# PBRF Layer 2 NBM v0.7 Initial Simulation Plan ## 1. Objective This document outlines the plan for initial computational experiments using the PBRF Layer 2 formalism DCIN v0.7, as defined in [[archive/projects/PBRF/0244_PBRF_L2_NBM_Definition_v0.7]]. While the core update rules are unchanged from v0.6, the focus shifts based on the elaborated physical interpretation and the need to understand cluster dynamics. The goals are: 1. **Parameter Space Mapping:** Systematically explore the parameter space (`α_S, α_P, β, ε, λ, w_max`) to map different regimes of cluster formation (e.g., stable clusters, transient patterns, homogeneous state). 2. **Quantitative Cluster Characterization:** Develop and apply metrics to quantify the properties of emergent clusters (size, density, persistence, boundary characteristics, total `S` as mass analogue). 3. **Cluster Interaction Probes:** Use specific initial conditions (e.g., two pre-formed clusters) to observe qualitative interactions mediated by the existing flow and dynamic weight rules, informing the need for explicit cluster dynamics rules in later versions. 4. **Interpretation Refinement:** Use simulation results to refine the physical interpretation proposed in v0.7. ## 2. Formalism Recap (DCIN v0.7 - Rules same as 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`. `Δt = 1`. * **Default Parameters:** `γ = 0.1`, `δ = 0.1`. Others varied per experiment. * **Initial Conditions:** * `S_i(0)`: Uniform random noise `U(0, 0.1)` or specific patterns (e.g., two Gaussian peaks). * `P_i(0)`: `P_i(0) = δ`. * `w_ji(0)`: Uniformly low `w_ji(0) = 0.1`. * **Network Topology:** 2D grid (e.g., 40x40 or 50x50 for better statistics), nearest-neighbor (8) connections, periodic boundaries. ## 4. Proposed Experiments **Experiment 1: Parameter Sweep for Aggregation Regimes** * **Objective:** Map the parameter space (`α_S`, `α_P`, `λ`, `w_max`) to identify regions leading to stable clusters vs. homogeneous states vs. other patterns. * **Network:** 40x40 grid, periodic. * **Parameters:** Fix `β=0`, `ε=0`. Systematically vary: * `α_S` (e.g., 0.1 to 5.0) * `α_P` (e.g., 0.1 to 5.0) * `λ` (e.g., 0.001, 0.01, 0.1) * `w_max` (e.g., 5.0, 10.0, 20.0) (Use a grid or sampling method for the parameter space). Random initial `S_i(0)`. * **Procedure:** Run simulations for each parameter combination until a quasi-steady state (e.g., T=3000). * **Expected Outcome:** A phase diagram showing parameter regions corresponding to: No aggregation (uniform `S`), stable isolated clusters, network-spanning structures, potentially dynamic patterns. * **Analysis:** Classify final state morphology (visual inspection + quantitative metrics like variance of `S`, average cluster size). Plot phase diagram. **Experiment 2: Influence of `β` and `ε` on Cluster Morphology** * **Objective:** Investigate how resistance (`β`) and context (`ε`) modify the clusters formed by the aggregation mechanism. * **Network:** 40x40 grid, periodic. * **Parameters:** Choose a parameter set (`α_S, α_P, λ, w_max`) known from Exp 1 to produce stable clusters. Systematically vary: * `β` (e.g., 0, 0.5, 1.0, 5.0) with `ε=0`. * `ε` (e.g., -1.0, -0.5, 0, 0.5, 1.0) with `β=0`. Random initial `S_i(0)`. * **Procedure:** Run simulations to quasi-steady state (e.g., T=3000). * **Expected Outcome:** Refine understanding from v0.6 results. `β` should lead to potentially denser, more static clusters. Negative `ε` should lead to sharper boundaries/segregation. Positive `ε` might destabilize clusters or lead to different morphologies. * **Analysis:** Compare cluster statistics (size, density, persistence, boundary sharpness) across different `β` and `ε` values. Visualize final states. **Experiment 3: Quantitative Cluster Characterization** * **Objective:** Develop and apply metrics to characterize emergent clusters as potential particle analogues. * **Network:** 50x50 grid, periodic. * **Parameters:** Select a few parameter sets from Exp 1 & 2 that produce distinct, stable clusters. * **Procedure:** Run simulations to quasi-steady state. Identify individual clusters (e.g., connected components above an `S` threshold). * **Expected Outcome:** Quantitative data for individual clusters. * **Analysis:** For each identified cluster: * Calculate Total Mass Analogue: `M = Σ_{i∈Cluster} S_i`. * Calculate Average Persistence: `P_avg = Avg(P_i)` for `i∈Cluster`. * Calculate Size/Volume: `V = Number of nodes` in cluster. * Calculate Average Density: `S_avg = M / V`. * Calculate Boundary Properties: Average weight `w` of edges connecting cluster nodes to non-cluster nodes. * Study correlations between these properties (e.g., does `M` correlate with `P_avg`?). **Experiment 4: Cluster Interaction Probe (Two Clusters)** * **Objective:** Observe qualitative interactions between pre-formed clusters using existing rules. * **Network:** 50x50 grid, periodic. * **Parameters:** Choose a parameter set known to form stable clusters. * **Initial Condition:** Instead of random noise, initialize `S_i(0)` with two distinct Gaussian peaks separated by some distance. Initialize `w_ji(0)` low. * **Procedure:** Run simulation (e.g., T=3000). Observe the evolution of the two peaks/clusters. * **Expected Outcome (Hypothesis):** Depending on parameters (`α`, `λ`, `β`, `ε`) and separation distance: * Clusters might remain separate and stable. * Clusters might attract each other (due to `α` terms strengthening inter-cluster weights) and merge. * Clusters might repel or segregate further (if negative `ε` dominates). * Clusters might deform due to mutual influence. * **Analysis:** Visualize the time evolution. Track the distance between cluster centers-of-mass analogue. Analyze the evolution of edge weights *between* the clusters. ## 5. Outputs and Analysis * Parameter space map (phase diagram) from Exp 1. * Visualizations (heatmaps) of `S`, `P`, `w` for different parameter regimes and initial conditions. * Quantitative cluster statistics (M, P_avg, V, S_avg, boundary `w`) and their correlations (Exp 3, 5). * Time evolution plots/animations of cluster interactions (Exp 4). * Refined interpretation of parameters based on observed effects. ## 6. Scope and Limitations * Focus on characterizing clusters formed by v0.6 rules and probing interactions without explicit motion/interaction rules. * Static node topology. * Physical interpretation remains qualitative. * Limited exploration of initial conditions. Results will provide a detailed map of the pattern-forming capabilities of DCIN v0.6/v0.7, quantify the properties of emergent structures, and guide the development of explicit cluster dynamics rules and a more quantitative physical interpretation in v0.8. **Next Step:** Implement and execute these experiments. Create node `0246_PBRF_L2_NBM_v0.7_Initial_Sim_Results`.