Thinking about this in terms of a "direct competitor" is the wrong way to frame a potential business and nearly impossible at this point, I think.

LinkedIn is fantastic at being LinkedIn.

The only way to bring them down would be to approach a part of the problem they solve in a completely different way and in a compelling enough way that would make a piece of LinkedIn irrelevant.

You can only compete by making LinkedIn irrelevant, not by building a better LinkedIn.