The network components

Complex network theory suggests that new networks start off with several small components joining up as links or edges are added either randomly or by following specific rules, such as preferential attachment, to form a large or giant connected component (Barabási et al., 2002).

When each node has an average of one link (Barabási, 2003), the disparate segments unite to form this dominating component. This may happen very suddenly and resemble a ‘tipping point’ (Gladwell, 2000) or a moment of critical mass (Ball, 2004) or ‘a phase transition’ in physics (Newman, 2003b; Watts, 1999a). Once the large

107 component has formed, the remaining components are not random in size, but appear to follow a power law pattern.

To illustrate this, Table 2 presents the component breakdown of the 2004 New Zealand stock exchange data for the combined male and female directors’ network (Stablein et al., 2005). There is one large component of 639 directors (66.2% of the nodes) and 54 smaller components. This data was then plotted using Microsoft Excel 2007 and a trend line fitted by regression. This is shown with its associated equation and R² value, in Figure 12.

The trend line with the R² value closest to 1.00 has the best fit to the data. Here, this was a power law trend line, a regression trend line that is used to compare measurements that increase or decrease at a specific rate (Dodge & Stinson, 2007). For the 2004 NZX network components, excluding the largest component, the resulting power law trend line equation was y = 22.68x-0.46 and R² = 0.927. Concerns with the reliability of regression analysis to calculate power law exponents accurately (Goldstein, Morris & Yen, 2004; Newman, 2003c) requires that the assigned power law be confirmed with other methodologies.

As an alternative methodology avoiding these problems, this data was converted to a log-log plot and a linear trend line fitted with its associated equation and R² values. Inspection of Figure 13 (a graph of the log-log scores) shows the left slanting straight line characteristic of a power law. The resulting best fitting linear equation is y = -0.527x + 1.449 with R² = 0.932. This methodology is developed further in this thesis and used to examine the other director networks for the size of the remaining components in the network and to determine whether they follow the same pattern of a few large components and many smaller components that decrease in size in a specific ratio to each other.

108

Table 2.

Stablein et al.’s (2005) 2004 NZX director-director network.

Figure 12.

Stablein et al. (2005) 2004 NZX male & female directors’ network components with power law trend line (largest component excluded).

Mixed Genders Ranking from Largest to Smallest Component Size by # of Directors Components # of 1 639 1 2 21 1 3 15 1 4 14 1 5 13 1 6 10 1 7 8 4 8 7 6 9 6 11 10 5 10 11 4 9 12 3 9 Total

109

Figure 13.

Stablein et al. (2005) 2004 NZX male & female directors’ network components log-log plot with linear trend line and equation (largest component excluded).

A explanation for this pattern can be found in the recent work done on microeconomic models of theories of the firm (Ball, 2004), where models of firm growth are derived from the aggregation of many worker’s or agent’s actions following their own agenda to best suit their own advantage, resulting in firms whose sizes follow power law distributions. Component size may be linked through board size and numbers of connector directors on a board to company size. Axtell (2001), using the entire population of tax-paying firms in the United States, showed that a Zipf distribution (an alternative name for the Pareto or power law distribution) characterizes firm sizes with the probability that a firm is larger than size s being inversely proportional to s. Axtell (2001) finds that:

Firm sizes in industrial countries are highly skew, such that small numbers of large firms coexist alongside larger numbers of smaller firms. Such skewness has been robust over time, being insensitive to changes in political and regulatory environments, immune to waves of mergers and acquisitions, and unaffected by surges of new firm entry and bankruptcies. It has even survived large-scale demographic transitions within work forces (e.g., women entering the labor market in the United States) and widespread technological change (Axtell, 2001, p.1818).

110 Rules of assortativity and preferential attachment would suggest that connector directors prefer boards not only with directors of the same degree, but of the same size company, giving rise to a similar power law in the smaller components in the director network. This contention is not tested in this thesis but noted here as a possible reason for the power law relationship in the components of director networks.

The power law driving the component distribution is examined further in this thesis and the location of women directors in the network components is also considered. It is not known if women directors are to be found clustered in the large component (if one is present), or are to be found in the peripheral and smaller components. If business women are seen as marginalised in an economy and peripheral on boards of directors, who appoint them with reluctance, then it can be hypothesized that the few women directors in the network would be found on the periphery of the director network in the smaller components.

However, Glass Network theory makes the opposite prediction that women directors are more likely to be found in the largest or giant component. Larger companies are likely to be clustered in the large or giant component as they have larger boards and directors of higher degree. Women directors tend to be found in larger companies with larger boards, often as the token woman (Nguyen & Faff, 2007; Farrell & Hirsch, 2005; Carter et al., 2003). In addition, women directors may prefer larger established companies, with connector women directors cherry picking the more prestigious and safer boards appointments.

3.2.2 Assortativity, network degree and betweenness in glass networks