The network is not (only) your graduating peers, it's primarily the alumni. If you see a job you want and the hiring manager also went to Harvard, they will have a more favorable opinion of you. You may ask how this is any different from any other college, but, due to prior network effects, the prior Harvard graduates themselves are more likely to be in higher positions of power. Prestige begets prestige, in a way that lower universities do not.
But the difference might not be between Harvard and a local community college. People don't come out of "slums" to Harvard and immediately join the upper class.
But you'll be exposed to different people and opportunities by going to a state school instead of a community college, or a better school out of state with a scholarship vs. a state college, etc. The small differences matter.
I literally got my first job because my roommate's dad knew of someone looking for a software developer job. Would I have met that person in community college? Maybe! Lots of people with good networks go to community colleges, too - but I'd bet fewer than more expensive schools.
