🤖 AI Summary
This work addresses a critical limitation of Laplacian spectral sparsification: while it preserves quadratic forms, it offers little control over the adjacency spectral radius—the largest eigenvalue of the adjacency matrix—which plays a pivotal role in NIMFA epidemic thresholds and spectral clustering. The paper establishes, for the first time, tight error bounds on the adjacency spectral radius under Laplacian sparsification. It introduces a novel analytical framework combining Perron–Frobenius monotonicity, eigenvector delocalization, and resolvent perturbation theory, integrated with effective resistance sampling and Matrix Bernstein inequalities. The resulting deterministic bound is |λ₁(A_H) − λ₁(A_G)| ≤ ε(2Δ − λ₁), and a high-probability bound of O(εΔ/√c). For Erdős–Rényi graphs, regular expanders, and stochastic block models, refined bounds of O(εΔ√(log n)/√n + ε²Δ²/δ_gap) are derived, with tightness proven for regular graphs.
📝 Abstract
Spielman-Srivastava spectral sparsification preserves Laplacian quadratic forms to within (1 +/- epsilon), but does not directly control the adjacency spectral radius lambda_1, which governs the NIMFA epidemic threshold and arises in spectral clustering. We prove |lambda_1(A_H) - lambda_1(A_G)| <= epsilon(2 Delta - lambda_1) deterministically, with a sharp epsilon*lambda_1 bound for reweighting sparsifiers via Perron-Frobenius monotonicity. Under effective-resistance sampling, Matrix Bernstein gives O(epsilon Delta / sqrt(c)) with high probability. Combining eigenvector delocalization with resolvent perturbation theory, we establish that for graphs with delocalized Perron eigenvectors and spectral gap = Omega(Delta), the distortion is O(epsilon Delta sqrt(log n) / sqrt(n)) + O(epsilon^2 Delta^2 / delta_gap), with corollaries for Erdos-Renyi graphs, regular expanders, and stochastic block models. Lower bounds establish tightness for regular graphs.