Spectral Radius Thresholds and Higher Homotopy in Random Simplicial Complexes

Introduction: We develop a spectral–topological framework linking the spectral radius of data matrices to the emergence of higher homotopy groups in associated simplicial complexes. Using a Hurewicz-type principle, we show that spectral growth governs both the birth and collapse of higher-order topological structure. The approach provides a computationally tractable alternative to persistent homology, with rigorous results for low dimensions and principled conjectures in general.

Methods: Given a sequence of symmetric matrices A_n, we constructed clique complexes X_n via thresholding. Spectral radius ρ(A_n) was used as the governing control parameter. Algebraic-topological tools (Hurewicz theorem, homology–homotopy correspondence) and probabilistic asymptotics were combined to study limiting behavior as n → ∞.

Results: Theorem 1 Spectral Threshold for π₂: There exist deterministic thresholds 0 < ρ₂^– < ρ₂⁺ such that π₂(X_n) is nontrivial with high probability if and only if ρ(A_n) ∈ (ρ₂^–, ρ₂⁺). Full formal proof will be provided in the paper.

Theorem 2 Limiting Law for Homotopy Rank: For k = 2, n^-1rank(π₂(X_n)) converges in probability to a continuous function of ρ(A_n). Full formal proof will be given in the paper.

Conjecture 1: Spectral thresholds (ρ_k^–, ρ_k⁺) exist ∀k ≥ 2.

Conjecture 2: A deterministic limiting law holds for n^-1rank(π_k).

Conjecture 3: Centered homotopy ranks satisfy asymptotic normality.

Conclusion: The results establish spectral radius as a unifying scalar invariant governing higher-order topology. The framework connects spectral graph theory, Hurewicz-type arguments, and random topology, opening a path toward scalable inference of homotopy without full persistence. Related frontier work includes Kahle (random complexes), Linial–Meshulam models, and recent spectral-TDA (topological data analysis) connections in applied topology.