Structural hole centrality: evaluating social capital through strategic network formation

Computational Social Networks

Table 3 Node centrality measures

Measure	Symbol	Formula
Shortest-path betweenness centrality	bc	\(c_w = \sum _{u\ne v \ne w}\sum _{\ell } {\mathbb {1}}(m_{\ell uv}>0)f_{\ell uwv}\)
Shortest-path closeness centrality	cc	\(c_w = \frac{n-1}{\sum _v \ell _{wv}}\)
Eigenvector centrality	ec	\(c_w = \frac{1}{\lambda _1} \sum _v a_{wv}e_v\)
Katz centrality	kc	\(c_w = \sum _v \left( ( \text {I} - \delta \text {A}^T)^{\tiny {-1}}-\text {I}\right) _{wv}\)
Degree centrality	dc	\(c_w = \sum _v a_{wv}\)
Harmonic centrality	hc	\(c_w = \frac{1}{n-1}\sum _v \frac{1}{\ell _{wv}}\)
Constraint centrality	conc	\(c_w = \sum _v a_{wv}(p_{wv} + \sum _{u} p_{wu} p_{uv})^2\)

Note that \(\ell _{wv}\) is shortest path distance between w and v, \(\delta \) is the benefit of a direct link, such that a connection along a length \(\ell \) path gets benefit \(\delta ^\ell \), \(e_v\) is the eigenvector of \(\text {A}\) corresponding to the dominant eigenvalue \(\lambda _1\); \(m_{\ell uv}\), \(p_{wv}\) and \(f_{\ell u w v}\) are as defined in “A bonding and bridging strategic game” section