Fig. 6: Dynamics evolving on configuration model graphs.

a, d Glauber dynamics, b, e SIS dynamics, and c, f Cowan dynamics. We used the configuration model (see Eq. (7)) to generate multigraphs of varying sizes and degree distributions. In the top row, we generated multigraphs with geometric degree distribution of size N = 1000 and with M = 2500 edges (see Fig. 7a). In the bottom row, we used the degree distribution of real networks: d Little Rock Lake food web⁹⁵, e European airline route network⁹⁶, f C. Elegans neural network⁹⁷. The parameters used to generate the time series are the same in the top and bottom panels (see Table 1), except in f the time series length is T = 5000 while in the others T = 2000. Similar to Fig. 5, U(G ∣ X) is shown in blue (left axis) and U(X ∣ G) is shown in orange (right axis). We show, for each dynamics, the uncertainty coefficients as a function of the coupling parameter: J for Glauber, λ for SIS, and ν for Cowan. Each shaded area indicates a range of couplings over which duality was observed. The vertical dotted-dashed lines correspond to the phase transition thresholds of each dynamics, which are estimated from Monte Carlo simulations (see Section IX of the Supplementary Information). For the Cowan dynamics, the forward and backward branches are shown with their corresponding thresholds and dual regions (see main text).