How do you "maintain" your professional network?

I don't have a great answer, but I think it's a great question :)

In my experience, most people do a really bad job of this -- it's easy to forget about past relationships, and the default ending state of most work relationships is that they peter out without consistent effort. A few thoughts on what the options you suggested:

• For the people you care about (e.g. you may want to do a startup with them, or you want to work with them in the future), you should set a reminder to catch up with them regularly, ideally in person. Even better is if you could do a shared, low-stakes project together, or brainstorm ideas. Having a shared activity is the best way of keeping the relationship alive.
• For groups, I haven't found anything that really works to keep in touch. The only model that had some success is the Pinterest Alumni angel investing community. This was something that people were broadly interested in ("Potential to make more money?? I'm in."). There were activities every few months with prominent angel investors, and as a byproduct of the events, I was able to catch up with people.

Another way of framing this is to accept that most working relationships will decay, but you want the ability to quickly pick things up at some arbitrary point in the future. To enable that, I'd recommend that you have something interesting to say/share, and be able to show receipts. Maybe you published an app that got tens of thousands of users, started a cool YouTube channel, or started angel investing. Being able to show people what you did will make past relationships much warmer.