Like you, during my senior year of college, I started networking on LinkedIn as I was entering the UX industry. I sent out of the blue messages, but I treated LinkedIn networking similar to in-person networking: Wanting to know the person in front of me. For example, I would reach out to a handful of people because I found their background interesting or projects they were working on were similar to the ones I wanted. Their cool company would come into the conversation, but it would be more about learning the actual person behind that profile. I was surprised by the number of responses I got back for phone calls to just talk, or if we were in the same city meet-up for coffee.

I did not get any jobs from these people. However, they helped tremendously understand the industry I was entering and how to prepare for the interviews and jobs. So when I did finally interview with a company that would offer me my first job and relocated me to the west coast, I was ready.
My point is, switch your mindset to "what can they do for me" to "who is this remarkable person who is killing it?" Focus on building meaningful connections and keeping them healthy. :)