It sounds like you are assuming and expecting stuff that she isn't aware of. Leaving her to take initiative... as said earlier is a slippery slope, but also not fair. Expectations that people don't know about are not fair. Why not tell her that you expect that she take a role in creating opportunities to connect with you. That you want a balance there. Don't assume that she doesn't already feel like there is enough connection. Maybe she has expectations of you that you are not filling. Ask her if that is the case and see what she says. That's what I would do anyway.
Anyone want to be friends on Facebook?
Send me your name via PM