Social Media Mining: An Introduction

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

70 Network Measures

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

v 1

v 3 v 2

balanced balanced balanced balanced

unbalanced unbalanced unbalanced unbalanced

• 
  

  –

  
  

   - – – –
  
      -

  
  


• –

+ ––

+

+

+ + +

+

+

++

+ +

Figure 3.12. Sample Graphs for Social Balance Theory. In balanced triangles, there are

an even number of negative edges.

Social Balance Theory

STRUCTURAL This theory, also known asstructural balance theory, discusses consistency

BALANCE

THEORY

in friend/foe relationships among individuals. Informally, social balance

theory says friend/foe relationships are consistent when

Thefriendofmyfriendismyfriend,

The friend of my enemy is my enemy,

The enemy of my enemy is my friend,

The enemy of my friend is my enemy.

We demonstrate a graph representation of social balance theory in Fig-

ure3.12. In this figure, positive edges demonstrate friendships and negative

ones demonstrate enemies. Triangles that are consistent based on this theory

BALANCED are denoted asbalancedand triangles that are inconsistent asunbalanced.

AND

UNBALANCED

TRIANGLES

Letwijdenote the value of the edge between nodesviandvj. Then, for

a triangle of nodesvi,vj, andvk, it is consistent based on social balance

theory; that is, it is balanced if and only if

wijwjkwki≥ 0. (3.56)

This is assuming that, for positive edges,wij=1, and for negative edges,

wij=−1. We observe that, for all balanced triangles in Figure3.12, the

valuewijwjkwkiis positive, and for all unbalanced triangles, it is negative.

Social balance can also be generalized to subgraphs other than triangles. In

general, for any cycle, if the product of edge values becomes positive, then

the cycle is socially balanced.