From equality to diversity: A bottom-up approach for hierarchy growth

Hierarchical topology stands as a fundamental property of many complex systems. In this work, we present a simple yet insightful model that captures hierarchy growth from bottom to top. Our model incorporates two key dynamic processes: the emergence of local leaders through promotions, where successful agents advance to higher hierarchical levels by attracting followers, and the natural degradation of agents to the lowest level. From an initial flat structure where all agents occupy the bottom level, the system evolves toward a stationary state characterized by an exponential distribution of agents across levels—a pattern remarkably similar to those observed in diverse real-world hierarchies, from hunter-gatherer societies and mammalian groups to online communities. Notably, while the average hierarchy level and the fraction of ground-level agents remain independent of system size, the maximum height of the hierarchy grows logarithmically with the total number of agents. In the stationary state, agents maintain a significantly smaller number of followers compared to their peak influence at the promotion moment. Results from numerical simulations are supported by analytical solutions derived based on the rate equations.