Revealing dynamics, communities and criticality from data

The paper states the hub reduction theorem exactly in Appendix D and explicitly defers its rigorous proof to a separate work ([19] in the paper’s references); it provides only an informal statement within this PDF. The candidate solution offers a plausible, alternative proof sketch via high-temperature (weak-coupling) SRB/Gibbs mixing, concentration for Lipschitz functionals, and a bias control argument, yielding the same scaling T ≤ exp[C ξ^2 Δ] and the same form x_j(t+1) = G_j(x_j(t)) + η_j(t) with |η_j(t)| < ξ. The statement in the paper matches the candidate’s conclusion word-for-word (up to notation) and constants, see Appendix D, Theorem 1: x_j(t+1) = G_j(x_j(t)) + ξ_j(t) with |ξ_j(t)| < ξ for 1 ≤ T ≤ exp[C ξ^2 Δ] and a good set of initial conditions of measure ≥ 1 − T/exp[C ξ^2 Δ] . The model’s proof sketch assumes standard mixing/decoupling and linear-response hypotheses that are not spelled out in the paper but are compatible with its assumptions.