# Transitivity in Structural Models of Small Groups

@article{Holland1971TransitivityIS, title={Transitivity in Structural Models of Small Groups}, author={Paul Holland and Samuel Leinhardt}, journal={Small Group Research}, year={1971}, volume={2}, pages={107 - 124} }

Our purpose here is to show how various deterministic models for the structure of interpersonal relations in small groups may all be viewed as special cases of a single model: namely, a transitive graph (t-graph). This exercise serves three purposes. First, the unified approach renders much of the mathematical discussion surrounding these various models quite transparent. Many of the arguments boil down to nothing more than defining certain equivalence relations and looking at the resulting…

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## References

SHOWING 1-10 OF 16 REFERENCES

### Organizations as Semilattices

- Business
- 1967

and second, toward rigorously defining their interrelationships. The second problem is the statement of a formal theory of interaction in terms of communication, consensus, and coordination. In the…

### On dominance relations and the structure of animal societies: I. Effect of inherent characteristics

- Economics
- 1951

Societies are considered in which a non-transitive dominance relation exists between every pair of members, such as the peck-right in a flock of hens. A one-dimensional measure of the structure of…

### A Method for Detecting Structure in Sociometric Data

- Computer ScienceAmerican Journal of Sociology
- 1970

This model contains as special cases a number of previously suggested models, including the structural balance model of Cartwright and Harary, Davis's clustering model, and the ranked-clusters model of Davis and Leinhardt.

### Clustering and Hierarchy in Interpersonal Relations: Testing Two Graph Theoretical Models on 742 Sociomatrices

- Education
- 1970

Growth of a Policy (Washington: U.S. Government Printing Office, 1946), p. 138. 33 U.S. Department of Health, Education, and Welfare, Digest of Education Statistics, 1968 Edition (Washington: U.S.…

### Structural balance: a generalization of Heider's theory.

- PsychologyPsychological review
- 1956

Heider (19), reflecting the general field-theoretical approach, has considered certain aspects of cognitive fields which contain perceived people and impersonal objects or events.

### The Structure of Positive Interpersonal Relations in Small Groups.

- Physics
- 1967

A rotatable wire wrap tool of the type having an apertured tip end operative to engage a terminal to wrap the terminal with wire as the tool is rotated, is provided with a bobbin carrying a supply of…

### The psychology of interpersonal relations

- Psychology
- 1958

The psychology of interpersonal relations , The psychology of interpersonal relations , کتابخانه دیجیتال و فن آوری اطلاعات دانشگاه امام صادق(ع)

### The principle of congruity in the prediction of attitude change.

- EducationPsychological review
- 1955

### Attitudes and cognitive organization.

- PsychologyThe Journal of psychology
- 1946

A comparison of attitudes and cognitive Organization in the context of war and post-war Europe shows marked differences in the attitudes of men and women towards one another and towards Europe in general.