That's a huge question. Honestly, I'd say a bit of both.

Let's start at the beginning. Christianity started out as a tiny sect in the first century Mediterranean world and by the fifth century, it was one of the biggest forces in Europe. However, the growth didn't come from anywhere and it didn't explode over night. It grew exponentially over the course of three centuries. Five factors were important in the spread of Christianity: 1. Jewish diaspora, 2. Cities and Urban movements, 3. Women, 4. Epidemics, 5. Upperclass/government.

The Jewish diaspora helped to create the essential network early Christians would need. With the destruction of the Temple in 70 AD, Jews dispersed throughout the empire, creating communities in Roman cities (we'll get to urban movements later). Social networks were key for making connections and allowed the spread of ideas within the circles of friends and family. Though Judaism and Christianity did ultimately split, these early Jewish communities were crucial in establishing networks within cities. The diaspora spread ideas in other ways too, particularly slaves and women. After 70, there was a huge influx of Jewish slaves, and those slaves carried their religion along with them.

Early Christianity centered in urban spheres. Cities acted as crossroads of ideas as people moved through trade routes. Ironically, though Christianity appealed to the oppressed masses, the elites were often the ones to convert in cities. Elites had the money and time to seek out their intellectual quests and journeys. Mystery religions were very popular in elite circles, as was the study of philosophy. It also helps that they met the literary requirement that the lower classes didn't have and gave them the skills to lead ceremonies in the early Christian meetings (wouldn't call them officially churches yet, IMO). By around the beginning of the fourth century, Christians made up around 25% of Rome's population, which was around 10% of the total empire. Much of the countryside still remained "pagan" until the next few centuries.

Now, still focusing on the urban spheres, if we think back to the first point on social networks the nuclei of social networks in cities was the home. Though Rome was a patriarchal society, women essential held control over domestic life. Early Christianity allowed women opportunity as leaders as the first meetings were often held in houses, the domain of the woman. Rich widows often attracted the roving preachers and their homes acted as the first churches. The control of the domicile also meant the control of the household gods...or god in this case. Mothers were often responsible for the children's religious beliefs, and even converted their husbands too. We can look at the Christian prohibition of infanticide as a correlation too. Roman families desired male children than female, females were often cast away. However, this Christian prohibition of female infanticide meant that the female population grew within Christianity, enlarging the cycle we just talked about with familial conversion.

Disease was rampant in cities, especially during the hot summers. However, in an ironic sense, plagues helped Christianity to grow. Now, usually when a plague hits a city, the normal reaction is to get the fuck out until it blows over. Unfortunately, only the rich had a nice villa in the countryside, and the rest had to be left behind. In fact, being left behind seemed to be the more common solution. The normal pagan reaction was the leave the sick alone. On the flipside, the Christians stayed behind to take care of the sick, because of tenets. This meant that the sick were more likely to survive and the caretakers could build an immunity. Because of this, survivors and pagans saw Christianity as powerful, and conversion increased. Think about it, you're a Roman during a plague in ancient times, and you see a group of people staying behind that not only take care of the sick, but they survive as well. That's pretty profound for them. We can also, once again, harken back to the social network. If friends and family are sick, they are taken care of by... you guessed it, other friends and family.

Now this brings us up to the 300s, which means Constantine. Constantine essentially Romanized Christianity. I personally don't place much evidence on the battle of Milvian Bridge, but rather the events after Constantine's victory. The Edict of Milan extended religious toleration, though not official sanction, which still meant the end of Christian persecution. Constantine worked hard to mesh Christianity with Roman society. However, I think I should emphasize that he didn't do everything solely for Christians. It was during this time that Sunday became the day off for the week for Christians, but it had already been so for pagans (mithraism and sun gods were very prevalent), and that's where we get the day for Christmas. Basilicas, seen as ecclesiastical buildings today, were nothing more than courthouses at first, but many Roman institutions were Christianized.

In the 300s, Christianity started to meld within the empire itself. Bishops, which in the early days of Christianity were nothing more than elders of the community, started to emulate Roman bureaucracy. Constantine held a preference for Christian leaders and many Bishops held roles similar to what we think of as mayors. Ambrose of Milan is a prime example, as he led the city as a bishop. In fact, the Roman Catholic diocese modeled itself of the imperial diocese, which "diocese" simply means something like "district." The church continue to emulate Roman imperial culture when worship stopped being a private event and became public. Meetings became more elaborate, and held in buildings like basilicas. That's where we get the Catholic priests and their robes, as well as the swinging of the incense. Christianity copied and adapted Roman imperial traditions, making it look more regal. Art is a good example, and if i can get some examples later, I might link some. Jesus in the early days was portrayed as a "good shepherd" that was poor and young. Centuries on, he is pictured as an older, more regal man with a beard and purple tunic. I should emphasize the purple tunic and toga, because that's a color solely reserved for emperors. That's a pretty profound reflection, I'd say.

Constantine helped to unite-sort of- Christianity. His involvement in the Council of Nicea meant that he was trying to get rid of the infighting and showed the role of government intervention into theology. I'm not really going to delve in the theological matters of the Council of Nicea, as I don't think explaining Arianism and trinitarianism will be too beneficial, but do take away the fact that gov't started playing a role in religious policies.

With that said, it wasn't until the time of Theodosius in the last two decades of the fourth century, where Christianity was finally made official. Not only was it official, but it led to the persecution of the pagan religions.

Now, what about the borders beyond the empire? Missionaries often traveled across the borders into Germania and the lands of the barbarian tribes. Many like the Goths and Franks, acted as foederati for the empire. Essentially buffer states on the border of the empire. They provided the defense against invaders. So with that, it was natural for Roman culture to leak into Gothic and Frankish lands. In fact, it was quite common at the Rhine and Danube, where Roman forts were establish. Forts became centers of trade and towns grew alongside of them (most of the cities in Germany along the Rhine were most likely founded on Roman forttowns). Roman culture and products spread behind the border, Christianity was no different. Tribes like the Goths and Franks wished to emulate Rome, as many wanted to be a part of the Roman world, and Christianity was a means of doing so. However, some like the Goths were Arian Christian, which didn't settle well with the orthodox Christianity of the Roman Empire, but that's another discussion. The development of the Frankish and Gothic kingdoms paralleled the Romanization of Christianity under Constantine. As the Franks and Goths married into Roman nobility, they also infiltrated into bishoprics. And missionaries continued to spread beyond the lands.

So, tl;dr: social networks and the Romanization of Christianity. I can probably give you a list of books to read but I think the most important one would be Rodney Stark's The Rise of ChristianityI don't personally have knowledge on anything really after the seventh century as that wasn't what I studied, but I hope this at least gives you a basis.