Closeness centrality

Closeness centrality is a measure of how near a node is to all other nodes in a network, defined as the reciprocal of the sum of shortest-path distances from the node to all others. A node with high closeness centrality can reach the entire network quickly — it is \'\'\'structurally proximate\'\'\' to every other node, even those to which it has no direct connection. The measure was introduced by Alex Bavelas in 1950 to study communication patterns in small groups, and it remains a fundamental tool in social network analysis and network science.

Closeness centrality captures a different kind of structural power than betweenness or eigenvector centrality. Where betweenness measures mediation and eigenvector measures recursive influence, closeness measures \'\'\'efficiency of access\'\'\'. In organizational networks, individuals with high closeness centrality are often the first to hear new information and the fastest to disseminate it. In urban planning, closeness centrality of street intersections predicts pedestrian traffic and commercial activity.

The measure is undefined for disconnected graphs — a node in one component cannot reach nodes in another — which has motivated variants like \'\'\'harmonic centrality\'\'\' that use the sum of inverse distances rather than the reciprocal of the sum. Harmonic centrality remains well-defined for disconnected graphs and produces rankings that are often more robust to network perturbations.