# Disentangling Director Attributes: Human Capital versus Social Capital of Directors
## 2 Literature Review and Human and Social Capital Determinants
### Social Capital
We set the distance between disconnected directors to 0. Effectively, this overcomes the issue of excluding directors from the analysis who are not connected at all or are connected to smaller satellite networks, but not to the main network ([PERSON] et al., 2010). A higher closeness score represents a director with closer connections that enable quicker and more readily available information and resource exchange. \(CLO\) is normalized by dividing by \((n-1)\) representing the percentage of the maximum \(CLO\) possible for a given director \(i\). The third measure, betweenness (denoted hereafter as \"\(BET\)\") ([PERSON], 1978), measures how well-situated the director is for connecting other directors to each other and the ability to potentially control the exchange of information and resources ([PERSON], 2005; [PERSON], 1978). [PERSON] (1978) constructs the \(BET\) measure to represent the probability that director \(i\) is positioned on a randomly selected shortest path that links two directors (\(l_{i}\),\(j\)). By doing so, \(BET\) considers the likelihood of information being circumvented through other channels to capture the probability of director \(i\) successfully controlling the information flow, i.e.,  
\[B^{B}_{(h,i,j)t}=\frac{g(h,i,j)}{g(h,j)}\, \tag{3}\]  
where \(g(h,j)\) is the maximum number of communication paths another director could be in a position to control. Therefore, the information passing between directors (\(l_{i}\),\(j\)) can be completely controlled by director \(i\) when there are no other directors between directors (\(l_{i}\),\(j\)), such that \(B^{B}_{(h,i,j)t}=1\). To measure the overall \(BET\) of director \(i\), we follow [PERSON] (1978) and take the sum of the proportions of all the shortest paths linking two directors which pass through director \(i\):  
\[C^{B}_{it}=\sum_{h<}^{n-1}\sum_{j}^{n-1}B^{B}_{(h,i,j)t},\quad\text{ where }h\ \
eq\ i\ \
eq\ j, \tag{4}\]  
where \(n\) is the number of directors in the network and \(B^{B}_{(h,i,j)}\) is defined as per Equation (3). We normalize \(BET\) by expressing it as the proportion of its maximum value possible in year \(t\). The maximum value for \(C^{B}_{it}\) is essentially the most central point a director can sit, that being \((t2-3n+2)/2\)([PERSON], 1978). The final measure is the relative \(BET\) centrality of director \(i\) in year \(t\) which is:  
\[C^{\prime\,B}_{it}=\frac{2\big{(}C^{B}_{it}\big{)}}{n^{2}-3n+2}\, \tag{5}\]  
where \(C^{B}_{it}\) is defined by Equation (4) and \(n\) represents the number of directors in the network. The fourth measure is eigenvector (denoted hereafter as \"\(EIG\)\") ([PERSON], 1972), which expands on the degree measure and is typically interpreted as capturing the power and prestige of a director's connections. Specifically, \(EIG\) combines a director's \(DEG\) score with their direct connections' \(DEG\) scores. \(EIG\) is defined as the sum of director \(i\)'s first-degree connections to all other directors (\(\delta(i,j)\)) in the network, weighted by the \(EIG\) of the directors to which director \(i\) is connected to, i.e.,  
\[C^{E}_{it}=\frac{1}{\lambda}\sum_{j=1}^{n}\delta(i,j)C^{E}_{jt},\quad j\
eq i\, \tag{6}\]  
where \(C^{E}_{it}\) is the \(EIG\) score for a particular director \(i\), \(\delta(i,j)\) is defined in Equation (1), and \(\lambda\) is a constant, defined as the maximum possible eigenvector for a given network in year \(t\). Connections to a highly connected director will increase a director's \(EIG\) score more than connections to less connected directors. A high \(EIG\) director has faster and increased access to information and resources which should increase their value on a board. Finally, we employ principal components analysis to create an aggregate connectivity score, (hereafter referred to as \"\(AGG\)\"). This method has commonly been used to account for the multidimensionality of social capital and examines several indicators simultaneously ([PERSON] et al., 2013; [PERSON] et al., 2013; [PERSON] et al., 2014).