Do there exist a family of graphs with the property: $$\left\alpha\left({G \boxtimes \bar{G}}\right)  \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right = O(\log(N_{G}))$$ where $G$ is the graph, $\bar{G}$ is its complement, $\boxtimes$ denotes the strong product, $\alpha(G)$ denotes the independence number of $G$ and $N_{G}$ is the number of vertices of $G$?
Since $$\alpha(\overline{K_n \boxtimes \overline{K_n}}) = n$$ and $$\alpha(K_n \boxtimes \overline{K_n}) = n$$ it follows that the family of complete graphs satisfies your assumptions. To see the first equality observe that $K_n \boxtimes \overline{K_n}$ is the disjoint union of $n$ copies of $K_n.$ Hence in the complement every disjoint $K_n$ is an independent set any pair of vertices from disjoint copies of $K_n$ is adjacent. 

