Information slows down hierarchy growth

Abstract

We consider models of growing multi-level systems wherein the growth process is driven by rules of tournament selection. A system can be conceived as an evolving tree with a new node being attached to a contestant node at the best hierarchy level (a level nearest to the tree root). The proposed evolution reflects limited information on system properties available to new nodes. It can also be expressed in terms of population dynamics. Two models are considered: a constant tournament (CT) model wherein the number of tournament participants is constant throughout system evolution, and a proportional tournament (PT) model where this number increases proportionally to the growing size of the system itself. The results of analytical calculations based on a rate equation fit well to numerical simulations for both models. In the CT model all hierarchy levels emerge but the birth time of a consecutive hierarchy level increases exponentially or faster for each new level. The number of nodes at the first hierarchy level grows logarithmically in time, while the size of the last, "worst" hierarchy level oscillates quasi log-periodically. In the PT model the occupations of the first two hierarchy levels increase linearly but worse hierarchy levels either do not emerge at all or appear only by chance in early stage of system evolution to further stop growing at all. The results allow to conclude that information available to each new node in tournament dynamics restrains the emergence of new hierarchy levels and that it is the absolute amount of information, not relative, which governs such behavior.