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When I speak of violence I don't just mean its graphic representation, I also include its specter. Tony Soprano or the guys from Goodfellas weren't whacking everyone all the time but you knew it was coming at some point and that creates suspense, allure, etc. The violent nature of these characters work the same way: they're interesting, watchable, largely due to what we know they're capable of.
It's true, of course, that you don't need violence to make a great drama or that it's the only ingredient that makes a great drama with violence great. But I do think it's something that can enhance a drama simply by it's incorporation and enable it reach levels it might not otherwise reach. Movie after movie and TV show after TV show (in the US at least) includes violence on some level for a reason: violence sells. (I think it'd be interesting to see what women have to say about generally acclaimed violent dramas and how that compares to men. That may put things in clearer perspective.)
As I've stated before, this doesn't mean a director who uses violence isn't great but that his touch can never really transcend the violence he depicts, that it's inextricably tied to it.