It certainly depends upon what part of the empire you are speaking of. Constantine moved the Roman capital to Constantinople and legalized Christianity by the Council of Nicaea in 325, leading up to that Christianity was illegal in the Pagan Roman Empire. Catholicism did not come until much later on. The sixth and seventh centuries saw "The Fall of Rome", or how many historians see it, the change into more of an agrarian empire, moving from centralized state power. This was a dark period for Byzantium, and besides the repeated invasions, supposed mass volcanic eruptions left the Romans, and much of Europe in a state of chaos, due to the volcanic ash crops and food sources suffered immensely, spreading the empire out into various farm-like communities with localized armies meant to protect crops. The Islamic invasions were also vastly underestimated by the Romans until it was too late and they began to fall. There was also much division between the West and East Roman empire, their versions of Christianity varied when it came to the discussion of the Trinity. Centralized Roman powers in the West wanted to keep State power and promoted a Christian hierarchy that gave the Pope the divine power, whereas the East had Bishops who possessed divine powers, of which pissed off the Pope. This may not be able to answer exactly what you are asking, but a big thing to consider was the division between the Eastern and Western Roman Empire, which weakened both sides and allowed for numerous invasions. I will post the sources a little later as I am not home at the moment to do so. This is also just a very broad overview, please, anyone, correct this or add!