Recursive Hierarchical Networks and the Law of Functional Evolution: A Universal Framework for Complex Systems

Abstract

Understanding and predicting the evolution of across complex systems remains a fundamental challenge due to the absence of unified and computationally testable frameworks. Here we propose the Recursive Hierarchical Network(RHN), conceptualizing evolution as recursive encapsulation along a trajectory of node $\to$ module $\to$ system $\to$ new node, governed by gradual accumulation and abrupt transition. Theoretically, we formalize and prove the law of functional evolution, revealing an irreversible progression from structure-dominated to regulation-dominated to intelligence-dominated stages. Empirically, we operationalize functional levels and align life, cosmic, informational, and social systems onto this scale. The resulting trajectories are strictly monotonic and exhibit strong cross-system similarity, with high pairwise cosine similarities and robust stage resonance. We locate current system states and project future transitions. RHN provides a mathematically rigorous, multi-scale framework for reconstructing and predicting system evolution, offering theoretical guidance for designing next-generation intelligent systems.