Tore Opsahl

Thesis: 4 Evolution of Networks

Networks evolve as a result of the joining and leaving of nodes, and the creating, reinforcing, weakening, and severing of ties. For example, the online social networks created from the virtual community grew when people joined the community and shrunk when people left it. If ties are defined in terms of online messages, they are formed when a person sends a message to another for the first time, and reinforced if multiple messages are subsequently sent (Panzarasa et al., 2009). Since, online interaction datasets generally do not contain information about the severing or weakening of ties, it is often assumed that social relationships are severed if no is sent during a certain amount of time (Kossinets and Watts, 2006). Similarly, a weakening is assumed to occur if the rate of messages exchanged decreases.

Up until recently there have been only a few network datasets where the exact evolution of the network could be mapped (e.g. Hall et al., 2001; Holme et al., 2004; Kossinets and Watts, 2006; Onnela et al., 2007; Opsahl and Panzarasa, 2008; Panzarasa et al., 2009). Thus, a substantial part of the empirical work on social networks has been conducted on cross-sectional or static networks (e.g. Bernard et al., 1988; Fararo and Sunshine, 1964; Foster et al., 1963; Gouldner, 1960; Katz and Proctor, 1959; Lazarsfeld and Merton, 1954; McPherson et al., 2001). In these studies, a host of measures have been proposed to detect features of the network structure. Based on the existence of a particular feature, it has been speculated that certain mechanisms have underpinned tie creation. For example, the clustering coefficient (for a review, see Chapter 2) measures the extent to which triangles occur in a network. The coefficient for an observed network can be compared to the one found in a corresponding random network (Erdos and Renyi, 1959; Newman, 2003; Panzarasa et al., 2009; Solomonoff and Rapoport, 1951). If it is higher than the randomly 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 (Heider, 1946; Holland and Leinhardt, 1971).

These measures fail to assess multiple effects that may influence the decision-making process of the nodes in a network. By using a conditional logistic regression framework, Powell et al. (2005) studied the effects of different mechanisms on tie generation among organisations. However, the framework that they developed was not general as their network was not a prototypical social network. In particular, they combined a one-mode network (i.e., one set of nodes and ties among those nodes) with a two-mode network (i.e., two sets of nodes with ties only between nodes in the different sets). Moreover, the network was only recorded at yearly intervals and, therefore, they could not account for the dependence among ties occurring in the same year. In addition, the network was undirected. This implies that the decision to form a tie does not rest with one node, but with both the nodes which are connected by the tie. In this Chapter, we develop a general and flexible framework for one-mode networks. We focus on the individual choices that a single node makes when creating a tie, and thereby we concentrate on directed networks. This enables us to assess mechanisms that cannot be tested in undirected networks, such as reciprocity.

The rest of this Chapter is organised as follows. First, we highlight a number of mechanisms thought to guide tie generation in social networks. In Section 4.2 we describe methods used to study combinations of these mechanisms in binary cross-sectional networks. We then turn our attention towards longitudinal network data, where the exact sequence of ties is known, and propose to use an established regression framework to examine the decisions made by individual nodes at the local level to initiate a social tie (binary analysis). We empirically test the proposed method on the online social network outlined in Chapter 1. We are able to use this network as each message (or tie) was recorded with the exact time at which it was created. This allows us to extract the exact sequence in which nodes and ties were added to the network. In Section 4.4 we extend our analysis to weighted networks and conduct a second assessment of the online social network by also taking into consideration messages used to reinforce existing social ties. The following section tests the sensitivity to a methodological and computational constraint. Finally, we highlight the contribution to the literature and offer a critical assessment of the main results.