DETERMINISTIC AND STOCHASTIC AUTOCATALYTIC GROWTH: ENTROPY, BIFURCATIONS, AND QUANTUM EXTENSIONS FOR NETWORK SELF-ORGANIZATION
Résumé
We investigate the dynamics of self-organizing networks through two autocatalytic models: a deterministic growth model and a stochastic extension with bounded noise. Both are governed by the inheritance rule x l+1 = kx l , where each node replicates according to a fixed parameter k. The deterministic model yields exponential growth characterized by a Lyapunov exponent λ = log k, ensuring structural predictability. In contrast, the stochastic model introduces uniform noise and leads to entropy amplification, with an effective Lyapunov exponent λ eff ≈ log k + 1 2 log 1 + ϵ 2 k 2 . Using time series analysis, we study the evolution of trajectories over discrete inheritance steps and quantify their sensitivity through bifurcation diagrams and entropy metrics. We demonstrate a transition from linear to chaotic growth as k varies, supported by Lyapunov and entropy computations. Additionally, we extend the model into the quantum regime by associating system states with density matrices and computing von Neumann entropy. This unified framework reveals deep connections between network propagation, complexity growth, and quantum information dynamics, with implications for quantum computing, secure communication, and crisis-resilient decentralized systems.
Dynamical systems • Network self-organization • Autocatalytic processes • Deterministic growth • Discrete-time systems • Structural stability • Time Series analysis
This work is dedicated to the memory of Felix Shmidel, Ph.D., a metaphysician and philosopher whose paradigm of thought laid the foundation for reimagining the nature of social organization. His intellectual legacy, expressed in works such as The Metaphysics of Meaning and Will to Joy, profoundly shaped the conceptual development of this research.
With bounded noise, x l+1 = kx l + ϵξ l , ξ l ∼ U(-1, 1), the system retains autocatalytic growth while exhibiting increased entropy. The effective Lyapunov exponent becomes:
showing that even small noise amplifies the complexity and chaotic potential of the system.
We extend the stochastic inheritance model into the quantum domain by representing the evolving system as a quantum mixed state ρ l , with informational complexity measured by the von Neumann entropy:
Simulations using Qiskit confirm oscillatory and non-monotonic entropy patterns, reflecting transitions between coherence and decoherence. These findings bridge classical autocatalysis and quantum chaos, enabling applications in quantum circuit complexity, secure quantum communication, and entropy-regulated information processing.
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