What is centrality and why do we care about it?
It seems simple enough — when we look at a network, some vertices seem more "central" than others — but turns out we can understand different things by "central." So let's start instead with why we care about it.
Consider the question of interpersonal influence — in a group of people, which individuals' attitudes are likely to have the largest effect on other individuals' attitudes? Or consider cities in the global cultural system —
Distinction: Centrality as a vertex metric vs. centralization as a network metric.
Types of centrality
Degree Centrality
Version 1: centrality equals how many other vertices a vertex is connected to.
(1)But 1: the raw number will vary depending on the size of the network and so is not comparable from one network to another
Fix 1: divide $C_D$ by the maximum possible degree (n-1) to get fraction of vertices in a graph to which the focal vertex is connected.
For a directed graph we can separately consider in-degree centrality and out-degree centrality. When the relationship is something like friendship or collaboration, in-degree can be interpreted as popularity or prestige, and out-degree as generosity or gregariousness.
In these data were collected by Davis1 et al in the 1930s. They represent observed attendance at 14 social events by 18 Southern women.
; Breiger R. (1974). “The duality of persons and groups.” Social Forces, 53, 181-190.
http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Affilation%20Networks.pdf
Tore Opsahl Defining Two-mode Networks
Borgatti, Stephen P.; Everett, Martin G. (2005). "A Graph-Theoretic Perspective on Centrality". Social Networks (Elsevier) 28: 466–484 (Science Direct)
Stephen P. Borgatti (2005). "Centrality and Network Flow". Social Networks (Elsevier) 27: 55–71. (Science Direct)