Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.

While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is positive (I have a 7x7 counterexample which I can post if somebody cares for it), I wonder if under some additional assumptions (maybe on the value of the negative entries), this eigenvector can be proved to be positive?