Sums, products and dilates on sparse graphs

Let $A \subset \mathbb R$ and $G \subset A \times A$. We prove that, for any $\lambda \in \mathbb R \setminus \{-1,0,1\}$, \[ \max \{|A+_G A|, |A+_G \lambda A|, |A\cdot_G A|\} \gg |G|^{6/11}. \]