Why Scale-Free Networks Are Spectrally Compressible: A Latent Sufficiency Theorem for Network Dynamics

We prove that the Latent Theorem — which guarantees finite sufficient spectral representations for smooth systems — explains the dimensional reduction of dynamics on scale-free networks. For a scale-free network with degree exponent γ and graph Laplacian \(L\), we define the spectral analyticity parameter \(\rho(\gamma) = (\gamma - 1)/2\) and prove three results:

1. Supercritical regime (γ > 3, ρ > 1): The network dynamics have a finite-dimensional Latent of dimension \(N = O(\log(1/\varepsilon) / \log \rho)\). The Gao–Barzel–Barabási one-dimensional effective dynamics is the grade-1 Latent projection. Higher grades provide systematic corrections: grade-2 captures degree-degree correlations (assortativity), grade-3 captures clustering.

2. Critical regime (γ = 3, ρ = 1): The Latent dimension diverges logarithmically with network size. The Barabási–Albert model sits exactly at this critical point.

3. Subcritical regime (γ < 3, ρ < 1): No finite Latent exists. The dynamics cannot be reduced to a finite-dimensional representation, explaining the failure of mean-field theory for networks like the Internet (\(\gamma \approx 2.2\)) and protein interaction networks (\(\gamma \approx 2.4\)).

The phase classification is proved in Lean 4 with 0 axioms for the algebraic core (29 theorems across 5 files). The mean-field closure is formalized as 1 audited axiom.