From equality to diversity -- bottom-up approach for hierarchy growth
Agnieszka Czaplicka, Janusz A. Hołyst
TL;DR
The paper addresses how hierarchical structures can emerge from bottom-up growth starting from equal peers, incorporating turnover via degradation. It develops a simple dynamical model where promotions to the next level occur when local leaders are elected and agents degrade back to the bottom, analyzed with rate-equation methods and supported by simulations. In the stationary state the level distribution is exponential $P(h) \propto e^{-\lambda h}$, with the average level and the fraction at the lowest level being independent of system size, while the maximal height scales as $H_{\max} \sim \log N$, and the average number of followers per agent at stationarity is much smaller than at promotion time. These results illuminate how real-world bottom-up hierarchies can self-organize and evolve under simple promotion-degradation dynamics, with potential implications for understanding organizational and social structures.
Abstract
The hierarchical topology is a common property of many complex systems. Here we introduce a simple but generic model of hierarchy growth from the bottom to the top. Therein, two dynamical processes are accounted for: agent's promotions to next hierarchy levels when local speakers are elected and followed by other agents and agent's degradations to the lowest hierarchy. Following the initial stage when all agents are at the bottom level in the course of time the system approaches a stationary state where new hierarchies no longer emerge and the distribution of agents at different levels is exponential. In the stationary state the average hierarchy level and the fraction of agents at the lowest level are independent from the system size however the height of hierarchy, i.e. maximal number of observed hierarchy levels grows logarithmically along the total number of agents. The average number of followers of an agent in the stationary state is much smaller than the number of followers he possessed at the promotion moment. Results from numerical simulations are confirmed by an analytical treatment based on the rate equation.