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Social structure may have changed from hierarchical to egalitarian and back along the evolutionary line of humans. Within the tradition of sociophysics, we construct a mathematical model of a society of agents subject to competing cognitive and social navigation constraints and predict, using statistical mechanics methods, that its degree of hierarchy decreases with encephalization and increases with group size, hence suggesting human societies were driven from hierarchical to egalitarian structures by the encephalization during the last few million years and back to hierarchical due to fast demographic changes during the Neolithic. In addition, applied to a different problem, the theory leads to the following predictions for modern pre-literary humans: (i) an intermediate hierarchy degree in mild climates. In harsher climates, societies will be (ii) more egalitarian if organized in small groups (of less than 100 persons) but (iii) more hierarchical if in larger (of more than 1000 persons) groups. The predicted bifurcation, characteristic of a phase transition, is also seen in the empirical cross-cultural record (248 cultures in the Ethnographic Atlas).
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