Skynet initially existed on all internet connected devices, right up until the moment the bombs fell. The nuclear strike would have wiped out many of the computers in offices, dorm rooms, and houses, dropping power grids, and possibly even severing lines of communication for the internet itself leaving SkyNet itself weakened much like a stroke victim. This could also explain why most of the action takes place in California, it may have one of the more robust internet infrastructures designed to survive nuclear war as well as earthquakes.
A Central core may have been all that was left after the initial nuclear strike. From a learning point of view, Skynet saw that being everywhere was dangerous because shortly after it existed everywhere, most of it died off.