[quote=So Boerd]
I don't follow. Not sure what physical influence means in this context.
[/quote]

Physical influence as in the extent of what could be their area of operations in a sense, obviously AoO is a more modern term, but it's fitting. Sort of like putting a border on the lands the Huns controlled at their height, in that it wasn't really "the" lands they controlled, but more or less the extent of their influence as far as we could tell.