Glassy nature of hierarchical organizations

Abstract

Abstract 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 a statistical mechanics-type approach. We look for the optimum of the efficiency function $${E}_{eff}=1/N{\sum }_{ij}{J}_{ij}{a}_{i}{a}_{j}$$ E e f f = 1 / N ∑ i j J i j 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 eff has a similar structure to that of the energy as defined for spin-glasses. Unconventionally, we assume that J ij -s can have the values 0 (no interaction), +1 and −1; furthermore, a direction is associated with each edge. The essential and novel feature of our approach is that instead of optimizing the state of the nodes of a pre-defined network, we search for extrema for given a i -s in the complex efficiency landscape by finding locally optimal network topologies for a given number of edges of the subgraphs considered.