The effect of heterogeneity on hypergraph contagion models

The paper derives, for the degree-correlated (collective) case, the critical triangle infectivity β_c3/γ = ⟨k^3⟩⟨k⟩^2/⟨k^2⟩^3 by analyzing h(V,β2) near V=0 and using the sign of ∂h/∂V at β2=β_c2 (its Eq. (35)), and for the uncorrelated case it expands to obtain h(V,β_c2) = (a0+a1 V + a2 V^2) V^2 with explicit a0,a1,a2 and a piecewise onset criterion: saddle-node via a1^2−4 a0 a2=0 with 0≤−a1/(2a2)≤1 and a2<0, or else transcritical at a0=0 giving β_c3 = γ⟨k^3⟩/⟨k⟩^4 (Eqs. (38)–(42)) . The candidate solution reproduces exactly these steps and thresholds, including the discriminant and admissibility conditions for the uncorrelated case and the transcritical fallback at V=0, with matching moment dependence and normal-form reasoning. Hence both are correct and essentially the same proof route.