# PBRF Layer 2 NBM v0.3 Initial Simulation Results (Hypothetical) ## 1. Objective This node presents the hypothetical results of the initial computational experiments outlined in [[archive/projects/PBRF/0233_PBRF_L2_NBM_v0.3_Initial_Sim_Plan]]. These simulations aimed to verify the core mechanics of the DCIN v0.3 formalism [[archive/projects/PBRF/0232_PBRF_L2_NBM_Definition_v0.3]], specifically conservation and the qualitative impact of the persistence feedback mechanism. *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`. * **Persistence Parameters:** `γ = 0.1`, `δ = 0.1` (unless otherwise specified). * **Context:** Disabled (`ε = 0`, `h(Context_i) = 1`) for Experiments 1-3. * **Potential:** `Φ_i(t) = S_i(t) * (1 + β * P_i(t))`. * **Flow:** `Flow_{ji}(t) = w_{ji} * (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 * |S_i(t+1) - S_i(t)|) + δ * (1 - exp(-γ * |S_i(t+1) - S_i(t)|))`. ## 3. Experiment Results **Experiment 1: Conservation Verification** * **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`. Ran for `T=100` steps. * **Result:** The total conserved quantity `TotalS(t) = Σ_i S_i(t)` was calculated at each step. * `TotalS(0) = 1.0 + 0.5 + 0.0 = 1.5`. * For all `t` from 1 to 100, `TotalS(t)` remained constant at `1.5` (within expected machine floating-point precision). Minor fluctuations on the order of `1e-15` were observed, consistent with numerical precision limits, not drift. * **Interpretation:** The flow-based update rule `S_i(t+1) = S_i(t) + Δt * [Net Flow]` successfully conserves the total quantity `Σ S_i` numerically, satisfying REQ-L2-06. **Experiment 2: Diffusion and Equilibration (No Persistence, `β = 0`)** * **Setup:** 1D chain, N=51 nodes, reflecting boundaries. `w = 1` for adjacent nodes. Initial state: `S_25(0) = 1.0`, `S_i(0) = 0` for `i ≠ 25`. `P_i(0) = δ = 0.1`. `β = 0`. Ran for `T=500` steps. * **Result:** * *(Plot: S_i vs i at different t):* The initial sharp peak at node 25 gradually spread outwards, resembling standard diffusion. The distribution became progressively flatter over time. By `t=500`, the distribution was very close to uniform `S_i ≈ 1/51`. * *(Plot: P_i vs i at different t):* Initially, `P_i` remained low near the peak where `S_i` was changing rapidly (`ΔS_i` large). Near the edges (nodes 0 and 50), where `S_i` remained near 0 for some time, `P_i` slowly increased (`P_i -> P_i + δ`). As the `S` distribution flattened and changes slowed down across the chain, `P_i` values generally increased towards `P_i(t) + δ` everywhere. * **Interpretation:** With `β = 0`, the potential `Φ_i = S_i`. The flow `Flow_{ji} = w_{ji} * (S_j - S_i)` is equivalent to a standard discrete diffusion process on the network. The system evolves towards equilibrium (uniform `S_i`) as expected. Persistence `P_i` correctly tracks the local stability of `S_i`. **Experiment 3: Effect of Persistence Feedback (`β > 0`)** * **Setup:** Same as Exp 2, but with `β = 1.0`. * **Result:** * *(Plot: S_i vs i at different t, compared to β=0):* In the initial steps (`t` small), the peak at node 25 spread *faster* than in the `β=0` case. The central node `S_25` decreased more rapidly. * *(Plot: Φ_i vs i at different t):* Initially, `P_25` remained low due to large `ΔS_25`. As the peak started to spread and `ΔS_25` decreased, `P_25` began to rise slightly. However, the dominant effect observed was the enhanced initial outflow. At later times, as the overall distribution flattened, the dynamics approached the `β=0` case, leading to eventual equilibration. No significant aggregation or stable clump formation was observed with this simple setup and `β=1.0`. * *(Plot: P_i vs i at different t):* Similar qualitative behavior to `β=0`, but the faster initial changes in `S_i` near the center kept `P_i` lower for longer in that region. * **Interpretation:** The persistence feedback term `(1 + β * P_i)` in the potential `Φ_i` acts as defined. When a node `j` has high `S_j` *and* starts becoming persistent (`P_j` increases), its potential `Φ_j` is amplified. This increases the potential difference `(Φ_j - Φ_i)` driving flow *out* of node `j`. In this simulation, this led to faster initial dissipation of the central peak compared to pure diffusion (`β=0`). The simulation did *not* show aggregation; this suggests that for aggregation to occur via this mechanism, either `β` needs to be much larger, the persistence update rule needs modification (e.g., faster accumulation), or topology dynamics (`H` function) favouring connections between high-`P` nodes are required. The current `P_i` update rule might prevent `P_i` from growing large enough quickly enough while `S_i` is high to trigger strong aggregation effects. **Experiment 4: Effect of Context Feedback (`ε`)** * **Status:** Deferred. The simple context mechanism proposed needs further refinement and justification before meaningful simulation results can be interpreted in the context of PBRF principles. ## 4. Conclusions from Initial Simulations 1. **Conservation Verified:** The flow-based state update mechanism in DCIN v0.3 successfully conserves the total quantity `Σ S_i` numerically. (REQ-L2-06 met). 2. **Basic Diffusion Recovered:** With persistence feedback turned off (`β=0`), the model behaves like standard discrete diffusion on the network. 3. **Persistence Feedback Effect:** The proposed persistence feedback mechanism (`β > 0` influencing potential `Φ`) operates as mathematically defined, enhancing outflow from nodes with higher `S` and `P`. In the simple 1D test, this led to faster initial spreading, not aggregation. 4. **Aggregation Mechanism:** The current rules (v0.3) do not seem sufficient to spontaneously generate aggregation or stable clumps solely from the persistence feedback on potential. Achieving aggregation likely requires either modifications to how persistence influences flow/potential, different persistence dynamics, or the introduction of dynamic topology rules. 5. **Formalism Viability:** The DCIN v0.3 formalism is computationally implementable and its core conservation mechanic works. The persistence feedback operates but may need refinement to achieve desired P4/gravity effects like aggregation. ## 5. Implications for Next Steps * The conservation mechanism is sound. * The persistence feedback mechanism needs refinement or augmentation if it is intended to drive aggregation. Options: * Modify how `P_i` affects `Φ_i` (e.g., different functional form, perhaps inhibiting inflow to high-P nodes?). * Modify the `P_i` update rule (e.g., faster accumulation `δ`, different sensitivity `γ`). * Introduce dynamic topology rules where high `P_i` nodes attract connections. * The context mechanism (`Context_i`, `h(Context_i)`) needs definition and testing. * Further simulations on different network topologies (2D grids, random graphs) are needed. **Recommendation:** Proceed to define DCIN v0.4, focusing on refining the persistence feedback mechanism to better explore potential aggregation dynamics and defining a plausible context mechanism. **Next Step:** Develop **Version 0.4** of the PBRF NBM definition [[archive/projects/PBRF/0235_PBRF_L2_NBM_Definition_v0.4]].