The early Christians
Rome was obviously not the place where Christianity had been born and was not the cultural center of the world. Christianity first spread in Palestine and Syria, then east to Armenia (the first country to convert) and to Greece, that was the cultural center of the empire. When the apostles spread, Peter went to Rome, but others went elsewhere. Notably, Taddeus went to Armenia.
Later, Paul went to Greece. The first community to call themselves Christians was in Syria. The man whom most (including Paul) considered the head of Christianity was James the Just, who remained in Palestine. These were all equal centers of Christianity. It was only after the Roman conversion that the Roman branch of Christianity became the official one, and the lineage back to Peter (the popes) was recognized as the only lineage worth knowing. It was then that only four gospels (probably written in Greece between 66 and the end of the second century) were accepted as true, even if for centuries several others had circulated. It was then that competing branches of Christianity were persecuted and annihilated.
What accounts for the rapid spread of Christianity around the Roman empire? It is not clear how many Christians there really were before Constantine forced the entire Roman empire to convert to Christianity, but it is reasonable to assume that at least a good number of them lived in Rome and in various provinces of the middle East. In the year 70, following a Jewish rebellion, Roman legions destroyed Jerusalem and expelled the Jews. That act may be responsible for the spread of Christianity: Jews of the Christian faith certainly ended up (as slaves) in Rome and probably (as refugees) in several middle-eastern provinces. While it is a mystery how they could make so many proselytes so quickly, it is quite normal that they could be found all around the eastern Roman empire. The number of proselytes (if it was indeed as high as the Church wants us to believe) could be explained in a simple way by assuming that there were already many Christians in Palestine itself, which, of course, would be possible only if Christianity was widely more popular than the official gospels admit and if Christianity predated Jesus.