The problem with your search is that Byzantium was not a subject city until quite late, which will muck up most online resources. Byzantium was a free city, like many other Greek cities, until Vespasian finally incorporated it as a provincial city. Like most other free cities Byzantium had been a Roman ally during the Macedonian Wars, and never switched sides, unlike some of the Greek cities. It's often hard to say precisely when various Greek free cities really became part of the empire. The best we can often say is that at some point between the establishment of an alliance with the Romans and eventual integration as a provincial city (as most free cities eventually were centuries later) political control passed over to the Romans through domination of all the surrounding area, since free cities were, legally speaking, not part of the Roman empire. In the case of Byzantium I would say, not quite arbitrarily (since there's no real legal basis for what I'm saying), that by the time of the war against Antiochus III, during which Byzantium was still allied to the Romans but firmly within their influence, the city was, for all intents and purposes, provincial. By the latest of course, all the Greek cities, free or not, came firmly under Roman control by the time of Perseus' defeat in the Third Macedonian War, which dissolved the kingdom of Macedon and established Macedonia as a provincial holding