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

1. Deterministic Network Growth Model

The majority of existing models of network growth, such as preferential attachment [1] and random graph processes [5], rely on stochastic dynamics, probabilistic rewiring, or agent-based interactions to explain the establishment of large-scale connectivity. These models have contributed significantly to our understanding of scale-free behavior [1,20], small-world effects [20], and emergent robustness in complex networks [12]. However, determinism and predictability are typically compromised in such frameworks due to inherent randomness.

In contrast, we propose a fundamentally new approach: a deterministic structural inheritance rule that specifies the evolution of world-wide network structure without recourse to randomness, heuristics, or behavioral assumptions. Our model offers an exact, rule-based foundation for network growth, supporting full analytical control over its topological and dynamical evolution.

1.1. The Inheritance Rule

We define a discrete-time deterministic growth law in which each node at time l generates exactly k new nodes at time $l+1$, each of which inherits the structure of its parent node. Let $N\left(l\right)$ denote the total number of nodes at iteration depth l. The evolution of the network is governed by the exponential law:

$$N\left(l\right)=k^l,\mathrm{where}k\in \mathbb{N},k\ge 1,l\in {\mathbb{N}}_0.$$

Here, k is the deterministic inheritance parameter—the constant number of new nodes each existing node gives rise to—and l is the discrete generation or iteration step. This differs markedly from the randomness inherent in classical growth models [1,5], and draws conceptually from deterministic chaos theory [17].

1.2. Topological Interpretation

From the graph-theoretical viewpoint, the construction establishes a rooted level-based tree topology in which nodes at level l link only to their unique parent nodes at level $l-1$. Unlike stochastic tree-growth models, our construction is built upon structural replication rather than probabilistic edge addition. Parent node subgraphs are thus recursively duplicated and inherited, yielding highly regular, scalable architectures.

The deterministic rule ensures:

• Full structural predictability: The topology at any future iteration is entirely predetermined by the initial configuration and the constant parameter k.
• Scalability without coordination: There is no interaction or communication among nodes, so the model is well-suited to distributed, trust-free environments such as secure knowledge propagation or autonomous data transmission networks [6,11].
• Parameter-driven complexity: The single control parameter k governs both the topological scale and dynamical divergence, and this relationship can be quantified analytically via Lyapunov exponents and entropy metrics [22,30].

2. Numerical Simulation

We simulate the trajectory $x_l=k^l$ for various values of k over the interval $l=0$ to 5, illustrating the exponential growth behavior of the deterministic inheritance process.

Table 1. Network Size Evolution for $k=50$.

Level l	$\mathit{N}\left(\mathit{l}\right)$	${log}_{10}\left(\mathit{N}\left(\mathit{l}\right)\right)$	Growth Rate
0	1	0.00	–
1	50	1.70	50.0
2	2,500	3.40	50.0
3	125,000	5.10	50.0
4	$6.25\times {10}^6$	6.80	50.0
5	$3.13\times {10}^8$	8.50	50.0

Table 1. Network Size Evolution for $k=50$.

Level l	$\mathit{N}\left(\mathit{l}\right)$	${log}_{10}\left(\mathit{N}\left(\mathit{l}\right)\right)$	Growth Rate
0	1	0.00	–
1	50	1.70	50.0
2	2,500	3.40	50.0
3	125,000	5.10	50.0
4	$6.25\times {10}^6$	6.80	50.0
5	$3.13\times {10}^8$	8.50	50.0

Table 2. Comparison with Classical Network Models.

Model	Growth Function	Predictability	Architectural Constraints
Erdős–Rényi	$P\left(k\right)\propto e^{-\lambda }{\displaystyle \frac{{\lambda }^k}{k!}}$	Probabilistic	None
Barabási–Albert	$P\left(k\right)\propto k^{-\gamma }$	Probabilistic	Preferential
Small-World	$P\left(k\right)\approx \delta (k-k_0)$	Semi-deterministic	Rewiring
Our Model	$N\left(l\right)=k^l$	Deterministic	$H(a,v)=0$

Table 2. Comparison with Classical Network Models.

Model	Growth Function	Predictability	Architectural Constraints
Erdős–Rényi	$P\left(k\right)\propto e^{-\lambda }{\displaystyle \frac{{\lambda }^k}{k!}}$	Probabilistic	None
Barabási–Albert	$P\left(k\right)\propto k^{-\gamma }$	Probabilistic	Preferential
Small-World	$P\left(k\right)\approx \delta (k-k_0)$	Semi-deterministic	Rewiring
Our Model	$N\left(l\right)=k^l$	Deterministic	$H(a,v)=0$

Table 3. Parameter Sensitivity Analysis.

k	$\mathit{N}\left(4\right)$	$\mathit{N}\left(5\right)$	Global Reach	Empirical Support
40	2.56M	102.4M	Level 5	Consistent
50	6.25M	312.5M	Level 5	✓Facebook: 4.74
60	13.0M	777.6M	Level 5	✓Consistent

Table 3. Parameter Sensitivity Analysis.

k	$\mathit{N}\left(4\right)$	$\mathit{N}\left(5\right)$	Global Reach	Empirical Support
40	2.56M	102.4M	Level 5	Consistent
50	6.25M	312.5M	Level 5	✓Facebook: 4.74
60	13.0M	777.6M	Level 5	✓Consistent

Analysis: For $k\ge 50$, global connectivity is achieved within 5 iterations, consistent with empirical findings from Facebook’s large-scale network analysis showing an average of 4.74 degrees of separation [2]. This demonstrates the critical role of the inheritance parameter in determining convergence rates and validates our theoretical predictions.

Results confirm:

• Perfect monotonicity
• Constant growth rate k
• Deterministic predictability
• Structural inevitability
• Empirical consistency with real-world networks

3. Dynamical Analysis and Chaotic Transitions

Although the deterministic formula $N\left(l\right)=k^l$ governs periodic and predictable exponential network development, it omits key features commonly observed in self-organizing systems such as uncertainty, heterogeneity, and adaptation. In nature and engineered networks, structural evolution is often regulated by environmental noise, topological disruption, or local adaptation [17,19,22]. These dynamics are integral to understanding complexity in neural systems, social communication, and even molecular biology [4,8].

To incorporate such effects while maintaining analytical tractability, we introduce a bounded stochastic perturbation into the inheritance rule, resulting in a hybrid deterministic-stochastic model.

3.1. Noisy Stochastic Model

We enhance the deterministic rule with an additive noise component ${\eta }_l$, leading to the recursive relation:

$$N(l+1)=k·N\left(l\right)+{\eta }_l,$$

where:

$${\eta }_l=ϵ·{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1),$$

and $ϵ>0$ is the noise amplitude controlling the strength of the perturbation. The random variable ${\xi }_l$ introduces uniform fluctuations at each iteration step.

This formulation results in a noisy exponential map [23], preserving the multiplicative backbone of deterministic inheritance while embedding bounded stochasticity. Such perturbations are particularly relevant in modeling:

• External disturbances: e.g., signal degradation or environmental uncertainty in decentralized systems.
• Spontaneous variation: arising from contextual or spatial heterogeneity in large-scale systems.
• Adaptive mutations: deviations from strict inheritance due to feedback, learning, or evolutionary pressures [27,30].

3.2. Bifurcation and Chaos Transitions

In order to study how the system crosses from stable to chaotic regimes as a function of k and $ϵ$, we perform a bifurcation analysis of the stochastic map. We define the normalized form:

$$x_{l+1}=k{x}_l+ϵ{\xi }_l,$$

(1)

where $x_l={\displaystyle \frac{N\left(l\right)}{N_0}}$ is the normalized population size.

Depending on k and $ϵ$, the system 1 has various regimes:

• For $k<1$, the system goes to zero even when it is noisy.
• For $k=1$, the system is nearly a noisy random walk and fluctuation-dominated.
• For $k>1$, the deterministic part dominates, but chaotic divergence and random bursts are induced by additive noise.

We determine the effective Lyapunov exponent ${\lambda }_{\mathrm{eff}}$ in order to estimate the mean rate of divergence:

$${\lambda }_{\mathrm{eff}}\approx log\left|k\right|+{\displaystyle \frac{1}{2}}log\left(1+{\displaystyle \frac{{ϵ}^2}{k^2}}\right).$$

This word demonstrates the way in which noise increases the entropy and instability of the system and induces transitions from organized to random expansion.

4. Fixed Point Analysis of the Noisy Inheritance Model

We consider the noisy discrete-time network growth model defined by the recursive relation:

$$N(l+1)=k·N\left(l\right)+{\eta }_l,$$

where ${\eta }_l=ϵ·{\xi }_l$, and ${\xi }_l\sim \mathcal{U}(-1,1)$ represents a uniformly distributed stochastic fluctuation, while $ϵ>0$ controls the amplitude of noise. This formulation models both deterministic inheritance and bounded environmental or contextual randomness.

4.1. Existence and Stability of Fixed Points

In the absence of noise ($ϵ=0$), the fixed points of the system are determined by solving:

$$N^{*}=k·N^{*}\Rightarrow (1-k)N^{*}=0.$$

Hence, the system has a unique fixed point at $N^{*}=0$, which is:

• Stable for $0<k<1$,
• Marginally stable for $k=1$,
• Unstable for $k>1$.

When noise is introduced ($ϵ>0$), the concept of a fixed point becomes statistical. We instead examine the long-term average behavior across many realizations to determine whether the system stabilizes, fluctuates, or diverges.

4.2. Numerical Experiment and Results

We perform numerical simulations of the model for $ϵ=0.1$, initial condition $N\left(0\right)=1$, and three representative values of the inheritance parameter:

$$k\in \{0.5,1.0,1.5\}.$$

Each simulation was run for 100 iterations and averaged over 100 independent realizations.

4.3. Interpretation

Figure 3 confirms the theoretical expectations of the noisy inheritance model under different discrete inheritance intensities:

• For $k=1$, the system exhibits a bounded stochastic regime. The inheritance and noise balance out, producing a fluctuating but non-divergent trajectory. This reflects a marginally stable phase, similar to a noisy random walk.
• For $k=5$, the deterministic term $kN\left(l\right)$ begins to dominate over the noise, resulting in clear upward drift. The average trajectory starts to diverge, though at a slower exponential rate.
• For higher values such as $k=20,50,100,145$, the system rapidly escapes any bounded region. The exponential growth completely overshadows the noise term, and the average trajectory reflects deterministic divergence. This illustrates the system’s transition to an autocatalytic regime where inherited structure becomes explosively self-amplifying.

These findings highlight the critical role of k as a threshold parameter: when $k>1$, the system enters a regime of irreversible exponential growth. This suggests that in empathic, self-organizing, or trust-independent networks, even modest deterministic amplification can lead to rapid and uncontrolled expansion—unless bounded by structural, cognitive, or environmental constraints.

5. Phase Portraits of the Noisy Inheritance Model

To analyze the temporal structure of system trajectories under different inheritance intensities, we generate phase portraits for the model:

$$N(l+1)=k·N\left(l\right)+{\eta }_l,\mathrm{where}{\eta }_l=ϵ·{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1),$$

with noise amplitude fixed at $ϵ=0.1$ and initial condition $N\left(0\right)=1$. We vary the inheritance parameter across a wide range $k\in \{1.5,5,20,50,100,145\}$ and simulate each trajectory over 100 iterations.

Interpretation

The phase portraits in Figure 4 illustrate the evolution of the noisy inheritance model across discrete empathic steps $l\in \mathbb{N}$ for several whole-number inheritance rates $k\in \{1,5,20,50,100,145\}$. The behavior diverges dramatically depending on whether the system is near, moderately above, or far above the critical threshold $k=1$.

• For $k=1$: The system behaves as a noisy random walk. Since the deterministic component is neutral (no growth or decay), the evolution is dominated by fluctuations due to ${\eta }_l\sim \mathcal{U}(-ϵ,ϵ)$. This reflects marginal stability and high sensitivity to initial conditions and noise.
• For $k=5$: We observe the beginning of exponential divergence. After a brief initial plateau, the deterministic amplification outweighs noise, leading to rapid nonlinear growth in $N\left(l\right)$.
• For $k\ge 20$: The model enters a strongly supercritical regime. Growth becomes explosive within a few empathic steps. The additive noise becomes negligible compared to the exponential term, and the system saturates numeric space extremely quickly.
• For large values like $k=50,100,145$: The trajectory reaches astronomical scales by iteration $l=100$, revealing the system’s autocatalytic inevitability. Even minimal empathic inheritance quickly escalates to global saturation, modeling the diffusion potential of recursive trust-irrelevant systems.

These results reinforce the mathematical claim that any $k>1$ generates unavoidable divergence under deterministic inheritance. The higher the value of k, the earlier this divergence occurs — which is critical when designing systems for information spread, epidemic containment, or crisis-resilient communication. The portraits also demonstrate that despite the stochastic term ${\eta }_l$, the qualitative dynamics are dominated by the inheritance law once k exceeds unity.

This supports the claim that deterministic inheritance—when not counterbalanced by saturation or regulatory dynamics—leads to autocatalytic growth with critical implications for both network design and cryptographic applications [4,11].

5.1. Implications for Self-Organizing Networks

Adding noise introduces critical behavior and complex dynamics with phase transition similarities into self-organizing networks [4,17]. This type of transition is extremely significant in:

• Knowledge diffusion with uncertainty: where message copying is occasionally corrupted [12].
• Crisis networks: where speed is important even when there is signal noise and corrupt links [19].
• Biological or neural systems: where development is active but not necessarily deterministic [5,22].

Such a stochastic extension renders the model more realistic without changing its mathematical form. It also provides the foundation for entropy-based statistical analysis, and we talk about it in the following section.

6. Noisy Inheritance Model Bifurcation Analysis

To study the onset of chaos in the self-organizing networks context, we examine a noisy version of our proposed deterministic model. The system is defined by the evolution rule:

$$N\left(l\right)=k^l+{\eta }_l,$$

where $k\in [0.5,4.0]$ is the inheritance parameter and ${\eta }_l\sim \mathcal{U}(-ϵ,ϵ)$ is an amplitude $ϵ=0.1$ uniform noise component. This formula entails both deterministic core of inheritance-based growth and random perturbations arising in the dynamics of real networks, like spontaneous contacts or environmental fluctuations.

Figure 5 illustrates a definite qualitative trend with changing regimes of the inheritance parameter k. For subcritical cases (with $k<1$), the term $k^l$ is exponentially small, so stochastic noise becomes the dominant dynamics. This results in a horizontal band of values clumped around zero, i.e., network size fluctuates around small scales solely due to randomness.

In the crossover point $k=1$, there is a dynamical threshold. Here, the deterministic component grows linearly with l, to counteract the influence of noise and create a perturbation- and initial-condition-sensitive system—a hallmark of self-organized criticality.

In the supercritical regime ($k>1$), the deterministic growth $k^l$ faster and faster dominates, and $N\left(l\right)$ diverges. Due to the extremely rapid growth, the values on the plot are outside the range of visualization, creating the impression of the disappearance of structure in the bifurcation plot. Such a limitation necessitates log-scaling or normalization in order to show the whole dynamic complexity.

More generally, this bifurcation analysis exhibits a phase transition in network growth dynamics under the influence of the inheritance factor k, exhibiting areas of randomness, criticality, and deterministic growth. This effect is well away from the traditional probabilistic or equilibrium paradigm, emphasizing the emergent chaotic features introduced by our stochastic perturbation and deterministic inheritance model.

6.1. Bifurcation Behavior in the Low-k Regime and Implications for Image Encryption

In this experiment, we investigate the dynamics of the noisy exponential growth model

$$N\left(l\right)=k^l+{\eta }_l,{\eta }_l\sim \mathcal{U}(-ϵ,ϵ),$$

in the low-inheritance regime where $k\in [0.05,0.9]$. Here, $ϵ=0.1$ controls the uniform noise amplitude, and the simulation is executed over 30,000 iterations, of which the final 100 values of $N\left(l\right)$ are plotted for each value of k, sampled over 500 evenly spaced steps. The resulting bifurcation diagram is shown in Figure 6.

The graph indicates a dense, irregular point set distribution across the entire range of k, with no observable periodic windows—high entropy and inheritance parameter sensitive dependence being the hallmark [21]. Such chaotic behavior, particularly the randomness and unpredictability of $N\left(l\right)$, are highly suggestive of patterns employed in chaotic-based image encryption methods [23]. Within such schemes, pixel values are typically permuted or mapped based on trajectories generated by chaotic maps, in which small variations in parameters lead to extensive state divergence—offering cryptographic diffusion and confusion [22].

By controlling k and the noise level $ϵ$, the system’s entropy features can be varied and hence tailored to encryption requirements such as key sensitivity, avalanche effect, and resistance to statistical effects [11]. The analogy of the current system’s behavior with classical models of encryption like the logistic or tent map opens a door to the use of this exponential inheritance model as a new hope for secure communication schemes [7].

7. Entropy Analysis of the Noisy Inheritance Model

Entropy serves as a central concept in characterizing disorder and unpredictability in complex systems [5,17]. In the context of stochastic discrete-time dynamical systems, such as our noisy inheritance model,

$$N(l+1)=k·N\left(l\right)+{\eta }_l,{\eta }_l=ϵ·{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1),$$

Shannon entropy offers a quantitative measure of how dispersed or uncertain the system’s evolution becomes due to the combined effects of deterministic amplification and random perturbations [12].

In particular, Shannon entropy:

• Measures the distribution of outcomes over time in terms of informational uncertainty;
• Helps assess how quickly and unpredictably a system evolves under variation in parameters;
• Provides insight into the cryptographic or communication-theoretic potential of a given process [11,23].

We compute the entropy of trajectories $\left\{N\right(l\left)\right\}$ for various integer values of the inheritance parameter k over 100 iterations using a logarithmic binning of the transformed values $log(1+N(l\left)\right)$. The probability distribution of occurrences across bins is used to calculate the Shannon entropy $H=-\sum p_i{log}_2{p}_i$ [21].

Interpretation

Figure 7 provides a compelling visualization of how unpredictability—quantified via Shannon entropy—varies with inheritance intensity:

• Low k regime (e.g., $k=1$): Entropy remains low ( 4.95 bits), consistent with marginally stable dynamics where the additive noise has only moderate impact and the system remains in a weakly fluctuating state.
• High k regime ($k\ge 20$): Entropy increases sharply and plateaus around 5.62 bits. This reflects a transition into stochastic divergence, where deterministic growth is fast enough to amplify the effect of even small noise, producing a highly disordered trajectory.

These findings validate the bifurcation-based and Lyapunov analysis [17,22,23]: once $k>1$, the deterministic structure expands exponentially, and the presence of even bounded randomness leads to high entropic complexity. Importantly, this plateau also implies a potential maximum entropy under fixed noise amplitude $ϵ$, supporting the idea that k can act as an entropy-control knob.

This behavior aligns with known principles in secure communication and chaotic encryption [25,29]. Systems with high entropy trajectories exhibit superior diffusion and confusion properties, making the noisy inheritance model applicable to domains requiring controlled unpredictability.

8. Quantum Entropy Extension of the Noisy Inheritance Model

The noisy inheritance model’s sensitivity to initial conditions in the system and amplification of perturbations, quantified by classical Shannon entropy, suggests an extension to quantum systems. More specifically, the sensitivity of the system’s initial conditions and the amplification of disturbances for the model render it well adapted to explaining entropy increase in quantum computation and quantum thermodynamics [25,26,30].

To this end, we consider how the model can be interpreted in a quantum setting with the aid of quantum entropy, namely von Neumann entropy:

$$S\left(\rho \right)=-\mathrm{Tr}(\rho log\rho ),$$

where $\rho$ is the density matrix for a quantum system encoding the evolving state $N\left(l\right)$. This entropy is the quantum analogue of Shannon entropy and carries with it both classical uncertainty and quantum entanglement [29].

Conceptual Mapping

We recommend depicting the state $N\left(l\right)$ as the occupation number or amplitude of a quantum register:

• The deterministic update $N(l+1)=kN\left(l\right)$ can be expressed in terms of a unitary scaling operator (e.g., a quantum walk or controlled-phase shift).
• The random component ${\eta }_l$ may be introduced via controlled quantum randomness—either via measurement feedback or noisy channel.
• The density matrix ${\rho }_l$ produced at each step corresponds to the statistical mixture or entangled evolution of the system [30].

Entropy Measurement and Interpretation

Computing $S\left({\rho }_l\right)$ at each empathic step l, we can:

• Track the rise in quantum informational complexity in comparison to classical entropy plots;
• Identify phase transitions where quantum action switches from coherence to decoherence;
• Compare if entropic saturation takes place under bounded quantum noise, just like the Shannon entropy plateau in Section Figure 7 [25,29].

This correspondence allows the inheritance model to be studied as a quantum dynamical system—providing applications in:

• Quantum circuit complexity,
• Modeling decoherence in open quantum systems [30],
• Quantum chaos and information scrambling [28],
• Quantum cryptography where entropic unpredictability is vital [27].

This quantum generalization of the classical model unlocks the door to simulate trust-insensitive or empathic dynamics in systems of quantum communication and computation. Future work will be comprised of applying the model to quantum simulators (e.g., Qiskit, Cirq) and measuring entropy via partial trace operations in order to isolate $S\left(\rho \right)$ through time.

9. Theoretical Computation of von Neumann Entropy via Quantum Chaos Approach

To extend the entropy calculation of our noisy inheritance model to the quantum domain, we reformulate the recursive dynamics

$$N(l+1)=kN\left(l\right)+{\eta }_l,{\eta }_l=ϵ·{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1)$$

as a quantum dynamical system driven by deterministic amplification and stochastic perturbation. In this theoretical extension, we use notions of quantum chaos and quantum information theory to derive an entropic approach suitable to study the complexity evolution of the system in Hilbert space [26,28,30].

Quantum-State Representation and Mixed Dynamics

We begin by casting the classical trajectory $N\left(l\right)$ into a quantum state vector in a discrete Hilbert space:

$$|{\psi }_l\rangle =\sum _{i=1}^n{\alpha }_i^{\left(l\right)}|i\rangle ,$$

where the amplitudes ${\alpha }_i^{\left(l\right)}$ are normalized complex coefficients representing occupation levels over quantized states $|i\rangle$, which correspond to bins or energy-like levels of the system. The deterministic component $kN\left(l\right)$ is expressed in terms of a unitary operator ${\widehat{U}}_k$, and the stochastic perturbation ${\eta }_l$ through a decohering noise channel ${\mathcal{N}}_{ϵ}$, e.g., a depolarizing or amplitude-damping map [30].

Each time step results in a mixed state in the form of a density matrix:

$${\rho }_l=\sum _j{p}_j^{\left(l\right)}|{\psi }_j^{\left(l\right)}\rangle \langle {\psi }_j^{\left(l\right)}|,$$

corresponding to an ensemble of possible system evolutions with respective probabilities $p_j^{\left(l\right)}$ [25,29].

Entropy via Spectral Decomposition

The von Neumann entropy of the system at time step l is:

$$S\left({\rho }_l\right)=-\mathrm{Tr}({\rho }_{l}log{\rho }_l),$$

which can be computed by diagonalizing ${\rho }_l$ and summing its eigenvalues $\left\{{\lambda }_i^{\left(l\right)}\right\}$:

$$S\left({\rho }_l\right)=-\sum _{i=1}^n{\lambda }_i^{\left(l\right)}log{\lambda }_i^{\left(l\right)}.$$

In the classical limit when ${\rho }_l$ is diagonal and purely stochastic, this reduces to the Shannon entropy [25]. However, in the quantum situation, coherence and entanglement create off-diagonal structure and lead to an increase in entropy due to superposition and interference [27,30].

Connection to Quantum Chaos and Complexity Growth

The entropy dynamics under such combined dynamics is closely related to results on quantum chaos. As shown by Zurek and Paz [30], and more formally proved in recent works on quantum ergodicity and operator spreading, quantum chaotic systems exhibit linear increase of entropy at short times, saturating to a value determined by the effective dimension of the Hilbert space:

$$S\left({\rho }_l\right)\lesssim logd,$$

where d is the dimension of the Hilbert space, here a function of the number of occupation bins used in discretizing $N\left(l\right)$.

In this context, the entropy plateau of the classical model is the analogue of the quantum entropy saturation limit, where the system is maximally mixed within its accessible subspace.

Theoretical Implications

This theoretical advance implies that:

• The von Neumann entropy in the quantum formulation of the inheritance model captures both phase coherence loss and statistical disorder;
• The point of saturation and entropic rate of increase reflect the trade-off between quantum noise ${\mathcal{N}}_{ϵ}$ and deterministic inheritance k;
• The model belongs to a family of quantum stochastic processes where dynamical entropy characteristics are evident, with applications in quantum thermodynamics and scrambling [27,30].

Thus, our model connects discrete-time classical chaos and quantum information complexity. It opens up the possibility of future quantum simulation using open-system quantum maps or Lindblad dynamics, where $S\left({\rho }_l\right)$ can be directly measured from quantum circuits via partial trace operations or tomography [25,29].

10. Time Series Analysis of the Stochastic Inheritance Model

We now perform a time series analysis of the stochastic inheritance model governed by the recursive relation:

$$x_{l+1}=k{x}_l+ϵ{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1),$$

where $k>1$ controls deterministic amplification and $ϵ>0$ sets the noise amplitude. This model exhibits both exponential growth and randomness, making it suitable for time series characterization techniques. We focus on key statistical features such as stationarity, autocorrelation, and spectral density under various values of k and $ϵ$.

10.1. Simulated Time Series

We simulate the time series $\left\{x_l\right\}$ for 100 iterations using different values of the inheritance parameter k and fixed noise amplitude $ϵ=0.1$. The initial condition is set to $x_0=1$. For each configuration, we generate 100 realizations and compute the mean trajectory.

Figure 9. Simulated trajectories of the stochastic inheritance model for $k=5$, $ϵ=0.1$. The gray lines show individual realizations; the red line indicates the mean over 100 trials.

Figure 9. Simulated trajectories of the stochastic inheritance model for $k=5$, $ϵ=0.1$. The gray lines show individual realizations; the red line indicates the mean over 100 trials.

11. Autocorrelation Analysis for the Stochastic Inheritance Model

To quantify memory and persistence in the stochastic inheritance model, we analyze the autocorrelation function (ACF) for a single realization of the process:

$$x_{l+1}=k{x}_l+ϵ{\xi }_l,{\xi }_l\sim \mathcal{U}(-1,1),$$

with parameters $k=5$, $ϵ=0.1$, and initial condition $x_0=1$. We simulate the process for $L=100$ iterations and compute the ACF up to lag 20 using the FFT-based method.

Discussion and Analysis

As shown in Figure 10, the autocorrelation decays sharply after just a few lags, indicating a lack of long-term memory in the process. This behavior is consistent with the exponential amplification driven by the deterministic term $k{x}_l$, which rapidly overshadows past values. The noise term $ϵ{\xi }_l$, although bounded, contributes only marginally to persistence due to the dominance of deterministic growth.

This confirms that for values of $k>1$, the system transitions quickly to a regime dominated by growth rather than recurrence or feedback, and the time series becomes essentially nonstationary. These results align with the entropy and Lyapunov-based interpretations developed in earlier sections.[31,34]

12. Dynamical Properties: Time Series Analysis with Gaussian Noise

To further investigate the behavior of our stochastic autocatalytic model, $x_{l+1}=k{x}_l+ϵ{\xi }_l$, we performed numerical simulations incorporating Gaussian noise. In this setup, ${\xi }_l$ is sampled from a standard normal (Gaussian) distribution with a mean of zero and a standard deviation of one, and is scaled by the noise amplitude $ϵ$. [36]

12.1. Simulation Setup

We simulated 100 independent trajectories, each for 50 time steps. The inheritance parameter was set to $k=1.1$, and the noise amplitude was $ϵ=0.1$. All trajectories commenced from an initial value of $x_0=1.0$.

12.2. Time Series Plot (Figure 1: Stochastic Inheritance Model: Gaussian Noise Trajectories)

The generated plot (Figure 11) displays a selection of 10 representative trajectories.

12.3. Commentary and Analysis

The time series plot reveals several key characteristics of the model’s dynamics under the influence of Gaussian noise:

• Underlying Exponential Growth: Consistent with the deterministic component of the model ($k=1.1>1$), all trajectories exhibit a clear trend of exponential growth. This reinforces the autocatalytic nature of the system, where the current state contributes multiplicatively to the subsequent state.[33,35]
• Significant Divergence Due to Gaussian Stochasticity: While the growth trend is shared, the individual trajectories diverge substantially from one another as time progresses. The use of Gaussian noise, which allows for theoretically unbounded deviations (though with decreasing probability), contributes to a potentially wider spread of outcomes compared to strictly bounded uniform noise, especially at later time steps when the $k{x}_l$ term amplifies even small initial noise perturbations. This highlights the inherent unpredictability in the exact future state of any given realization.
• Path Dependence and Amplification of Noise: The divergence demonstrates a strong path dependence: minor differences introduced by the random noise in early steps are significantly amplified by the exponential growth mechanism, leading to widely disparate values of $x\left(l\right)$ in later steps. This property is crucial for understanding how small, random fluctuations can lead to large-scale differences in the long-term evolution of self-organizing systems.
• Implications for Chaos and Complexity: For these parameters, the system does not converge to a bounded attractor; rather, it continuously grows with increasing variability. The ’chaos’ in this context arises not from boundedness and recurrence but from the extreme sensitivity to initial conditions and the stochastic element, leading to a complex and unpredictable ensemble of possible trajectories. This behavior is fundamental to understanding emergent complexity in systems where deterministic growth interacts with random influences.

This analysis visually supports the role of stochasticity in shaping the dynamics of autocatalytic growth, providing empirical context for the theoretical discussions of entropy amplification and network self-organization within our framework.