📂Graph Theory

Proximity Centrality in Network Theory

Definition ¹

Let’s call the geodesic distance of a network $\left( V, E \right)$ as $d$. The following defined $C_{C} : V \to \mathbb{R}$ is referred to as the Closeness Centrality of node $v \in V$. $$C_{C} := {{ 1 } \over { \sum_{u \ne v} d \left( u, v \right) }}$$

Description

Intuitive Meaning

Closeness Centrality serves as a measure of ‘how easily can one node reach another,’ and its intuition can be understood from the derivation of the formula. Assuming that the number of nodes in $\left( V, E \right)$ is $n$, the average distance from $v$ to the other nodes is $${{ 1 } \over { n }} \sum_{u \ne v} d \left( u, v \right)$$ However, as this value increases, it paradoxically means that $v$ has reduced accessibility to other nodes. Therefore, by simply taking the reciprocal $${{ n } \over { \sum_{u \ne v} d \left( u, v \right) }}$$ to make it a ’the higher, the better’ metric, and since the network size $n$ is a constant meaning nothing in this context, we eliminate it to arrive at the formula for closeness centrality as follows. $$C_{C} = {{ 1 } \over { \sum_{u \ne v} d \left( u, v \right) }}$$

See Also

Various Centralities of Network

• Degree Centrality
• Betweenness Centrality
  • Stress Centrality
• Closeness Centrality
• Eigenvector Centrality