Maybe give a bit more info when posting these. Answer the questions the people trying to find out whats going on are asking you:
What does the F1 console say, and how are you starting Rust?
Doing this should show you an error or exception message we can use to search on, its a valid question
Do you play fullscreen or windowed?
Also a valid question!
the F1 console one is probably the most important, the easiest way to fix problems like these is to check as it will tell you exactly why you disconnected, then there is a good change that error has been posted about and potentially resolved already here in the forums.
Also maybe post some info about your setup as well, what your a/v is and any software firewalls like BitDefender if you have one.
Elix didn't say don't buy early access if you cant handle not playing, he said this sort of shit is to be expected in early access, and it is a valid point. At one stage I couldn't run the new version for over a month until I could work out how to fix it. If you can post up the answers to the questions we will take a look and see if we can help you out.