We study a new set of duality relations between weighted, combinatoric invariants of a graph $G$. The dualities arise from a non-linear transform $\mathfrak{B}$, acting on the weight function $p$. We define $\mathfrak{B}$ on a space of real-valued functions $\mathcal{O}$ and investigate its properties. We show that three invariants (weighted independence number, weighted Lovász number, and weighted fractional packing number) are fixed points of $\mathfrak{B}^{2}$, but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.