Decoding how higher-order network interactions shape complex contagion dynamics

The paper’s Proposition 3.3 asserts that for the 4th-order mean-field SIS model ẏ = (C2−γ)y + (C3−C2)y^2 + (C4−C3)y^3 − C4y^4, if C3 < γ and C4 > γ, then the (C2,y) equilibrium curve has two fold points, citing the identity C2(y)=γ/(1−y)−C3 y−C4 y^2 and a discussion of an inflection when C4>γ as its proof sketch . However, differentiating gives F(y)=dC2/dy=γ/(1−y)^2−C3−2C4 y, which is strictly convex, so two folds occur if and only if min F(y)<0. This yields the necessary and sufficient threshold C3>3γ^{1/3}C4^{2/3}−2C4; otherwise there are zero folds (a cusp at equality). The paper’s proof only observes an inflection point and does not establish the two-fold condition, and in fact the unconditional claim is false (e.g., γ=1, C3=0.5, C4=2 gives no folds). By contrast, the candidate solution derives the correct threshold and full stability picture; its local transcritical classification also matches the paper’s 3-body case (Prop. 3.1) and the 4-body local result at C2=γ .