Date Published: April 18, 2019
Publisher: Public Library of Science
Author(s): Sandro Claudio Lera, Didier Sornette, Saber Elaydi.
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.
Throughout most of Homo sapiens 300’000 year record, humans have lived in small-scale, mostly egalitarian hunter-gatherer societies, comprising around 30-50 or, at most, a few hundred individuals [1–3]. Following the strong warming of Earth by 5 to 10 °C from about 15’000 years ago leading to the end of the last ice age, settled communities emerged around 10’000 years ago, together with agriculture and animal domestication. These societies have been mostly structured into hierarchical societies. Over the past millennia, even more complex, large scale interconnected societies have evolved, shaped into cultural, economic, political and corporate hierarchies [3, 4]. Explanations for the benefits of hierarchical organisation are manifold, such as advantages in warfare and multilevel selection [3, 5], optimal search properties , robustness , effective use of resources  and so on. But a framework to quantitatively relate the specific hierarchical structures to the functions and constraints facing different types of society has been lacking.
In section I-2.2, we suggested a derivation of Dunbar’s number ∼ 100 − 250, describing the maximum number of people with whom one can develop stable social relationships. But Dunbar’s number is actually just a part of the full story. In 2005, Zhou et al.  discovered the general existence of a discrete hierarchy of group sizes with a preferred scaling ratio close to three: humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3 − 5, 9 − 15, 30 − 45, 90 − 140, 250 − 400 and so on. This finding has been corroborated in many different contexts [9, 21, 23, 24] as well as for various groups of animals . These works quantify the qualitative anthropological studies showing that societies, from primates  to humans , tend to arrange into discrete hierarchical structures, with group sizes ratios between hierarchical levels that typically range from 2 to 4 . Within our framework embodied in Eq (4), this observation finds a natural explanation, as we will now show.