Chapter 2. Theoretical Background on Network Formation Models
2.2. Strategic Network Formation Models
In the context of strategic network formation models, researchers try to identify the effects of individuals’ strategic interactions on the upcoming network structures. Three major categories of topics in this area are network games (Galeotti, Goyal et al. 2010),
28 public good provision (Rittenberg 2009), and bargaining and power in networks (Jackson 2008). Each of these topics can be represented with different theoretical models.
Game theoretic models have been proposed to analyze the strategic interactions of individuals in a network. From this point of view, each individual in a system obtains a utility due to its interaction with others in the network. The utility is defined through a utility function (utility function) and it describes the level of an individual's utility from interactions with others. For example, the utility function might consider the number of connections or the distance between the source and the destination. The utility can also be used to measure the social welfare (i.e., a society’s collective utility), which represents a level of well-being of an entire society.
A comprehensive introduction to social and economic networks has been offered by Jackson (Jackson 2008). He presented different strategic link establishment models, in which choices of individuals have an impact on the topological features of the network (Jackson and Wolinsky 1996). Two examples are the symmetric connection model and the co-author model [8].
Symmetric connection model is a model, in which individuals directly communicate with their direct contacts and the obtained utility of an individual through interactions with others are proportional to his or her distance to the target players. As this distance increases, the obtained utility decreases. The symmetric connection model always produces positive externality for the individuals within the network. Variations of symmetric connection models are also presented in the literature. They are degree- distance-based versions or have negative externalities (Morrill 2011, Möhlmeier, Rusinowska et al. 2013).
29 Co-author model focuses on the collaboration among the individuals. All the obtained utilities for the individuals in the co-author model are due to the collaboration with direct contacts, and there is no utility for communicating with indirect contacts. In the co-author model (Jackson and Wolinsky 1996), the utility function is defined in a way to encourage the cooperation between two individuals. That means, two isolated individuals do not receive any utility in case of no cooperation but, as soon as they establish a link with each other, the co-author utility function assigns a positive utility to both parties. However, the co-author model negatively changes the pay-off of other indirect neighboring individuals within the network. For example, if a new individual joins the network and establishes a link with an existing node, the receivers of the link gain a credit, while the neighbors of the existing node face a lower utility due to the negative externality caused by the co-author utility function. Therefore, contrary to the symmetric connection model, individuals do not receive positive externality from establishment of the indirect links in the network.
Strategic complements and strategic substitutes are also two specific classes of network games, in which every player wishes to adjust her action in response to the activity of other players (Galeotti, Goyal et al. 2010). The decisions of two or more players are called strategic complements, if they mutually reinforce one another, and they are called strategic substitutes, if they mutually offset one another.
A strategic complement model shows the tendency of the network participants to follow actions or beliefs of others. It is based on the idea that a given player’s relative utility of taking an action increases in the context of neighbors who took actions. In fact, the decision of some people in a network can have a positive effect on the choices of other members of the network.
30 Under a strategic substitute model, network participants tend to take the opposite choices of others. They do not adopt their decisions. Lamberson presented a model of friendship-based games played on a social network (Lamberson 2011). It allows games of strategic complements and strategic substitutes. Finally, a comprehensive tutorial on strategic substitute models has been introduced by Michael et al. (Michael and Battiston 2009).
In addition to these game theoretic models, some studies linked individuals' incentives for establishing links with the target node’s structural importance. Therefore, in such cases, the utility function depends on the centrality measure. Konig et al. (König, Tessone et al. 2009) presented a link establishment model, in which links are formed on the basis of agents’ centrality. Other studies investigated centrality measures even further (Freeman 1978, Borgatti 2005, Borgatti and Everett 2006, Buechel 2008, Newman 2008, Bei, Chen et al. 2009, Gallo 2012), showing some instability of the network structure.
Buechel (Buechel 2009) proposed a model, in which individuals strive for two types of benefits measured by closeness and betweenness centrality.
Compared to these studies, our proposed interaction model in chapter 5 is based on complex adaptive system theory and aims to captures the changes in the utility gain of individuals in an evolving network. Changes in the network structure (network evolution) in our interaction model are consequences of four important factors: which are mainly: (1) initial underlying network structures; (2) process of network growth; (3) adoption of strategic interactions of individuals; and (4) network visibility. Furthermore, strategic interactions of the individuals are modeled as a utility maximization behavior in response to what others have performed within the network.
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