Deterministic and Stochastic Autocatalytic Growth: Entropy, Bifurcations, and Quantum Extensions for Network Self-Organization

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 xl +1 = kxl , 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 λeff ≈ log k + 1/2 log (1 + ϵ2k2). 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.