# Fuzzy Closeness Centrality

#### Definition

Closeness centrality measures how close a node is to all
other nodes. This definition closely relates to fuzzy concepts:
1) the concept of close is a fuzzy linguistic variable, and the
concept of all other is a fuzzy universal quantifier.

A generalized fuzzy centrality measure is defined as First, there is a path closeness function for a given node v:μ

A generalized fuzzy centrality measure is defined as First, there is a path closeness function for a given node v:μ

_{close}: (v, u) → [0; 1] which maps possible information flow between a node-pair into a value in the unit interval. Secondly, an aggregation function λ : [0, 1]^{n}→ [0, 1], which aggregates these information flows into a single value. Here λ is chosen from the set of andness-directed operators. Following, the AIWA operator can be plugged in for λ, to get the aggregate measure of closeness to all other nodes, as seen in: When ρ < 0:5 the measure increases the ranking of nodes which have many close nodes, and for ρ > 0:5 ranking is increased for nodes close to all nodes. For ρ = 0:5 the ranking is equal to that of the classic closeness centrality measure.#### References

- S. A. Davidsen and M. Padmavathamma. A fuzzy closeness-centrality with andness-direction to control degree of closeness. In Proc. 1st Int. Conf. on Networks and Soft Computing, pp. 225-230, 2014.