# PBRF Layer 2 NBM v0.6 Initial Simulation Plan ## 1. Objective This document outlines the plan for initial computational experiments using the refined PBRF Layer 2 formalism, the Dynamic Causal Influence Network (DCIN) v0.6, as defined in [[archive/projects/PBRF/0241_PBRF_L2_NBM_Definition_v0.6]]. The primary goals are: 1. **Verification:** Verify conservation (`Σ S_i`) with the saturated weight update rule. 2. **Saturation Effects:** Investigate the impact of the maximum edge weight `w_max` on aggregation dynamics and cluster properties. 3. **Aggregation Drivers:** Compare the effectiveness and resulting morphology of state-driven (`α_S`) versus persistence-driven (`α_P`) aggregation. 4. **Parameter Interplay:** Further explore the combined effects of aggregation (`α`), resistance (`β`), and context (`ε`) parameters on pattern formation and stability. 5. **Cluster Characterization:** Begin quantifying the properties (size, density, persistence, boundary weights) of emergent clusters. ## 2. Formalism Recap (DCIN v0.6) * **Network:** Directed graph `G=(V, E)` with dynamic edge weights `0 ≤ w_ji(t) ≤ w_max`. * **States:** Node state `S_i ∈ ℝ`, Node persistence `P_i ∈ ℝ≥0`. * **Context:** `Context_i(t) = Avg(S_k)` over incoming neighbors (using `w_{ki}(t)`). * **Potential:** `Φ_i(t) = S_i(t) * (1 + ε * Context_i(t))`. * **Flow:** `Flow_{ji}(t) = [ w_{ji}(t) / (1 + β * P_i(t)) ] * (Φ_j(t) - Φ_i(t))`. * **State Update:** `S_i(t+1) = S_i(t) + Δt * [Net Flow]`. * **Persistence Update:** `P_i(t+1) = P_i(t) * exp(-γ * |ΔS_i|) + δ * (1 - exp(-γ * |ΔS_i|))`. * **Weight Update:** `Growth = α_S*S_j*S_i + α_P*P_j*P_i`. `Decay = λ*w_{ji}`. `w_{ji}(t+1) = min( w_max, max(0, w_{ji}(t) + Δt * [ Growth - Decay ]) )`. ## 3. Simulation Setup * **Environment:** Standard scientific computing environment. * **Time:** Discrete steps `t = 0, 1, 2, ...`. `Δt = 1`. * **Default Parameters:** `γ = 0.1`, `δ = 0.1`, `β = 0`, `ε = 0`, `α_S = 0`, `α_P = 0`, `λ = 0.01`, `w_max = 10.0` (unless varied). * **Initial Conditions:** * `S_i(0)`: Uniform random noise `U(0, 0.1)` unless specified. Ensure `Σ S_i(0)` is tracked. * `P_i(0)`: `P_i(0) = δ`. * `w_ji(0)`: Uniformly low `w_ji(0) = 0.1`. * **Network Topology:** 2D grid (e.g., 30x30 for better cluster stats) with nearest-neighbor connections (8 neighbors, Moore), periodic boundaries. ## 4. Proposed Experiments **Experiment 1: Conservation Verification (v0.6)** * **Objective:** Verify `Σ S_i` conservation with `w_max` active. * **Network:** Small closed network (e.g., 5x5 grid, periodic). * **Parameters:** Activate all mechanisms (`α_S>0, α_P>0, β>0, ε≠0, λ>0, w_max` finite). Random initial `S_i(0)`. * **Procedure:** Run T=200 steps. Track `TotalS(t) = Σ_i S_i(t)`. * **Expected Outcome:** `TotalS(t)` constant within machine precision. * **Analysis:** Plot `TotalS(t)`. **Experiment 2: Effect of Weight Saturation (`w_max`)** * **Objective:** Investigate how limiting connection strength affects aggregation. * **Network:** 30x30 grid, periodic. * **Parameters:** Set strong aggregation drive (e.g., `α_S = 1.0, α_P = 0.5, λ = 0.01`). Set `β=0, ε=0`. Vary `w_max` (e.g., 1.0, 5.0, 10.0, 50.0, Infinity). Random initial `S_i(0)`. * **Procedure:** Run simulation until quasi-steady state (e.g., T=2000). * **Expected Outcome (Hypothesis):** Lower `w_max` might limit the size or density of clusters, potentially leading to more numerous smaller clusters or preventing runaway aggregation. High `w_max` should resemble v0.5 behavior. The distribution of final weights `w_ji` will be capped by `w_max`. * **Analysis:** Visualize final `S` and `w` distributions for different `w_max`. Compare cluster statistics (size, number, density). Plot histogram of final `w_ji` values. **Experiment 3: `α_S` vs `α_P` Dominance** * **Objective:** Compare aggregation driven primarily by state vs. persistence. * **Network:** 30x30 grid, periodic. * **Parameters:** Set `β=0, ε=0, λ=0.01, w_max=10.0`. Compare three regimes: * State-Dominant: `α_S = 1.0`, `α_P = 0.1` * Persistence-Dominant: `α_S = 0.1`, `α_P = 1.0` * Balanced: `α_S = 1.0`, `α_P = 1.0` Random initial `S_i(0)`. * **Procedure:** Run simulation until quasi-steady state (e.g., T=2000). * **Expected Outcome (Hypothesis):** State-dominant might form clusters faster but potentially less stable initially. Persistence-dominant might form clusters slower but reinforce stable structures more strongly. Balanced might be most robust. Morphology might differ. * **Analysis:** Visualize final `S`, `P`, `w` distributions. Compare cluster statistics and formation time course. **Experiment 4: Interplay (`α, β, ε`)** * **Objective:** Explore combined effects on established clusters. * **Network:** 30x20 grid, periodic. * **Parameters:** Use a parameter set known to form clusters (e.g., Balanced `α_S=1, α_P=1, λ=0.01, w_max=10.0`). Run initial phase to form clusters (e.g., T=1000). Then, introduce `β` or `ε` and continue simulation: * Case A: Introduce `β = 1.0` (resistance). * Case B: Introduce `ε = -0.5` (segregating context). * Case C: Introduce `β = 1.0` AND `ε = -0.5`. * Control: Continue with `β=0, ε=0`. * **Procedure:** Run second phase (e.g., T=1000 more steps). * **Expected Outcome (Hypothesis):** `β` should slow down further evolution and potentially stabilize cluster boundaries. `ε` should enhance segregation, potentially sharpening boundaries or preventing merging. Combined effect should be strongest stabilization/segregation. * **Analysis:** Compare cluster morphology, boundary sharpness, and internal dynamics (e.g., fluctuations in `S` within clusters) across cases A, B, C, and Control. **Experiment 5: Cluster Characterization** * **Objective:** Quantify properties of emergent clusters. * **Network:** 30x30 grid, periodic. * **Parameters:** Use parameter sets from Exp 3 and Exp 4 that produce distinct stable cluster patterns. * **Procedure:** Run simulations to quasi-steady state. Identify clusters (e.g., using a density threshold on `S`). * **Expected Outcome:** Obtain quantitative data on cluster properties. * **Analysis:** For each parameter set: * Calculate cluster size distribution (number of nodes per cluster). * Calculate average internal density `Avg(S_in)`. * Calculate average internal persistence `Avg(P_in)`. * Calculate average internal edge weight `Avg(w_in)` and average boundary edge weight `Avg(w_boundary)`. * Relate these properties to the simulation parameters (`α_S`, `α_P`, `β`, `ε`, `w_max`). ## 5. Outputs and Analysis * Time series data for `S_i(t)`, `P_i(t)`, `w_ji(t)`. * Calculation of total `Σ S_i(t)` (for Exp 1). * Visualizations (heatmaps) of `S`, `P`, `w` distributions at key time points and final states. * Histograms of final `w_ji` values (Exp 2). * Cluster statistics: size distribution, average internal `S`, `P`, `w`, boundary `w`. * Comparison plots across different parameter regimes. ## 6. Scope and Limitations * Focus on testing v0.6 refinements (saturation, `α_S`/`α_P` comparison, interplay) on 2D grids. * Static node topology (no creation/deletion of nodes or edges, only weight changes). * Limited parameter space exploration. * Physical interpretation remains preliminary. Results will refine understanding of aggregation dynamics, the role of different parameters, and provide quantitative characteristics of emergent structures, informing v0.7 and the ongoing physical interpretation. **Next Step:** Implement the DCIN v0.6 model and execute these experiments. Create node `0243_PBRF_L2_NBM_v0.6_Initial_Sim_Results`.