The only thing I can think of would be if the GC was power cycled while connected to software, log book entries would report lost connection to GC. This is all software type and version dependent on where you would look in logbooks. There is nothing front panel of GC that I know of.
That's the only way I can think of it too. and through the front panel logs saying the instrments been started. The only other way is to go through the router to see if the GC IP has disconnected at all. unfortunately I can only do this on a few routers on a a few networks, so can't see all.