A Set of Measures of Centrality Based on Betweenness Linton C. Freeman, 1977

1. A Set of Measures of Centrality Based on BetweennessLinton C. Freeman, 1977 Advisor : Professor Frank Y.S. Lin Presented by: Tuan-Chun Chen Presentation date: Mar. 13, 2012

2. Agenda • Introduction • Measuring point centrality • Measuring graph centrality • Applications

3. Agenda • Introduction • Measuring point centrality • Measuring graph centrality • Applications

4. Introduction • Betweenness : A point in a communication network is central to the extent that it falls on the shortest path between pairs of other points. (Bavelas, 1948) • Another viewpoint by Shimbel (1953): If we count all of the minimum paths which pass through a site, then we have a measure of the ‘stress’ which the site must undergo during the activity of the network.

5. Agenda • Introduction • Measuring point centrality • Measuring graph centrality • Applications

6. Measuring Point Centrality • Shaw (1954) • Unordered pair of points {pi, pj} • {pi, pj} are Unreachable or there are one or more paths between them. pk pi pj

7. Measuring Point Centrality • A point falling between two others can facilitate, block, distort or falsify communication between the two. • But if it falls on some but not the shortest path connecting a pair of points , its potential for control is more limited. pk pi pj

8. Measuring Point Centrality • Define “partial betweenness” • If pi and pj are not reachable from each other, pk is not between them. let • If pi and pj are reachable.

9. Measuring Point Centrality • p2 and p4 each have a probability of ½ of falling between p1 and p3. p3 p2 p4 p1

10. Measuring Point Centrality • Determine overall centrality of a point:

11. Measuring Point Centrality • Its magnitude depends upon two factors: • 1) the arrangement of edges in the graph that define the location of pk with respect to geodesics linking pairs of points. • 2) the number of points in the graph.

12. Measuring Point Centrality • Problem ! ? • Example: • A graph containing 5 points, CB(pi)=6. A graph containing 25 points, CB(pj)=6. • They have the same potential for control in absolute terms, but differ markedly in their relative potential for control.

13. Measuring Point Centrality • Maximum Value: pk pi pj ph

14. Measuring Point Centrality • The relative centrality of any point in a graph, expressed as a ratio : • When C’B(pk)=1, the graph is a star or a wheel.

15. Agenda • Introduction • Measuring point centrality • Measuring graph centrality • Applications

16. Measuring Graph Centrality • A network is central to the degree that a single point can control its communication.(Measures of graph centrality based upon the dominance of one point.)

17. Agenda • Introduction • Measuring point centrality • Measuring graph centrality • Applications

18. Applications • Original application was in the study of communication in small groups. Speed, activity and efficiency in solving problems and personal satisfaction and leadership in small group setting(Leavitt 1951). • Study of the diffusion of a technological innovation in the steel industry(Czepiel 1974) • Examined the impact of centrality on urban growth(Pitts 1965). • Discussing the design of organization(Beauchamp 1965)(Mackenzie 1966)

19. Applications • Consider the relationship between point centrality and personal satisfaction in Leavitt’s(1951) study of small group problem solving. • Each participant had a piece of information necessary for the solution of a problem. Each could communicate only with designated others. • Leavitt measured point centrality as a function of the lengths of paths or the distance between points.

20. Applications

21. Thanks for your attention！