Neural Percolation Model (NPM): An Engineering Analogy Framework for Neural Network Information Propagation and Capability Emergence

We propose the Neural Percolation Model (NPM), an engineering framework that maps the physics of Pore Network Models (PNM) onto neural network information propagation. Starting from a single conservation principle—steadystate information flow satisfies nodal balance—we derive a unified master equation:
                                                                            aout = Weff · ain
where the effective conductance matrix Weff takes three distinct forms corresponding to feedforward networks (fixed weights), Graph Neural Networks (graph Laplacian, exact PNM analog, verified with zero numerical error), and Transformers (dynamic softmax attention, row normalization verified as flow conservation). We further introduce a percolation-based data density model in which emergent capabilities arise when data density ρ exceeds capability specific critical densities ρc, yielding a multi-threshold field view of emergence rather than a single critical point. A Participation Ratio (PR) method for quantifying data distribution breadth σ is proposed and validated (monotonicity confirmed). B-layer parameters (α, β, δ, k) are fitted against 14 public models and Kaplan scaling data, achieving Spearman correlation 0.70 with MMLU and exact matching of the empirical scaling exponent N 0.076. MIS-based pore analysis of NN weight matrices reveals a three-phase training transition (pressurization–breakthrough–reorganization) in Pythia-70m, analogous to percolation threshold crossing. The framework unifies 17 observed LLM phenomena under a single physical mechanism. We position NPM as an engineering analogy framework in the tradition of Darcy’s Law and the Reynolds number.