Let's suppose I have a generic directed graph $G$ and it's adjacency matrix $A$. I can add an arc wherever I want in the graph. (i.e. perturb the matrix $A$ changing a single $0$ into a $1$). Where should I put that one to have the highest increase in the biggest eigenvector as possible?
I suppose that the answer is "where you can connect the two largest strongly connected components".
These two papers should be helpful for you.
[1] J. G. Restrepo, E. Ott, and B. R. Hunt, Phys. Rev. Lett. 97, 094102 (2006)
[2] A. Milanese, J. Sun, and T. Nishikawa, Phys. Rev. E 81, 046112, (2010)