There is a massive range of time in which Rome has been a center of influence, but I'll assume you are talking about the range of 1st to 5th century in the time of the Apostolic Fathers such as Polycarpe, Clement[s], and Ignatius.

Contrary to popularized belief, the Christians of this time were not so much persecuted because they were Christian; but rather because they were not subservient to the Grecco-Roman practice of piety to the gods, and especially to the ruler of Rome.

The mentality that most peoples had was that the gods of Greece/Rome would only cause calamity if they were not properly respected. If, for example, there was a flood that had killed that season's main supply of crops: the flow of though would be as follows:

-Why would the gods punish us so? -They must have been disrespected, or crave the blood of a sacrificial animal -We have all been pious, so who has not been in our community? -These groups on the periphery are not adhering to the status quo, so lets [objectify and punish if they do not conform].

The popularized view of Cesarian oppression of Christians had been exemplified by the fanatical actions of Nero (lining his court with crucified christians, and blaming them for the great fire); but the more common and logical explanation is the perceived subversion of Christians to the absolutism of Roman rule.

After locations had out-ed a christian population, regional adjudicators would interview, [and was commonplace in regulations to torture before any kind of confession was accepted regardless of religion/cause of investigation], to identify the problem.

It was not that these people were Christian they were punished, but rather that they would not identify the Roman ruler as lord of the land. The impiety publishable was the refutation to acknowledge Roman domination as chief authority. When Christians were charged to identify the lord of the land, they would snuff the Roman and cite that the Lord God is the highest authority.

(There are many examples of Christians getting to walk away if they capitulate, but it was a growing trend in Christianity to stick to their guns and die in martyrdom to emulate the death of Jesus)

If you read into the Apostolic Fathers, (Polycarpe for example), you can see the possibility of life being snubbed and entire families dying for their insolence to Rome, rather than insolence to the gods.