A theory of discrete hierarchies as optimal cost-adjusted productivity organisations

Abstract

Hierarchical structures are ubiquitous in human and animal societies, but a fundamental understanding of their raison d’être has been lacking. Here, we present a general theory in which hierarchies are obtained as the optimal design that strikes a balance between the benefits of group productivity and the costs of communication for coordination. By maximising a generic representation of the output of a hierarchical organization with respect to its design, the optimal configuration of group sizes at different levels can be determined. With very few ingredients, a wide variety of hierarchically ordered complex organisational structures can be derived. Furthermore, our results rationalise the ubiquitous occurrence of triadic hierarchies, i.e., of the universal preferred scaling ratio between 3 and 4 found in many human and animal hierarchies, which should occur according to our theory when production is rather evenly contributed by all levels. We also provide a systematic approach for optimising team organisation, helping to address the question of the optimal ‘span of control’. The significantly larger number ∼ 3 − 20 of subordinates a supervisor typically manages is rationalised to occur in organisations where the production is essentially done at the bottom level and in which the higher levels are only present to optimise coordination and control.

Comparative analysis of layered structures in empirical investor networks and cellphone communication networks

“…The empirical hierarchical structures of the personal ego networks in SVEIN and SVCN are compatible with the structure of 3-5, 10-15, 30-50, 100-200 from the inner to the outer layer, which is close to the theoretical Dunbar Circle. And the average empirical scaling ratio is close to the previously found value 3, which can also be accounted for theoretically (Lera and Sornette, 2019).…”

Non-normal interactions create socio-economic bubbles

“…An adjacency matrix with such cyclical symmetry still gives rise to a normal adjacency matrix. The condition MM T ≠ M T M is instead satisfied when the network is directed and hierarchical, which are both intrinsic properties of socioeconomic systems 17,50,51 . Indeed, it is well-established in anthropology that people organize their social interactions along one or more of the four elementary modes: communal sharing, authority ranking, equality matching, and market pricing, with the latest one being particularly relevant in finance 52,53 .…”

“…We suggest that this new mechanism is much more general than previously existing ones and likely to be often the dominant process at work, because it does not require the fine-tuning close to or sweeping 15 of the system over a critical point. Our proposed mechanism is based on the fact that social influence is characterized by two critical properties: it is directed and hierarchical 16,17 . For instance, in our example of financial markets, the influence of a famous investor on a retail investor is likely much larger than the other way around.…”