Not a WI, but did Byzantium do most of the things that the Roman empire did; building roads/aqueducts/canals/public buildings/major region-protecting fortifications, maintian permanent navy and an army equipped by the state? Or did it become decentralised like fuedal Europe, but to a lesser degree?
Largely not.
Frist of all the Late Roman Empire changed considerably during the sixth century: it stopped building baths and ampitheatres and started building churches and monasteries, so the move to medieval archictecture and away from the Classical conception of public buildings had already started. This was once perceived as a process of decline but is not seen as a process of change.
After the Arab conquests the Byzantine state that emerged was radically different from the Late Roman one. Most importantly it was not aan urban state, the cities declined to small settlements centres around fortresses. This involved important political changes as well as demographic/economic ones - the cities had been self-governing and had dominated the countryside, and had been important sponsors of public building.
However, the governmenal legacy of the Late Roman state survived - the tax and military systems. This meant that the Byzantines did maintain permanent military forces, and also survived as a highly centralised state - officials were paid in gold that could be kept save in Constantinople, and so was much more valuable than lands that the Arabs raided each year, this encouraged loyalty to the centre.
There is a vigorous, if lopsided, debate about whether the thematic armies were paid in gold or lands and gold. Most historians plump for the latter. However the thematic armies were not peasant infantry as Ostrogorsky argued - they were heavy cavalry, just like western armies.
Rural Byzantium in the high period then was not unlike the west - it featured armoured horsemen supported by the land living in a largely rural society. The political superstructure - and heavy use of gold thanks to the imperial tax system, are very important differences.
One thing to consider is that Constantine VII Porphyrogenitus never travelled more than 100 km from Constantinople, to Cyzicus in fact, Western monarchs could never dream of this, they were forced to be peripatetic, to be presentr on the ground as much as possible. It is notable that the more successful later Emperors (the 3 great Comneni in particular) were extremely mobile, which is indicative of the decline of the centralised system. On which I shall rant more if you are interested.
