Figure 4: Bifurcation analysis for WTM contagions on a noisy ring lattice.

(a) We plot the critical thresholds for k=0 given by equation (1) (dashed curve) and equation (2) (solid curve) versus the ratio α=d^(NG)/d^(G) of non-geometric to geometric edges. These curves divide the parameter space into four qualitatively different contagion regimes, which we characterize by the presence versus absence of WFP and ANC. (b) Equations (1) and (2) for other values of k further describe WFP and ANC, and we show them for d^(G)=6. Note that the curves become lower with increasing k. Fixing (d^(G), d^(NG))=(6, 2), which yields α=1/3, we find four contagion regimes (which we label using the symbols I–IV), where increasing T corresponds to slower WFP and less-frequent ANC. (c) For N=200 and T∈{0.05, 0.2, 0.3, 0.45}, we plot the contagion size q(t) versus time t for one realization of a WTM contagion with cluster seeding (that is, q(0)=1+d^(G)+d^(NG)=9). We observe, as expected, that the growth rate decreases with T. In particular, for regime III (for example, T=0.3), the contagion spreads strictly via WFP, which initially spreads at a rate of 1 node per time step (both clockwise and counterclockwise along the ring) but eventually accelerates to d^(G)/2 nodes per time step. As we show using the labelled black lines, we predict and observe linear growth for q(t) when the contagion spreads by WFP and no ANC and either q(t)≈1 or q(t)≈N. (See the section ‘Bifurcation analysis’ and Supplementary Note 6.) (d) We plot the number of contagion clusters C(t) versus t. As expected, C(t) only increases above its initial value of C(0)=1+d^(NG)=3 for regimes I and II (for which T<T₀^(ANC)). There is no spreading in regime IV.