stochastic implementation of cont approx

1. my goal is to make a 🗄️table comparing the models on process from each paper.[[Space/Sources/Lab/📜Kauffman22_theory_of_adj_possible]] 2. from Kauffman Theory of the Adjacent Possible.pdf "we note that Steel et al.’s stochastic implementation from Steel20_BDproc_comb_innov.pdf is not a discretisation of the TAP equation itself, because it requires a calculational timestep to be small to keep the creation probabilities below unity; it is a stochastic discretization of a continuum approximation to the original equation. Their numerical analysis also makes the further assumption of a ﬁxed upper limit (usually i max= 4) in the summation." 3. based on 1 and 2, my ultimate goal is to support or falsify my hypothesis. My hypothesis is inherent need to consider discretization time in continuous approximation simulation (e.g. Forrester's solution for compartment-based simulation: Sample at finite intervals (ST > DT) from process noise and feature noise.md) continuous dynamics may cause the gap between discrete and continuous model. This gap from stochastic discretization of continuum approximation is well explained in | Aspect | Steel et al. (2020) | Cortés et al. (2023) | | --------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Core Model Type | Stochastic birth-death process | Deterministic TAP equation | | Base Equation | Mt+1 = Mt(1 - μ) + ∑(αi(Mt choose i)) | Same base equation | | Key Parameters | - μ: extinction rate - αi = Pα^i: efficiency of i-way combinations - P ∈ (0,1]: overall efficiency - α: base rate | - μ: extinction rate - α: single combination rate | | Growth Pattern | Stochastic with possible extinction; E[T] and Var[T] derived | Deterministic with guaranteed finite-time explosion | | Implementation Steps | 1. Remove items: u ~ Poisson(μMt) 2. Calculate expected combinations: si = P×αi×(Mt choose i) 3. Generate new items: ri ~ Poisson(si) 4. Update: Mt+1 = Mt - u + ∑ri | Direct calculation: Mt+1 = Mt(1 - μ) + α(2^Mt - Mt - 1) | | Variable Definitions | - Mt: number of items at time t - si: expected new items from i-way combinations - ri: actual new items (stochastic) - u: number of items removed | - Mt: number of items at time t - No intermediate variables needed | | Continuum Treatment | Stochastic discretization of continuum approximation | Direct analysis of discrete equation | | Numerical Method | Poisson distribution sampling at each step | Direct iteration of TAP equation | | Time Step Constraints | si must stay < 1 for valid Poisson sampling, requiring small Δt | No constraints | | Combination Limit | i ≤ 4 to keep computation tractable | No limit | | Process Meaning | - u captures random extinctions - si represents potential new combinations - ri introduces randomness in successful combinations - Sum of ri gives total innovations | Single deterministic step combining: - Extinction: Mt(1 - μ) - All possible combinations: α(2^Mt - Mt - 1) |