A global synchronization theorem for oscillators on a random graph

The paper defines ΔL := L − p J_n + n p I_n (its equation (2)) and explicitly asserts E[L] = p J_n − n p I_n for Erdős–Rényi graphs . Under that stated ΔL, any vector orthogonal to 1 is an eigenvector with eigenvalue λ_i(L) + n p, hence ||ΔL|| ≥ n p for every graph with 0 < p < 1. This makes the paper’s key hypothesis (ii), namely ||ΔL||/(n p) < 1/4 in Theorem 11, impossible to satisfy . The same sign issue propagates into Lemma 4/5 and the probabilistic bounds in Section VI, where the paper claims with high probability ||ΔL|| is O(√(n log n))—contradicting the deterministic lower bound ||ΔL|| ≥ n p when ΔL is defined as in (2) . The model correctly identifies this inconsistency and explains that, with the paper’s ΔL, condition (ii) can never hold, so Theorem 11 becomes vacuously true.