"Networking" doesn't mean following someone on Twitter and adding someone on LinkedIn. Most internships afford you the opportunity of living in a city like New York, bustling with culture, shops, and hundreds of like-minded people. You're also probably going to meet plenty of people in the industry you would give your left arm to work for, so why not introduce yourself? At worst—you land yourself another internship, at best—you're one step closer to your dream job. Shaking someone's hand and meeting them in person is ten thousand times more legitimate than an @mention on Twitter.
Put yourself out there.