I second this from my own experience (in physics, undergrad from a state university). I met my grad school advisor at a conference, and would definitely not have been able to get without their support.
I was a physics undergrad too. At the time I thought getting into a top school was about being a genius. Now that I'm at a top school (for CS) that I got into through networking I wish someone had told me back then because I might've not given up on physics. It's almost criminal that undergrads are fed stories about needing first author papers in top tier journals and 4.0s and 990s on subject gres to get into these places.