Network Latent Structure

We establish a formal connection between the power-law exponent γ of scale-free networks and the analyticity radius ρ of their latent representations via the relation ρ = (γ−1)/2. This relationship reveals a phase transition at γ = 3 (equivalently ρ = 1): supercritical networks (γ > 3) have finite latent dimension with exponentially decaying graded corrections, while subcritical networks (γ < 3) exhibit divergent latent dimension. The Barabási-Albert model sits exactly at the critical point. We prove monotonicity of the phase mapping, characterize the three regimes (subcritical, critical, supercritical), and establish bounds on graded decay factors. The graph Laplacian structure of these networks is axiomatized with standard spectral properties. All results are formalized in Lean 4 with 12 verified theorems and 0 axioms beyond standard definitions.