Social Balance Theory

There are 8 possible configurations of the current sentiment relations described
in figure 1 (table 1). In the table we can see that there are 4 patterns of balance state and 4
others imbalance one in the relation of three agents (triad).

Table 1

Possible combination of relation sign in a triad and the
relation’s characteristic

For one group of individuals of more than three people, the formalization of
Heider’s theory by Cartwright and Harary (1956) resulting structural balance theory for
a graph represents interpersonal network. Sentiment relations among individuals can be
assumed as a social interpersonal network (Kadushin, 2004). The structural balance
theory shows how change of the dyadic sign cause change of the balance for the whole
graph. The structural balance theory as modeled as graph (with many nodes and edges)
can be summarized as two structure theorems (Hummon and Doreian, 2002 & see
Robert, 1999 for proof), i.e.:

Structure Theorem

A graph (network of individuals) within a large group of people is balanced if and only if the
group can be divided into two subgroups (two sets), wherein individual relations in the same
subgroup are all positive (all edges between vertices in the same set are ‘+’) and between
individuals in different subgroups are negative (all edges between vertices in the different sets are
‘-’)

In the next section, we represent social interpersonal network among individuals
in a large group as a network composed of possible triads, while in the other hand each
local triadic consisted of three dyadic (Pattison and Robins, 2001). By knowing the
initial pattern of all dyadic relations, we can determine whether the formed triads are
balanced or not. We use a global balance index, defined as the ratio of the number of
balanced triads divided by number of all possibly formed triads, and the local balance
index, defined as the ratio between the number of balanced triads and the existing
number of triads formed by each dyad. The index is used to measure the balance or
imbalance of a network (locally or globally). A globally balanced network will have
balance index of 1, means that the network are balanced collectively.

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