2.3.3.1 Density
‘Density’ is the total number of edges in a network divided by the total number of possible ties. It is a measure of the general linkage among nodes or the extent to which all possible relations are actually present (Scott, 2000). Density is also a proxy for inclusiveness. If board appointments favour one gender over another this should be reflected in that gender having more dense networks than the other. If affirmative action takes place, changes in density may be an indicator of increasing network openness or alternatively resistance to change. It is an easily calculated measure and can be reliably estimated from a sample and is one of the commonest measures in network analysis (Scott, 2000). Kogut and Walker (2001) reported that the finding of very low densities in company-company networks has led some authors like Useem (1984) to suggest that there is no basis for the belief in co-ordinated action in company networks. In other words, there is no evidence for the operation of old boy networks. There appear to be no studies measuring density in director-director networks. To establish base line measures density measures are included in this thesis.
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2.3.8.2 Reach centrality and Freeman’s betweenness centrality
Being embedded in a relational network imposes constraints on a director, but it also offers opportunities to the director by virtue of being connected to others. Network position determines which director is more favourably placed and therefore able to take better advantage of the available opportunities such as additional board appointments. Having a favoured position means that a director may extract better bargains in exchanges, have greater influence, and that the director will be a focus for deference and attention from those in less favoured positions (Hanneman & Riddle, 2005). Directors in ‘old boy’ networks or have links to each other beyond the shared board links, who are linked to each other through shared boards, are in a position to bring opportunities to the notice of their peers and to consolidate positions of power and influence as a result. Not only does the individual director benefit, but his peers do too.
Directors who can reach or link to many others will have a more central role in the network. This is known as ‘reach centrality’ and is a measure of the number of nodes each node can reach in k or less steps. Borgatti (2005) describes this measure as a natural metric for assessing each node when searching for key individuals who are well positioned to reach many people in a small number of steps. This measure may be used to identify influential connector directors and as an objective way of determining how influential female directors rank to other directors. As this measure ranks each director in the network in terms of reach centrality it is a proxy for inclusiveness of the network. This measure has not been used before in director network research and there is no information relating to the reach of male or female director networks.
In contrast, ‘betweenness centrality’ is a fundamental measurement concept in social network analysis (White and Borgatti, 1994; Newman, 2005a). It is a measure commonly used to measure how influential a given person is in a society (Goh et al., 2002). Freeman’s betweenness measure sums the proportion of shortest paths from one node to another that pass through a given node (Freeman 1979). Thus, a node with high betweenness is responsible for connecting many pairs of nodes via the best
88 path, but the assumption is that whatever flows through the network moves only along the shortest possible paths (Borgatti, 2005).
Goh et al. (2002) have demonstrated that not only does the degree probability distribution of a complex network show a power law, but so does the betweenness probability distribution. Betweenness is known to be strongly correlated with the node degree in most networks (Goh et al., 2003; Newman, 2005). The director equivalent is a connector director with multiple board seats and high degree who is therefore more likely to be on multiple paths between other pairs of directors.
If the two are strongly correlated, Newman (2005) asks, why go to the effort of calculating betweenness, when degree is almost the same and much easier to calculate? The answer is that betweenness is useful in identifying the small number of nodes for whom betweenness and degree are very different. Secondly, betweenness is more sensitive than degree in capturing the underlying structure of the network in a summary measure. Thirdly, networks that are robust to attack can be distinguished with this measure, as it measures the duplication of paths between nodes, offering an alternative if one is broken.
Only two studies measure director centrality, Davis et al. (2003) who used the Bonacich measure of centrality and Stablein et al. (2005), whose measure was not stated. Neither study looked at the betweenness distribution. Usefully, Davis et al. (2003) identified the top ten linchpin directors of the US Fortune 1000 in 1982, 1990 and 1999. These top ten directors were overwhelmingly male, although in 1990 one woman appears and is replaced by another woman in 1999. Inspection of the lists indicates that only one director was present in all three lists and the progress of that male director, Vernon Jordan Junior, is measurable up the ranking in that time period. Of the 2004 top ten directors in New Zealand listed companies, two were women and all directors were of European descent.
From this suite of measures, an overall view of director networks can be established. From the baseline of the mixed gender network, the female director network can be
89 extracted, mapped and compared at the global and national levels. Only then can the research set of questions be answered. Before this can be done, some fundamental questions have to be asked and the literature examined for past answers. This is the question of the origins of networks, where do they come from and how do they grow and change over time?
Only when the mechanisms involved are understood can change be successfully induced in a social network. The attempts to change the gender mix of director networks have not been successful. Some countries are resorting to quotas to solve the problem and finding unintended consequences. By reviewing past research some light may be thrown on how networks resist change and maintain their structure despite changing inputs and outputs. This will give fresh insight and more confidence when devising new interventions.