# PBRF Layer 2 NBM v0.3 Initial Simulation Plan ## 1. Objective This document outlines the plan for initial computational experiments using the proposed PBRF Layer 2 formalism, the Dynamic Causal Influence Network (DCIN) v0.3, as defined in [[archive/projects/PBRF/0232_PBRF_L2_NBM_Definition_v0.3]]. The primary goals of these *initial* simulations are: 1. **Verification:** Numerically verify the implementation of the core update rules and the conservation mechanism (REQ-L2-06). 2. **Mechanism Exploration:** Observe the qualitative effects of the persistence feedback mechanism (`β` parameter) on the dynamics of the conserved quantity `S` (related to P4/gravity hypothesis). 3. **Basic Behavior:** Explore fundamental behaviors like diffusion/equilibration and potential pattern formation tendencies on simple network structures. 4. **Parameter Sensitivity:** Gain initial insights into the sensitivity of the dynamics to key parameters (`β`, `γ`, `δ`, `ε`). These simulations are not intended to reproduce complex physics at this stage but to build confidence in the formalism's basic mechanics and consistency, informing further refinement towards v1.0 and Layer 3 modeling. ## 2. Formalism Recap (DCIN v0.3) * **Network:** Directed graph `G=(V, E)` with static edge weights `w_ji ≥ 0`. * **States:** Node state `S_i ∈ ℝ` (conserved quantity density), Node persistence `P_i ∈ ℝ≥0`. * **Potential:** `Φ_i(t) = S_i(t) * (1 + β * P_i(t)) * h(Context_i(t))` (where `h` and `Context` need final specification, start simple). * **Flow:** `Flow_{ji}(t) = w_{ji} * (Φ_j(t) - Φ_i(t))` * **State Update:** `S_i(t+1) = S_i(t) + Δt * [Net Flow]` (inherently conservative). * **Persistence Update:** `P_i(t+1) = P_i(t) * exp(-γ * |ΔS_i|) + δ * (1 - exp(-γ * |ΔS_i|))` ## 3. Simulation Setup * **Environment:** Standard scientific computing environment (e.g., Python with NumPy/SciPy, potentially NetworkX for graph handling). * **Time:** Discrete sequence steps `t = 0, 1, 2, ...`. Set `Δt = 1`. * **Initial Conditions:** * `S_i(0)`: Sampled from a distribution (e.g., uniform random around a baseline, or specific initial patterns like a single high-`S` node). * `P_i(0)`: Typically initialized low (e.g., `P_i(0) = δ`). * `w_ji`: Defined by the chosen network topology (e.g., uniform weights for simple tests). * **Context Function `h(Context_i)`:** For initial tests, simplify or disable context dependence. **Option 1 (No Context):** Set `h(Context_i) = 1` (i.e., `ε = 0`). **Option 2 (Simple Context):** Use weighted average `Context_i(t) = Σ_{k∈N_{in}(i)} w_{ki} S_k(t) / Σ_{k∈N_{in}(i)} w_{ki}` and `h(Context_i) = (1 + ε * Context_i)` with a small `ε`. Start with Option 1. ## 4. Proposed Experiments **Experiment 1: Conservation Verification** * **Objective:** Numerically verify that `Σ_i S_i(t)` remains constant over time. * **Network:** Small, closed networks (no boundary nodes with external flow). Examples: * Two connected nodes (`1 -> 2`, `2 -> 1`). * A 3-node cycle (`1->2->3->1`). * A small random graph (e.g., 5-10 nodes) with fixed topology. * **Parameters:** Choose arbitrary initial `S_i(0)`, `P_i(0)`, `w_ji`, `β`, `γ`, `δ`. Set `ε = 0`. * **Procedure:** Run simulation for T steps (e.g., T=100). Calculate `TotalS(t) = Σ_i S_i(t)` at each step. * **Expected Outcome:** `TotalS(t)` should remain constant within machine precision. * **Analysis:** Plot `TotalS(t)` vs `t`. Check for drift. **Experiment 2: Diffusion and Equilibration (No Persistence)** * **Objective:** Observe basic diffusion-like behavior when persistence feedback is off. * **Network:** 1D chain (line graph) with N nodes (e.g., N=50). Edges `i -> i+1` and `i+1 -> i`. Uniform weights `w=1`. Boundary conditions: Reflecting (no flow out of ends) or Periodic. * **Parameters:** Set `β = 0` (no persistence influence on potential). Set `ε = 0`. Choose `γ`, `δ`. Initial condition: A single peak, e.g., `S_k(0) = 1` for central node `k`, `S_i(0) = 0` for `i ≠ k`. `P_i(0) = δ`. * **Procedure:** Run simulation until near equilibrium. * **Expected Outcome:** The initial peak of `S` should spread out along the chain, eventually approaching a uniform distribution (equilibrium `S_i = 1/N` for reflecting, `S_i = 1/N` for periodic). `P_i` should evolve based on local `ΔS_i`. * **Analysis:** Plot `S_i(t)` and `P_i(t)` profiles at different times `t`. Plot total `Σ P_i(t)`. **Experiment 3: Effect of Persistence Feedback (`β`)** * **Objective:** Observe how non-zero `β` affects the flow and potential aggregation of `S`. * **Network:** Same as Exp 2 (1D chain) or a 2D grid. * **Parameters:** Repeat Exp 2 setup, but now with `β > 0`. Vary `β` (e.g., small, medium, large values relative to other scales). Keep `ε = 0`. * **Procedure:** Run simulation. * **Expected Outcome (Hypothesis):** With `β > 0`, nodes with high `S` that become persistent (`P` increases) will develop higher potential `Φ`. This should enhance outflow from persistent high-`S` nodes and potentially inhibit inflow, counteracting simple diffusion. Could this lead to sharpening of peaks, formation of stable clumps, or other non-diffusive behavior? Does high `P` in low-`S` regions "resist" inflow? * **Analysis:** Compare `S_i(t)` and `P_i(t)` profiles with the `β = 0` case. Look for evidence of aggregation, peak sharpening, or slowed equilibration. Analyze the evolution of `Φ_i(t)`. **Experiment 4: Effect of Context Feedback (`ε`) (Preliminary)** * **Objective:** Observe the basic effect of the simple context mechanism. * **Network:** Small graph with interesting topology (e.g., a node with high vs low incoming degree). * **Parameters:** Use `Context_i(t) = Avg(S_k)` and `h(Context_i) = (1 + ε * Context_i)`. Set `β` to 0 or a fixed value. Vary `ε`. * **Procedure:** Run simulation with non-uniform initial `S_i`. * **Expected Outcome (Hypothesis):** Positive `ε` means high-context nodes have higher potential, potentially enhancing outflow. Negative `ε` would do the opposite. How does this interact with flow dynamics? * **Analysis:** Compare dynamics for different `ε`. Focus on nodes with different contexts. (This experiment is more exploratory as the context mechanism is less developed). ## 5. Outputs and Analysis * Time series data for `S_i(t)`, `P_i(t)`, `Φ_i(t)` for all nodes. * Calculation of total `Σ S_i(t)` (for Exp 1). * Plots of spatial profiles `S_i`, `P_i`, `Φ_i` at different times `t`. * Comparison plots for different parameter values (`β`, `ε`). * Visualizations of network state evolution (optional). ## 6. Scope and Limitations These are *initial* tests on simple structures with simplified context. They aim to verify basic mechanics and explore core hypotheses (conservation, persistence feedback). They do not aim to: * Model complex emergent geometry. * Incorporate dynamic topology. * Use sophisticated context mechanisms. * Reproduce specific physical phenomena quantitatively. Results will inform the refinement of the DCIN formalism (e.g., refining `f_flow`, `f_context`, parameter choices) before more complex simulations are attempted. **Next Step:** Implement the DCIN v0.3 model in a chosen simulation environment and execute the experiments outlined in this plan. Create node `0234_PBRF_L2_NBM_v0.3_Initial_Sim_Results`.