2.4 Proofs of Chapter 2
3.1.1 Motivation and Research Questions
The examination of beneficial network positions is as old as social network analysis itself (see, e.g., Wasserman and Faust, 1994). But, until recently, the question has not been asked how incentive for central network positions affect the network structure. We introduce a model, in which agents strive for closeness and betweenness (centrality), while links are costly.
Three motivations justify such a model. First, it complements the theory of centrality that originally measured the effect of network positions on individual opportunities, but not the effect of individual behavior on network structure. Secondly, the centrality indices are based on network statistics that are relevant in many different applications – from ancient marriages (Padgett and Ansell, 1993) to R&D collaborations (Walker et al., 1997). In the same manner incentives for central positions are not restricted to single applications, but represent a general type of behavior in building networks. Third, there is empirical support for centrality being beneficial, e.g. Song et al. (2007) find that the centrality of a work unit has a positive impact on its creativity. But regardless of the empirical validity of centrality indices, there is justification to study network formation based on centrality incentives as long as there are researchers and businessmen who claim that central positions are desirable. In fact, this claim becomes more and more popular in the practice of business consulting, as argued in Section 1.1.
To cover the third aspect it is worthwhile to incorporate a centrality index that is well known. We chose the three indices based on Freeman (1979): degree centrality, closeness centrality and betweenness centrality. By this choice we cover the three most studied types of centrality measures according to the typology of Borgatti and Everett (2006).
This model is supposed to capture two types of linking incentives.21
Closeness stands for all incentives to access resources (information and support) by having many other agents in close reach. Variants of closeness were defined in Subsection 2.3.1. Betweenness stands for
21
We consider an environment where maintaining links is costly. Thus, degree centrality leading to costs (and benefits) of direct links is not considered as a goal in its own right.
the intermediation rents that stem from being a broker for others. Of course, there are other indices to measure the intermediation rents. Depending on the application it might be more appropriate to assume that only those players are getting intermediation rents who are essential for a pair of players.22
A second shortcoming is that betweenness does not account for the length of a path (as we will see in the definition). For instance, if a player is the only one between a pair of agents his betweenness rent is the same as if there are 5 agents in a row that are on the only shortest path. These two variations were studied by Goyal and Vega-Redondo (2007). In their utility function they do not only incorporate incentives for intermediation rents, but also incentives to avoid being brokered as well as incentives to be connected. In contrast, the model studied here shall allow us to decouple effects of incentives for connections from incentives for intermediation rents.23
This is not the first work to analyze network formation based on centrality incentives. Rogers (2006) models the formation of weighted graphs using an index of social influence (Bonacich index). The beneficial aspects in the models of Buskens and Van de Rijt (2008), Goyal and Vega-Redondo (2007) and Jackson and Wolinsky (1996) can also be interpreted as measures of centrality. What has not been done in the literature is (a) to contrast and (b) to combine the dynamics of “closeness-type” incentives to the dynamics of “betweenness-type” incentives. Considering the latter point, there is hardly any research on the interplay between different types of incentives to predict network formation processes, although it is likely that multiple incentives are simultaneously important.24
3.1.2 (A Centrality) Model
Consider a society (N, G, u). Closeness is formally introduced in Section 2.3: CLOSEi(g) = M
M−1 − P
j∈Ndij(g)
(M −1)(n−1). Its idea reaches back to the origins of social network analysis (Sabidussi,
1966). Dekker et al. (2003) argue that closeness increases accuracy of information. Song et al. (2007) provide empirical evidence for the importance of closeness for the knowledge processing of organizational units. Moreover, in the study of Powell et al. (1996) experienced firms are
22
Being essential here means that there is no path between the pair that the focal player cannot block. 23
Although this analysis is sometimes rather a thought experiment than an empirical model. Especially, for λ close to 1.
24
In fact, it seems difficult to find contexts where only one type of centrality is significant. For example, the Medici’s position in the marriage network was important for their trading abilities (see Padgett and Ansell, 1993). Here betweenness centrality is stressed, but closeness should not be ignored. At least for agents with low betweenness, it is important to be close to others.
Centrality 34 likely to occupy positions with high closeness.
Freeman (1979) clarifies that closeness measures one aspect of centrality, while it cannot sufficiently capture others. Some agents exhibit a mediating role between other agents, which can be beneficial for them. Burt (1992) emphasizes this idea by the term “tertius gaudens.” To measure the brokerage role of a certain agent he not only proposes some new measures, but also employs betweenness centrality (see Burt, 2002). Betweenness was introduced by Freeman (1979) and was shown to be beneficial in many studies thereafter (e.g. Song et al., 2007).
The betweenness of an agent i is proportional to the number of pairs j and k for whom i lies on the shortest path. If there are more than one shortest paths between j and k, the fraction of shortest paths going through i is used. Formally,
BET Wi(g) = 2 (n− 1)(n − 2) X j<k(j6=i,k6=i) τi jk(g) τjk(g) , (3.1)
where τjk(g) is the number of shortest paths between j and k, and τjki (g) indicates the number
of shortest paths between j and k that go through i; the fraction τjki (g)
τjk(g) is replaced by zero,
when τjk(g) = 0. The constant before the fraction normalizes betweenness to be between zero
(an agent is on no shortest path between two other agents) and one (an agent is on all shortest paths).
Besides closeness and betweenness, we also incorporate a player’s degree li(g), as in the
models (M1,M2,M3) before. Maintaining links is the source of costs in the network. On the other hand, degree can also be interpreted as a measure of centrality (see Freeman, 1979). We assume in our model that the costs of maintaining relationships exceed the benefits that are restricted to direct contacts such that the net benefit of degree is negative. Without this assumption every agent wants to be directly linked to every other agent, independently of the network structure.
Note that the network statistics closeness, betweenness and degree are interrelated. Trivially, an agent without any link, must have closeness and betweenness equal to zero. Such interde- pendencies already imply a trade-off between high closeness, high betweenness and low costs: while it is possible to reach two goals – having (a) high closeness and high betweenness (e.g. center of a star k has maximal closeness, maximal betweenness CLOSEk(g) = BET Wk(g) = 1,
but also maximal tie costs) or (b) having high closeness and low costs (e.g. a peripheral player in a star, his average distance is smaller than 2) or (c) having high betweenness and low costs
(e.g. the center of a line) – it is not possible to satisfy all three goals at once. Thus, agents have to weigh the three aspects against each other.
In brief, the centrality model is based on the following assumptions (where the numbers refer to the categories of assumptions introduced in Subsection 2.1.3):
A0 The relevant network statistics are closeness, betweenness and degree. A1 The evaluation function25
is increasing in closeness, increasing in betweenness and de- creasing in degree.
A4 Each player’s preferences are linear. A5 All players have homogeneous preferences.
Given these assumptions, we can represent the preferences of any agent i ∈ N by a utility function ui : G→ R with
ui(g) = (1− λ)CLOSEi(g) + λBET Wi(g)− cli(g).
The parameter c > 0 stands for the costs of one link (marginal costs). The parameter λ∈ [0, 1] stands for the weight of betweenness versus closeness in the benefits.26
We will analyze the model for all points (λ, c) in the parameter space [0, 1]×R+, as they represent different contexts.