I imagine it working similarly to IPFS, where the data is distributed across the network.

Think of it as more like email. You have an account on a specific server, but can send or receive emails from anyone on any server.

There's not one network with mastodon. Depending on which node you start in, you can get different graphs (from what I remember, mostly due to differences in value between valuing free speech vs preventing hate speech).

Ya'll are describing SSB protocol and ManyVerse.

I just looked into SSB and I really like the concept. It seems much more elegant than Mastodon. Why is Mastodon/the Fediverse so much more popular (or is it?)? What are the advantages of federation over total decentralization? I suppose one disadvantage of total decentralization is the need for ‘fat clients’, as another commenter pointed out.

As you can see with Twitter, more popular doesn't mean better.

There's a way to do this, but it requires fat clients https://indieweb.org/POSSE