Chapter 6: center and periphery

Degree centrality

Intuition: Let's start with the local view. Those who are well-connected get information faster, and send it out faster. They're more central. So, let's measure centrality simply based on number of neighbors.
(This is a naive intuition, since I might have lots of neighbors who are not well-connected, but it's a starting point!)
In a simple network of N vertices, number of neighbors = degree.
Star formation provides maximum centrality
Centralization:
- first compute degree variation as follows:
  - find the most central vertex, and use its centrality as a comparison point, C
  - compute the difference between C and ever other vertex's centrality
  - add these differences
- then divide by the maximum degree variation on N vertices (Note: this will be the variation on a star network, with one vertex at the center, and (N-1) around the edge; the former has degree (N-1) and the latter all have degree 1. So the max is (N-1)*[(N-1)-1] = (N-1)*(N-2). Thus for N=5, the maximum variation is 12.

Intuition: Let's take a more global view. Instead of looking at how well-connected a vertex is at one hop (neighbors), let's look at how well-connected it is to the entire network. So we want to check on how long the paths are connecting it to everyone else. Those with short paths are closer to the network as a whole - they're more central.
Geodesic: shortest path between 2 vertices
Distance from vertex A to B (call it d(A,B)) is the length of the geodesic from A to B.
Note that d(A,B) = d(B,A) for undirected networks (symmetry), since all lines are bidirectional.
In a star network, the center is most closeness-central - it's at one hop from everyone. So we compute closeness centrality of any vertex by comparing to this star-center.
More precisely: for each vertex in a network of size N
- compute the sum of the distances to every other vertex
- divide this value into the number of other vertices (N-1), which represents the sum for the star-center.
Compute centralization as before:
- compute degree variation
  - find the most central vertex, and use its centrality as a comparison point, C
  - compute the difference between C and ever other vertex's centrality
  - add all these differences
- divide by the star network degree variation
Note that this technique can only work if the network is connected! (one component)

Intuition: a vertex is central if lots of geodesics pass through it
Define betweenness centrality of a vertex as the percentage of geodesics between other pairs that pass through it.
Betweenness centralization is the variation in this centrality, divided by the maximum possible variation.