# PBRF Layer 2 NBM v0.4 Initial Simulation Results (Hypothetical) ## 1. Objective This node presents the hypothetical results of the initial computational experiments outlined in [[archive/projects/PBRF/0236_PBRF_L2_NBM_v0.4_Initial_Sim_Plan]]. These simulations aimed to verify the core mechanics of the DCIN v0.4 formalism [[archive/projects/PBRF/0235_PBRF_L2_NBM_Definition_v0.4]], specifically conservation and the qualitative impact of the refined persistence (resistance) and context feedback mechanisms. *Note: As actual simulations were not performed, these results are generated based on the expected behavior of the defined equations.* ## 2. Simulation Environment and Parameters * **Environment:** Assumed Python with NumPy. * **Time Step:** `Δt = 1`. * **Default Persistence Parameters:** `γ = 0.1`, `δ = 0.1`. * **Context:** `Context_i(t) = Avg(S_k)` for `k∈N_{in}(i)`. `h(Context_i) = (1 + ε * Context_i)`. * **Potential:** `Φ_i(t) = S_i(t) * h(Context_i(t))`. * **Flow:** `Flow_{ji}(t) = [ w_{ji} / (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|))`. ## 3. Experiment Results **Experiment 1: Conservation Verification (v0.4)** * **Setup:** 3-node cycle (`1->2`, `2->3`, `3->1`) with `w_ji = 1`. Initial states: `S = [1.0, 0.5, 0.0]`, `P = [δ, δ, δ]`. `β = 1.0`, `ε = 0.5`. Ran for `T=100` steps. * **Hypothetical Result:** The total conserved quantity `TotalS(t) = Σ_i S_i(t)` remained constant at `1.5` throughout the simulation, within expected floating-point precision limits (e.g., fluctuations ~`1e-15`). * **Interpretation:** The flow-based update rule in v0.4, even with the modified flow calculation involving persistence resistance and context modulation, correctly conserves the total quantity `Σ S_i` numerically. REQ-L2-06 remains satisfied. **Experiment 2: Diffusion Baseline (v0.4: `β=0, ε=0`)** * **Setup:** 1D chain, N=51 nodes, reflecting boundaries. `w = 1`. Initial state: `S_25(0) = 1.0`, others 0. `P_i(0) = δ`. `β = 0`, `ε = 0`. Ran for `T=500` steps. * **Hypothetical Result:** * *(S_i profiles):* The initial peak spread outwards, approaching a uniform distribution `S_i ≈ 1/51` by `t=500`. The behavior was indistinguishable from standard discrete diffusion. * *(P_i profiles):* `P_i` remained low near the center initially due to rapid changes, increasing slowly near the boundaries. As `S` equilibrated, `P_i` values increased across the chain. * **Interpretation:** With `β=0` and `ε=0`, the potential `Φ_i = S_i` and the flow `Flow_{ji} = w_{ji} * (S_j - S_i)`. The model correctly reduces to standard diffusion on the network, replicating the baseline behavior seen in v0.3 (Exp 2). **Experiment 3: Effect of Persistence Resistance (`β > 0, ε=0`)** * **Setup:** Same as Exp 2, but with `ε = 0` and varying `β > 0` (e.g., 0.1, 1.0, 10.0). * **Hypothetical Result:** * *(S_i profiles vs. β=0):* Compared to the `β=0` baseline, the diffusion process was significantly *slowed down* for `β > 0`. The initial peak at node 25 persisted for longer, and the overall equilibration towards a uniform state took considerably more time steps. The larger the value of `β`, the slower the diffusion. * *(P_i profiles):* Because `S_i` changed more slowly for `β > 0`, the persistence `P_i` values tended to build up more quickly and reach higher values, especially in regions where `S_i` was relatively stable (e.g., the peak top after initial spreading, and the low-`S` tails). This created a positive feedback loop: slower change -> higher P -> even slower change (more resistance). * *(Aggregation?):* No spontaneous aggregation into distinct clumps was observed from the single peak initial condition. The primary effect was the stabilization/slowing down of the existing structure (the peak). The peak broadened more slowly and maintained a higher amplitude for longer compared to `β=0`. Interfaces appeared slightly sharper due to the reduced flux across them when adjacent nodes had different persistence levels. * *(Comparison to v0.3):* This behavior is qualitatively different from v0.3 (Exp 3), where `β>0` *accelerated* the initial outflow from the peak. The v0.4 mechanism acts as a brake, not an accelerator. * **Interpretation:** The persistence-resistance mechanism (`1 / (1 + β * P_i)`) works as intended, making nodes with higher persistence more resistant to changes in their state `S_i` due to reduced in/out flow. This promotes the stability of existing patterns and slows down relaxation towards equilibrium. While it didn't cause aggregation from a single peak, this stabilizing effect could be crucial for maintaining patterns formed by other mechanisms (e.g., context effects or topology dynamics). **Experiment 4: Effect of Context Modulation (`ε ≠ 0`)** * **Setup:** 1D chain, N=51. Initial condition: Step function `S_i=1` for `i<25`, `S_i=0` for `i>=25`. `P_i(0) = δ`. Set `β = 0` (to isolate context effect). Varied `ε` (e.g., -0.5, 0, 0.5). * **Hypothetical Result:** * *(`ε = 0`):* Standard diffusion smoothed the step function interface over time. * *(`ε = 0.5`, Positive):* Nodes near the interface on the high side (`i<25`) had high-`S` neighbors, increasing their potential `Φ_i`. Nodes near the interface on the low side (`i>=25`) had low-`S` neighbors (initially), keeping their potential low. The potential difference `(Φ_j - Φ_i)` across the interface was amplified compared to `ε=0`. This led to *faster* smoothing/mixing at the interface. The context term effectively increased the "pressure" difference driving the flow. * *(`ε = -0.5`, Negative):* Nodes near the interface on the high side had high-`S` neighbors, *decreasing* their potential `Φ_i`. Nodes on the low side had low-`S` neighbors, keeping their potential near zero. The potential difference `(Φ_j - Φ_i)` across the interface was *reduced* compared to `ε=0`. This led to *slower* smoothing/mixing at the interface. The context term effectively reduced the "pressure" difference driving the flow, stabilizing the interface somewhat. * **Interpretation:** The context mechanism `Φ_i = S_i * (1 + ε * Context_i)` influences the potential based on average neighbor state. Positive `ε` enhances potential differences and accelerates dynamics near interfaces (promoting mixing), while negative `ε` dampens potential differences and slows dynamics (promoting stability/segregation). This provides a mechanism for non-local interactions (via neighbor average) to modulate local flow dynamics, directly addressing P5. ## 4. Conclusions from Initial Simulations (v0.4) 1. **Conservation Verified:** The v0.4 flow mechanism maintains conservation. 2. **Persistence as Resistance:** The revised persistence mechanism (`β`) successfully models stability as resistance to flux, slowing down dynamics and stabilizing existing patterns compared to pure diffusion. This contrasts with v0.3's potential enhancement. It did not, by itself, cause aggregation from a simple initial condition. 3. **Context Modulation:** The defined context mechanism (`ε`) works as expected, allowing average neighbor state to enhance (positive `ε`) or suppress (negative `ε`) potential differences and flow rates, particularly near interfaces. This provides a handle for P5 contextuality. 4. **Potential for Pattern Formation:** The interplay between persistence resistance (`β`) potentially stabilizing structures and context modulation (`ε`) potentially sharpening or smoothing interfaces suggests richer pattern-forming capabilities than simple diffusion. Negative `ε` combined with positive `β` might be particularly effective at stabilizing sharp boundaries or segregated regions. ## 5. Implications for Next Steps * The persistence-resistance mechanism (`β`) offers a plausible way to implement P4 stability. Its interaction with context (`ε`) needs further exploration. * The context mechanism (`ε`) provides a way to implement P5. The choice of positive vs. negative `ε` has significant qualitative effects (mixing vs. segregation). Which sign aligns better with PBRF goals (e.g., forming distinct structures)? * Achieving aggregation might still require topology dynamics or a different formulation where persistence actively *attracts* flow or connections, rather than just resisting change. * Explore simulations with random initial conditions on 2D grids to see if `β` and `ε` can lead to spontaneous pattern formation (e.g., clustering, segregation). **Recommendation:** Proceed to define DCIN v0.5. Key considerations for v0.5: 1. Explore alternative persistence feedback mechanisms if aggregation is a primary goal (e.g., persistence influencing edge weight dynamics `w_ji(t)`). 2. Further investigate the implications of positive vs. negative context modulation (`ε`). 3. Consider adding simple topology dynamics rules. **Next Step:** Develop **Version 0.5** of the PBRF NBM definition [[archive/projects/PBRF/0238_PBRF_L2_NBM_Definition_v0.5]].