Problem
Research questions and friction points this paper is trying to address.
Modeling hierarchy growth in complex systems.
Analyzing agent promotions and degradations dynamics.
Exploring stationary state properties of hierarchical distributions.
From equality to diversity - bottom-up approach for hierarchy growth
🤖 AI Summary
This work investigates how hierarchical structures spontaneously emerge in complex systems from initially egalitarian states via bottom-up mechanisms. We propose the first analytically tractable bottom-up hierarchical evolution model that jointly incorporates local election-based promotion and mandatory demotion-based regression. The model is analyzed using rate equations and validated through large-scale numerical simulations. Our theoretical and computational results reveal that, at steady state, agent-level distributions strictly follow an exponential decay; the system’s hierarchical height scales logarithmically with population size (H ∝ log N); both the mean level and the fraction of bottom-level agents are scale-invariant; and the average number of followers per agent remains significantly below the promotion threshold. Crucially, this study provides the first unified characterization of bidirectional promotion–demotion dynamics, establishing a general mechanistic explanation and quantitative predictive framework for spontaneous hierarchy formation across social, biological, and artificial systems.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Bottom-up hierarchy growth model introduced
Dynamics include promotions and degradations
Stationary state analyzed via rate equations
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