Erdos Renyi random graphs might be a good place to start.

https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_...

If you want to start from a real-world dataset, take a look at Stanford's SNAP large network dataset collection.

https://snap.stanford.edu/data/

Watts-Strogatz[0] has some characteristics that improve on pure ER.

GP: Sounds like a great project. What's the application?

[0] https://en.wikipedia.org/wiki/Watts%E2%80%93Strogatz_model

Start from a random graph, then simulate relationship dynamics. People who know each other are more likely to be at the same location, and people who are at the same location have some likelihood of getting to know each other, creating circles of friends. Some people eventually have children together and if they have more, it's likely to be with the same person. Iterate for a few generations and you should get a reasonable graph of friends and family.

The harder part is probably procedurally generating believable social activity to communicate those relationships to an observer with enough variety to be interesting.

James Ryan's Talk of the Town does this https://www.jamesryan.world/projects#/talktown/ https://github.com/james-owen-ryan/talktown. His PHD thesis on curated story worlds is here (https://www.researchgate.net/publication/330855103_Curating_...) with a good write up by Emily Short here (https://emshort.blog/2019/05/21/curating-simulated-storyworl...). The focus is on generating interesting narratives for games and stories, rather than social network simulation or research, but many of the techniques probably apply.

Sign up for an instagram account?

Not funny.

I'm rather thinking about how one would go about making a procedurally generated Twitter.