1. Point centrality:
Point centrality measures how centrally a point is located in a given graph. Usually the point at the center of a star or the hub of a wheel is the most central possible position. There are three approaches to measuring the point centrality:
1) Point centrality based on the degree: the point with the largest degree is the most central point in the graph.
Nieminen's (1974) measurement: count of the degree or number of adjacencies, for a point pk:
where n = number of points
a(pi,pk)=1 if and only if pi and pk are connected by a line
0 otherwise
If we control the size of the graph by dividing CD(pk) by n-1, which is the maximum degree of pk in any graph, and make the measurement comparable between graph, we get a "relative centrality" for pk, C'D(pk).
2) Point centrality based on betweenness: the point which falls on the largest numbers of geodesic paths (that is, the shortest path link a given pair, say, pi and pj) between pairs is the most central point in the graph.
The measurement developed by Anthonisse (1971) and Freeman (1977):
where gij = the number of geodesics linking pi and pj.
gij(pk)= the number of geodesics linking pi and pj that contain pk.
Like the centrality based on the degree, we divide CB(pk) by the maximum value in any graph, (n-2)(n-1)/2 =(n2-3n+2)/2, to get a "relative centrality" based on betweenness for pk, C'B(pk).
3)Point centrality based on closeness: the point which is closest to all other points in the graph is the most central point.
Sabidussi's (1966) measurement:
where d(pi,pk) = the number of edges in the geodesic linking pi and pk.
The "relative centrality" based on closeness is:
Different research interests leads to different measurement:
Concern with communication activity suggests a degree-based measure.
Interest in control of communication requires a measure based
upon betweenness. And concern with either independence or efficiency
leads to the choice of a measure based upon closeness.
2. Graph centrality (the centralization of the network): measures the differences between the centrality of the most central point and that of all others. The larger the relative differences are ( controlling for the size of the graph), the more central a graph (network) is.
The general formula for graph centrality is:
where n=number of points
CX(pi) = one of the point centralities defined above
CX(p*) = largest value of CX(pi) for any point in the network and
= the maximum possible sum of differences
in
point centrality for a graph of n points.
(Note: Since CX is a ratio of an observed sum of differences
to its maximum value, it will vary between 0 and 1.)