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

A Set of Measures of Centrality Based on BetweennessLinton C. Freeman, 1977Advisor : Professor Frank Y. S. LinPresented by: Tuan-Chun ChenPresentation date: Mar. 13, 2012

freemanBetweenness Centralitypaperfreeman1977paperA set of Measures of centrality based on betweenness1AgendaIntroductionMeasuring point centralityMeasuring graph centralityApplications

agenda2AgendaIntroductionMeasuring point centralityMeasuring graph centralityApplications

introduction3IntroductionBetweenness : 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.

betweennessFreemanbetweennessbavelas1948Betweenness

()shimbel1953

4AgendaIntroductionMeasuring point centralityMeasuring graph centralityApplications

betweennesspoint centrality5Measuring Point CentralityShaw (1954)Unordered pair of points {pi, pj}{pi, pj} are Unreachable or there are one or more paths between them.

pipjpkShaw1954PiPj()PiPj ()()6Measuring Point CentralityA 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.pipjpk()communicationpipj()Pkpipj7Measuring Point CentralityDefine partial betweenness If pi and pj are not reachable from each other, pk is not between them. let

If pi and pj are reachable.

gijThe number of geodesics linking pi and pj.gij(pk)The number of geodesics linking pi and pj that contain pk.partial betweenness()

bij of pk pkbetweennesspipjbij of pk = 0

()pipjbij of pk = gij gij of pk()gijpipjgij()Gij of pkpipjpk8Measuring Point Centralityp2 and p4 each have a probability of of falling between p1 and p3. p2p4p3p1

p2p4p1p3()b13= ()p1p3p2()p2

g13 = 2p2g13 of p2 =1

9Measuring Point CentralityDetermine overall centrality of a point:

CB(pk)An index of the over all partial betweenness of point pk.nThe number of points in the graph.betweenness centralitykijcentralitykbetweenness centralityN10Measuring Point CentralityIts 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.

graphedge11Measuring Point CentralityProblem ! ?

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.()25betweenness centrality 6

() Pipj25

12Measuring Point CentralityMaximum Value:

pipjpkph

nThe number of points in the graph.Freemangraphcentralitynormalization

() C max= nn-1 (n-1)N() pk()pk()pkcentrality

3

13Measuring Point CentralityThe relative centrality of any point in a graph, expressed as a ratio :

When CB(pk)=1, the graph is a star or a wheel.

01starwheelpk14AgendaIntroductionMeasuring point centralityMeasuring graph centralityApplications

graph centrality15Measuring Graph CentralityA 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.)

CB(pk*)The largest centrality value associated with any point in the graph.CB(pi)The centrality value of pinThe number of points in the graph.Graph centralitycentralityCb(pk*)centralitycentralityCb of pi picentrality valueN

graph centrality16AgendaIntroductionMeasuring point centralityMeasuring graph centralityApplications

17Applications 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)

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18ApplicationsConsider the relationship between point centrality and personal satisfaction in Leavitts(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.Leavitt1951

point centralityLeavittpathpoint centrality19Applications

Leavitt

leavittpoint centrality

betweenness centrality20Thanks for your attention, 21