Not all main characters have to have traditional arcs, but all stories must chronicle some kind of change via their main character. Either the world changes the character or the character changes his/her world. In this instance, it's Zuckerberg who literally changes the world. The brilliance in the way he's portrayed is that even though he changes the world, his invention doesn't change/satisfy him. In the opening scene he has the girl (Erica Albright), but wants something more - to be part of the social clubs, popularity, etc. But in the final scene, sitting alone in the deposition room - having created the greatest "social club" the world has ever known - all he wants now is the girl, as evident by him refreshing his friend request to Erica Albright as the credits roll. To me, it's one of the most brilliant endings ever, and such a sad look at the gaping hole in his soul, which still remains, despite his efforts.
But if you're looking for something more concrete, his "arc," if you want to call it that, is tied off pretty nicely by the two women who bookend the movie...
ERICA ALBRIGHT (in the opening scene): "You are probably going to be a very successful computer person. But you're going to go through life thinking that girls don't like you because you're a nerd. And I want you to know, from the bottom of my heart, that that won't be true. It'll be because you're an assh*le."
MARYLIN DEPLY (played by Rashida Jones in the final scene): "You're not an assh*le, Mark. You're just trying so hard to be."
In other words, it's Deply who takes Albright's "assh*le" quote (based on the deposition Deply heard in the beginning) and turns it on its head, finally/hopefully getting Zuckerberg to realize that his approach to gaining popularity was wrong all along. In short, he tried to do it by being an assh*le, not by simply being a good/nice/likeable person. So he doesn't have a traditional arc, per se, but *does* seem to come to some kind of realization, if only briefly, that he went about his goal all wrong. The Social Network is not a success story about a guy who becomes a billionaire, it's a tragedy about a guy who made the greatest social platform the world has ever seen, but still doesn't know how to make friends.
