Node Metrics I - Centrality

### 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 —

### Types of centrality

#### Degree Centrality

Version 1: centrality equals how many other vertices a vertex is connected to.

(1)
$$C_D(v) = deg(v)$$

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.

Bibliography
1. Davis, A. et al. (1941). Deep South. Chicago: University of Chicago Press
2. 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)