A dynamic structure is defined as one where the number of relations in the structure is changing over time. The theory of structural balance is outlined in the context of incomplete structures, as dynamic structures are necessarily incomplete at some instant. Theorems concerning the partitioning properties of incomplete symmetric and non-symmetric structures are considered. Finally, the theory of balance is placed on a probabilistic footing.
References
1.
J. O. Morrissetti , `An Experimental Study of the Theory of Structural Balance' , Human Relations,11, 1958.
2.
For a criticism of the axiomatics of this model see: R. M. Lambert , `An Examination of the Consistency Characteristics of Abelson and Rosenberg's Symbolic Psycho-Logic' , Behavioural Science,11, 1966.
3.
By a structure is meant a set of units (persons, roles, etc.) and a set of relations holding between some or all of these units.
4.
The degree of completion of a structure is the extent to which all the structural units are related to all the other units. A complete structure is one where all possible pairs of structural units are related.
5.
F. Harary has suggested that structures exhibit a tendency towards completion. See, F. Harary , `On the Measurement of Structural Balance' , Behavioural Science, 4, 1959.
6.
A decaying structure is one where the number of related pairs (ordered pairs) of units is decreasing over time. A fluctuating structure is one where the ordered pairs fluctuate over time.
7.
F. Harary, op. cit., ref. (5).
8.
F. Heider , `Attitudes and Cognitive Organization' , Journal of Psychology, 21, 1946.
9.
Op. cit., ref. (2).
10.
Peter Abell , `Measurement in Sociology: (I) Measurement Systems' , Sociology,2, 1967.
11.
Peter Abell , `Balance in Ordinal Structures', forthcoming.
12.
This example is taken from Claude Flament, Application of Graph Theory to Group Structure.Englewood Cliffs, New Jersey: Prentice-Hall , 1963.
13.
See, for instance, R. P. Abelson and M. J. Rosenberg, op. cit., ref. (2).
14.
See ref (11).
15.
In practice this will only be a relatively large time period compared with the time affective relations take to initially form.
16.
A bipartition is a classification of all the points in the set into two mutually exclusive and exhaustive classes.
17.
A proof of this theorem may be found in Claude Flament, op. cit., ref. (12).
18.
See, Claude Flament, op. cit., ref. (12).
19.
For the definition of a component see below.
20.
If the structure consists of only one, all positive, component then one class is empty.
21.
Isolated points are defined as components.
22.
φ is the empty set.
23.
A detailed study of this phenomenon is best carried out in the context of an empirical enquiry.
24.
All positive relation components must be regarded in the same manner as suggested under Theorem II.
25.
This expression assumes, of course, that the probability of an ordered pair <a, b> being positive is independent of <b, c> and <a, c>. One might expect, however, that given <a, b> and <b, c> are positively connected, the probability that <a, c> will be positive is significantly greater than one would expect on random grounds. Thus, the actual probability of occurrence of all positive 3-cycles would be greater than P3(p).
