for the purpose of analysing cross-sectional networks. However, networks evolve over time by the addition and removal of nodes, and the forming, strengthening, weakening, and ultimately, the severing of ties. For example, one network that is currently receiving a great deal of attention in the literature is the network of commercial airports (the nodes) that are tied together by scheduled flights (Amaral et al., 2000; Barrat et al., 2004; Guimera et al., 2005; Opsahl et al., 2008). This network grows when new airports open and shrinks when old ones close down. Ties are created when new routes are started and reinforced if the capacity of an existing route is increased. Weakening of a tie occurs when airlines cut capacity, which ultimately results in the severing of the tie if all airlines terminate flights on the route. Yet, the evolution of networks is typically not recorded. This implies that the dependency structure among ties is unknown.

Nevertheless, a number of measures aimed at examining the underpinning principles of tie generation have been developed. The first generation of methods aimed to detect a single structural feature in cross-sectional networks. For example, the clustering coefficient (for a review, see Chapter 2) identifies the extent to which triangles occur in a network. The coefficient obtained for an observed network can be compared to the expected value on a corresponding random network (Erdoos and Renyi, 1959; Newman, 2003; Panzarasa et al., 2009; Solomonoff and Rapoport, 1951). If it is higher than the expected one, then scholars have often concluded that there is a mechanism that increases the likelihood of forming a tie between two nodes if they have ties to the same other node (triadic closure). However, this approach could be biased as other mechanisms could contribute the generation of triangles. For example, it could be the case that similar people were more likely to form ties with each other than randomly expected(homophily; Lazarsfeld and Merton, 1954; McPherson et al., 2001). If this was the case, a set of similar nodes is likely to be create a tightly knit group that would increase the level of clustering in the network, without any triadic closure effect. In fact, it would be difficult to assess whether, and the extent to which, the tightly knit group was formed due to, for example, triadic closure or homophily mechanisms.

This issue has motivated the development of a second generation of methods ($p^*$ or Exponential Random Graph models). These methods allow for a multivariate analysis of mechanisms that might lead to tie generation (Holland and Leinhardt, 1981; Robins and Morris, 2007; Snijders, 2001; Wasserman and Pattison, 1996). They model the entire network and try to study how, and the extent to which, different mechanisms can be combined to produce the observed network. These mechanisms of tie generation include triadic closure, homophily, preferential attachment (Barabasi and Albert, 1999), and reciprocity (Gouldner, 1960; Plickert et al., 2007). However, these models suffer from criticism as they are difficult to interpret and extend, and have a range of unknown parameters (Hunter et al., 2008). Although there have been attempts to solve these issues (Snijders et al., 2006), a simpler, more flexible, methodologically sound framework is needed to improve the analysis of the underpinning principles of tie generation. This will ultimately improve our knowledge of the organisation and function of networks.