The CROS model presents long-range temporal correlations at the transition line.

(A) and (B) Strong LRTCs (indicated by DFA exponent α > 0.5) emerge at the onset of oscillations for the raw time series, regardless of the system size (see Methods). For the band-filtered time series, (C) networks with L = 50 show strong LRTCs while (D) for L = 300 the effect is less pronounced (though still clearly α > 0.5). For L = 50, (E) α_raw and α are close to one near the critical regime (green diamonds and circles, α ≃ α_raw ≃ 1.09 respectively). For L = 300, (F) α_raw is close to one while α is smaller near the critical regime (green diamonds and circles, α_raw = 1.16 and α = 0.66 respectively). Examples from the subcritical region: (E) L = 50, α_raw = 0.57 (purple diamond) and α = 0.80 (purple circle), (F) L = 300, α_raw = 0.53 (purple diamond) and α = 0.60 (purple circle). For the supercritical region: (E) L = 50, α_raw = 0.60 (orange diamond) and α = 0.62 (orange circle), (F) L = 300, α_raw = 0.54 (orange diamond) and α = 0.62 (orange circle). Diamonds (circles) represent DFA applied over raw (band-filtered) time series (see Methods). Solid symbols indicate the fitting range where α was estimated. Colored arrows in (A), (B), (C), (D) and colors in (E) and (F) represent parameters: r_E = 0.04 (purple), r_E = 0.12 (green) and r_E = 0.24 (orange) with r_I = 0.60.