# PBRF Layer 2 NBM v0.7 Initial Simulation Results (Hypothetical) ## 1. Objective This node presents the hypothetical results of the computational experiments outlined in [[archive/projects/PBRF/0245_PBRF_L2_NBM_v0.7_Initial_Sim_Plan]]. These simulations, using the DCIN v0.7 formalism [[archive/projects/PBRF/0244_PBRF_L2_NBM_Definition_v0.7]] (rules identical to v0.6), aimed to map parameter regimes for cluster formation, quantitatively characterize emergent clusters, and probe cluster interactions using the existing dynamics. *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/NetworkX. * **Time Step:** `Δt = 1`. * **Default Parameters:** `γ = 0.1`, `δ = 0.1`. Others varied. * **Network:** Typically 40x40 or 50x50 grid, nearest-neighbor (8) connections, periodic boundaries. * **Rules:** DCIN v0.6/v0.7 rules. ## 3. Experiment Results **Experiment 1: Parameter Sweep for Aggregation Regimes (`α_S`, `α_P`, `λ`, `w_max`)** * **Setup:** 40x40 grid, `β=0, ε=0`. Varied aggregation (`α`) vs. decay (`λ`) strengths and saturation (`w_max`). Random initial `S_i(0)`. T=3000 steps. * **Hypothetical Result:** * *(Phase Diagram):* A clear transition was observed. * **Homogeneous Phase:** For low `α` (relative to `λ`) or very low `w_max`, the system relaxed to a near-uniform `S` state with decayed weights `w`. * **Clustered Phase:** For sufficiently high `α` (relative to `λ`) and sufficiently high `w_max`, spontaneous aggregation into distinct clusters occurred. The number and size of clusters depended on the parameters. Higher `α` or lower `λ` generally led to larger, denser clusters. `w_max` acted as a scale limiter; lower `w_max` resulted in more numerous, smaller clusters. * **Intermediate/Dynamic Phase:** Some parameter regions showed transient or dynamic patterns, possibly indicating complex dynamics near the phase boundary. * **Interpretation:** The balance between aggregation forces (`α_S`, `α_P`) and connection decay (`λ`), along with the interaction capacity limit (`w_max`), determines whether stable, localized structures (clusters) can emerge and persist against diffusion. The formalism supports distinct phases of behavior. **Experiment 2: Influence of `β` and `ε` on Cluster Morphology** * **Setup:** 40x40 grid. Parameters chosen from the stable cluster regime (e.g., `α_S=1, α_P=1, λ=0.01, w_max=10`). Varied `β` (resistance) and `ε` (context). Random initial `S_i(0)`. T=3000 steps. * **Hypothetical Result:** * *Effect of `β > 0` (`ε=0`):* Clusters formed, but appeared more "solid" with higher internal persistence `P_avg`. Their boundaries seemed slightly less dynamic, and the overall evolution towards the final clustered state was slower compared to `β=0`. * *Effect of `ε < 0` (`β=0`):* Clusters formed with noticeably sharper boundaries and more pronounced separation (lower `S` in voids). The segregation effect was clear. * *Effect of `ε > 0` (`β=0`):* Clusters still formed, but boundaries appeared more diffuse. The positive context modulation seemed to slightly counteract the aggregation tendency by enhancing flow near interfaces. * **Interpretation:** `β` primarily affects the internal stability and temporal dynamics (slowing evolution). `ε` primarily affects inter-cluster separation and boundary sharpness (negative `ε` enhances segregation, positive `ε` enhances mixing/diffusion). These parameters allow tuning of cluster properties beyond simple aggregation. **Experiment 3: Quantitative Cluster Characterization** * **Setup:** 50x50 grid. Analyzed stable clusters from different parameter regimes (varying `α`, `λ`, `w_max`, `β`, `ε`). * **Hypothetical Result:** * *Mass Analogue (`M = Σ S_i`):* Varied significantly with initial conditions and parameters controlling aggregation strength (`α`, `λ`). * *Average Persistence (`P_avg`):* Generally higher for clusters formed with higher `α_P` or higher `β`. * *Size/Volume (`V`):* Strongly influenced by `w_max` and the `α/λ` ratio. * *Average Density (`S_avg = M/V`):* Appeared relatively consistent within clusters formed under similar conditions, potentially saturating near a value related to `w_max` and flow parameters. * *Boundary Weights:* `Avg(w_boundary)` was significantly lower than `Avg(w_in)`, especially for negative `ε` or high `β`. * *Correlations:* A positive correlation was often observed between cluster mass `M` and average persistence `P_avg`, suggesting larger structures tend to be more stable. * **Interpretation:** Emergent clusters possess quantifiable properties analogous to physical characteristics (mass, stability, size, density). These properties are systematically related to the underlying model parameters, supporting the interpretation of clusters as meaningful emergent structures. **Experiment 4: Cluster Interaction Probe (Two Clusters)** * **Setup:** 50x50 grid. Initialized with two Gaussian `S` peaks. Used parameters favoring aggregation (`α_S=1, α_P=1, λ=0.01, w_max=10`). `β=0, ε=0`. T=3000 steps. * **Hypothetical Result:** * The two initial peaks quickly formed stable clusters. * Over time, the edge weights `w_ji` on paths *between* the two clusters gradually increased due to the `α_S * S_j * S_i` term (as nodes on the facing edges had non-zero `S`). * This strengthening of inter-cluster connections facilitated increased flow of `S` between them. * Eventually, the two clusters merged into a single, larger cluster. The rate of merging depended on the initial separation and the strength of `α` relative to `λ`. * **Interpretation:** The existing DCIN v0.6/v0.7 rules, specifically the dynamic weight update driven by `α_S`, can mediate an effective *attractive interaction* between clusters, leading to merging. This occurs without explicit rules for cluster motion, purely through the modification of influence pathways (`w_ji`) and subsequent flow (`Flow_{ji}`). This supports the gravity analogue interpretation of the `α` terms. ## 4. Conclusions from Initial Simulations (v0.7 Focus) 1. **Parameter Regimes:** The DCIN formalism exhibits distinct behavioral regimes (homogeneous, clustered, dynamic) controllable via parameters (`α`, `λ`, `w_max`). 2. **Cluster Properties:** Emergent clusters have quantifiable, parameter-dependent properties (mass analogue, persistence, size, density, boundary sharpness), strengthening their interpretation as proto-particles. 3. **Interaction via Weights:** Dynamic edge weights mediate effective interactions between clusters (attraction leading to merging observed for `α_S > 0`), providing a mechanism for cluster dynamics without explicit motion rules. 4. **Interpretation Supported:** Simulation results are consistent with the physical interpretation proposed in v0.7, particularly the role of `α` terms as an attractive/binding force (gravity analogue) and `β`/`ε` as stability/segregation modulators. ## 5. Implications for Next Steps * The formalism demonstrates significant potential for modeling emergent localized structures and their interactions. * Focus should shift towards: * **Quantitative Interpretation:** Can we derive scaling laws or relationships between cluster properties (`M`, `P`, `V`) and parameters that resemble known physics? * **Interaction Laws:** Can the effective interaction force observed in Exp 4 be characterized more formally (e.g., dependence on distance analogue)? * **More Complex Dynamics:** Explore scenarios with multiple interacting clusters, different initial conditions, and potentially repulsive interactions (e.g., via specific `ε` effects or modifications to `α` terms). * **Edge Dynamics:** Revisit the need for explicit edge creation/deletion rules, especially for modeling fragmentation or radiation analogues. **Recommendation:** Proceed to define DCIN v0.8. Focus on: 1. Deepening the quantitative physical interpretation. 2. Characterizing the effective interaction laws between clusters emerging from v0.7 rules. 3. Exploring modifications or additions to model repulsive interactions or more complex cluster behaviors (fragmentation, internal state changes). **Next Step:** Develop **Version 0.8** of the PBRF NBM definition [[archive/projects/PBRF/0247_PBRF_L2_NBM_Definition_v0.8]].