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 $G$?



A question on graphs
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.


