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 $$c_w = \sum _v a_{wv}(p_{wv} + \sum _{u} p_{wu} p_{uv})^2$$
1. 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