Social Media Mining: An Introduction

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

62 Network Measures

v 1

v 2 v 3

v 4

v 6 v 7

v 8

v 10

v 9

v 5

Figure 3.6. Example for All Centrality Measures.

Example 3.8. For nodes in Figure3.5, the closeness centralities are as

follows:

Cc(v 1 )= 1 /((1+ 2 + 2 +3)/4)= 0. 5 , (3.39)

Cc(v 2 )= 1 /((1+ 1 + 1 +2)/4)= 0. 8 , (3.40)

Cc(v 3 )=Cb(v 4 )= 1 /((1+ 1 + 2 +2)/4)= 0. 66 , (3.41)

Cc(v 5 )= 1 /((1+ 1 + 2 +3)/4)= 0. 57. (3.42)

Hence, nodev 2 has the highest closeness centrality.

The centrality measures discussed thus far have different views on what

a central node is. Thus, a central node for one measure may be deemed

unimportant by other measures.

Example 3.9.Consider the graph in Figure3.6. For this graph, we compute

the top three central nodes based on degree, eigenvector, Katz, PageRank,

betweenness, and closeness centrality methods. These nodes are listed in

Table3.1.

As shown in the table, there is a high degree of similarity between

most central nodes for the first four measures, which utilize eigenvectors

or degrees: degree centrality, eigenvector centrality, Katz centrality, and

Table 3.1.A Comparison between Centrality Methods

First node Second node Third node

Degree Centrality v 3 orv 6 v 6 orv 3 v∈{v 4 ,v 5 ,v 7 ,v 8 ,v 9 }

Eigenvector Centrality v 6 v 3 v 4 orv 5

Katz Centrality:α=β= 0. 3 v 6 v 3 v 4 orv 5

PageRank:α=β= 0. 3 v 3 v 6 v 2

Betweenness Centrality v 6 v 7 v 3

Closeness Centrality v 6 v 3 orv 7 v 7 orv 3