Here is a map of the Swedish internet circa 1985 by Jörgen Städje
@icm
I'm pretty sure that's incomplete. "Sunet" was (and is) the national university network; there's several connections to there in the diagram. And Sunet isn't the entire network, just the academic bit of it.
This seems to be a specific network interconnecting a few universities, and from the look of it, physics departments in particular.
My memory is a bit hazy, but I recall connecting to my work at Ericsson in Stockholm from the University of Michigan in early 1983 via telnet, in order to read email. I guess it might have been a dial-up, then?