Closeness

Closeness is a radial measure of centrality that favors actors who are connected with many others via short paths. Intuitively, if the graph represents a transportation network, then a node with high closeness would make a good location for a warehouse since the average distance to all other locations (i.e., all other nodes in the graph) is relatively short. In information-spreading networks, a node with high closeness centrality would be a good choice to start a rumor since many others can be reached with relatively few intermediates.

Definition (simple case)

On directed, unweighted graphs $\displaystyle {G=(V,E)}$ that are strongly connected, the closeness centrality $\displaystyle {c_{C}(v)}$ 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 $\displaystyle {d_{G}(v,t)}$ denotes the length of a shortest directed path from $v$ to $t$. The definition for connected undirected graphs is identical with $\displaystyle {d_{G}(v,t)}$ being defined as the length of a shortest path.

Special cases

Edge weights and distances

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.