Social Media Mining: An Introduction

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CUUS2079-03 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 16:45

64 Network Measures

v 1

v 2

v 3

v 4 v 5

Figure 3.7. Group Centrality Example.

Group Closeness Centrality

Closeness centrality for groups can be defined as

Ccgroup(S)=

1

̄lgroupS , (3.45)

wherel ̄groupS =|V^1 −S|

∑

vj ∈SlS,vjandlS,vjis the length of the shortest path

between a groupSand a nonmembervj∈V−S. This length can be

defined in multiple ways. One approach is to find the closest member inS

tovj:

lS,vj=min

vi∈S

lvi,vj. (3.46)

One can also use the maximum distance or the average distance to

compute this value.

Example 3.10.Consider the graph in Figure3.7.LetS={v 2 ,v 3 }. Group

degree centrality for S is

Cdgroup(S)= 3 , (3.47)

since members of the group are connected to all other three members

in V−S={v 1 ,v 4 ,v 5 }. The normalized value is 1, since 3 /|V−S|= 1.

Group betweenness centrality is 6 , since for 2

( 3

2

)

shortest paths between

any two members of V−S, the path has to pass through members of S. The

normalized group betweenness is 1 , since 6 /

(

2

(|V−S|

2

))

= 1. Finally, group

closeness centrality – assuming the distance from nonmembers to members

of S is computed using the minimum function – is also 1, since any member

of V−S is connected to a member of S directly.

3.2 Transitivity and Reciprocity

Often we need to observe a specific behavior in a social media network.

One such behavior is linking behavior. Linking behavior determines how

links (edges) are formed in a social graph. In this section, we discuss two