The discussion that in my mind connected breadth requirements to social networks reminded me of my time as a graduate student at Berkeley, where there were lots of informal social opportunities for graduate students to meet and talk. The CS grad student association had a weekly bagel/donut hour that was always well attended (mmmm...donuts....); the theory students arranged a Friday afternoon seminar (I believe called TGIF) where theory students would present something (a practice talk, some ongoing research, a paper they liked), no faculty allowed; there was a regular Friday pre-weekend grad student get-together that would be at a semi-randomly chosen local bar (usually upwards of a dozen or so people would show, with a high percentage of theorists, who didn't have to spend the weekend coding or running jobs); a grad student poker game broke out once a month or so (again, with a high percentage of theorists, who'd add complexity to the games).
Besides helping create a more pleasant graduate student atmosphere and experience, these activities let graduate students get to know more about each other and what they were doing. And I believe expanding your own work-related social network pays dividends in the long run.
I don't know what the current state of graduate student life is like these days, but if you don't think there's enough of this sort of stuff where you are, I encourage you, take action! (Don't wait for the department to do it; you'll do it better anyhow.) Try to get a grad student poker game going. Or a biweekly pub night. Or start an informal seminar. Or an open problems session. Or whatever it is you want to do, where graduate students can just hang out together, without faculty, with it being about fun, instead of or combined with work. Besides having more fun, you'll open up more opportunities for the serendipitous events in your work-life that lead to exciting projects.