I think Eugene Smith is terribly overrated here. #4? Really?
Alec Soth is surely more influential than many of the names here. His "I'm living a life of ennui" portraits are now a mainstay among college photographers.
Ralph Eugene Meatyard, Fredrick Sommer, Keith Carter for God's sake?! Callahan, Metzger, Lee Friedlander!! All missing.
If influential, you mean taught and inspired, those names would all be in the top ten.