Social Media Mining: An Introduction

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CUUS2079-08 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:22

222 Influence and Homophily

v 1

v 2

v 3

v 4

Figure 8.3. A Modularity Example for a Bipartite Graph.

Example 8.1.Consider the bipartite graph in Figure8.3. For this bipartite

graph,

A=

⎡

⎢⎢

⎣

0011

0011

1100

1100

⎤

⎥⎥

⎦,=

⎡

⎢⎢

⎣

10

10

01

01

⎤

⎥⎥

⎦, d=

⎡

⎢⎢

⎣

2

2

2

2

⎤

⎥⎥

⎦,m=^4.

(8.13)

Therefore, matrix B is

B=A−ddT/ 2 m=

⎡

⎢⎢

⎣

− 0. 5 − 0. 50. 50. 5

− 0. 5 − 0. 50. 50. 5

0. 50. 5 − 0. 5 − 0. 5

0. 50. 5 − 0. 5 − 0. 5

⎤

⎥⎥

⎦. (8.14)

The modularity value Q is

1

2 m

Tr(TB)=− 0. 5. (8.15)

In this example, all edges are between nodes of different color. In other

words, the number of edges between nodes of thesame coloris less than the

expected number of edges between them. Therefore, the modularity value

is negative.

8.1.2 Measuring Assortativity for Ordinal Attributes

A common measure for analyzing the relationship between two variables

COVARIANCE with ordinal values is covariance. Covariance describes how two variables

change with respect to each other. In our case, we are interested in how

correlated, the attribute values of nodes connected via edges are. Letxibe

the ordinal attribute value associated with nodevi. In Figure8.4, for node

c, the value associated isxc=21.