I don't think whether it's on a network or not would matter--those waves are around you as long as your router is on, even if your computer isn't. But I don't know about risk just from the computer running. An LCD screen shouldn't be an issue (other than the ugly light it gives off). I don't know what happens with the innards of the laptop that could be troublesome. I've wondered about this too. I'll admit, though, that I've had my laptop at arm's reach just about every night as I nurse dd to bed. I used to for naps, too, but dh does naps now.