Glassy nature of hierarchical organizations

The question of why and how animal and human groups form temporarily stable hierarchical organizations has long been a great challenge from the point of quantitative interpretations. The prevailing observation/consensus is that a hierarchical social or technological structure is optimal considering a variety of aspects. Here we introduce a simple quantitative interpretation of this situation using an approach reminiscent of those developed for describing complex behaviour in terms of statistical mechanics. We look for the optimum of the efficiency function $E_{\rm eff}=1/N \sum_{i,j} J_{ij} a_i a_j$ with $J_{ij}$ denoting the nature of the interaction between the units $i$ and $j$ and $a_i$ standing for the ability of member $i$ to contribute to the efficiency of the system. Notably, this expression for $E_{\rm eff}$ has a similar structure to that of the energy as defined for spin-glasses. There is, however, an essential and novel feature of our approach: instead of optimizing by looking for a locally optimal state of the units in the nodes of a pre-defined network, we search for extrema in the complex efficiency landscape by finding locally optimal network topologies using a standard Monte Carlo method.