The Inheritance of Rome connects directly to Ward-Perkins’s book but approaches the period from a more conventionally historical perspective. It explains how the regions of the former empire remained connected to Rome and continued to understand themselves through Roman concepts, even though they were no longer integrated politically or economically in the same way.
It also shows that the East did not collapse as the West did, although it gradually became something that was not necessarily Rome in quite the same sense. The region remained wealthy, and the rising Arab empires were not creating an entirely new civilization from scratch so much as replacing and inheriting structures that already existed—at least, that is my reading of it.