We've just been doing half and half, with a bit of overlap for comparison's sake. But we don't want to move too quickly and look like we're deliberately trying to network with people, either.

[They want to compare notes on other people, but if those same others compare notes and discover that they've started taking an abrupt interest in getting to know the others, that could be awkward.]

Who have you gotten to know already? We can start from there and slowly expand your circle so it looks natural.