Closeness

Closeness is a radial measure of centrality that favors actors who are connected with many others via short paths.

The raw closeness score of a node $v$ is defined as the reciprocal of its total distance to those it can reach via directed paths,

$c_{C}(v)={\frac {\text{number of reachable nodes}}{\sum \limits _{{\text{reachable nodes }}t}d_{G}(v,t)}}$

where $d_{G}(v,t)$ denotes the length of a shortest directed path from $v$ to $t$. If a link strength has been selected, the length of an $(v,t)$-path is the sum of the corresponding attribute values of all links in the path.

Definition (simple case)

On directed, unweighted graphs $G=(V,E)$ that are strongly connected, the closeness centrality $c_{C}(v)$ of a node $v\in V$ is defined as

$c_{C}(v)={\frac {|V|-1}{\sum \limits _{t\in V\setminus v}d_{G}(v,t)}}$,

where $d_{G}(v,t)$ denotes the length of a shortest directed path from $v$ to $t$.