The necessary condition for subcriticality in inhomogeneous random graphs

Let $\kappa$ satisfy condition

. Let $r_{\kappa}$ denote the associated subcritical growth parameter, and let equation

be the fixed-point equation for the corresponding branching-process generating function. Necessary-and-sufficient condition for subcriticality. One has $r_{\kappa}>1$ if and only if equation

has an almost surely finite solution $f<0$. This is proposed as the converse to the sufficient condition in Proposition

; the equivalence is transparent for homogeneous Erdős–Rényi graphs, but its validity for general inhomogeneous kernels is left open.