Short answer: Yes, and it wasn't much.

Longer answer: The bishop of Rome was not specifically seen as any more important than that of the three main regions in the Empire as a whole: Antioch, Jerusalem (before destruction), and Alexandria. There were significant relationships fostered between many large cities in the Middle East, Persian, and Latin-ish, regions, almost all of which acknowledged the intellectual pride of place of Antioch and Alexandria.

It wasnt until Pope Leo's Tome became a major decising factor on the Nestorian crisis that the Papacy began to, itself, see its own role as much more than simply as occupying a senior position of equal bishoprics. By Alexandria's request for the Papacy to agree with their position contra Nestorius (who by that point was the bishop in the New Rome - Constantinople) concerning the nature(s) of Jesus, it was thence argued that, when Leo answered, he spoke for the whole church and it was to be final.

This can be seen in the Roman demands that Rome be seen as sovereign over the whole church based upon Christ's claim that Peter was the Rock upon which the church would be built (but, of course, this is heavily debated as to what Christ actually meant). Such demands were visible from as early as Nicea, but were more pronounced at the 381 Constantinoplitan Council. And, importantly, at that council, Rome was acknowledged as the first light and Constantinople as Rome's peer - almost as an equal. This, obviously, aggravated the other lights of the Greek church, but especially Alexandria, whose church had lost political influence to Constantinople - the seat of the Eastern Empire, as well as theological influence to Antioch - because the Antiochans basically got to be the Patriarch of the Greek church through Constantinople.

That all being said, however, the Latin-Greek split was ongoing almost since the inception of the church and this can be seen as early as Irenaus who effectively gave blessing for Roman dominance in the Latin (Western Empire) territories based upon the premise that the Roman church was the only Western church to have been founded directly by an apostle (Thomas, it was claimed, founded the church in Antioch, for example).

Thus, in the West it was basically always assumed Rome was the dominant bishopric due to its heritage and Apostolic claims (it also helped that Paul and Peter are credited with martyrdom there).

In the East, however, first Alexandria, then Antioch, and finally Constantinople all fought back against the universal claims of Roman ecclesiastical hegemony. A key component in this discussion for the Eastern Church is that, simply put, the majority of intellectually robust theologians in the first 300 years of the church are Greek-speaking, Greek-cultured, and at the very least familiar with Greek Philosophy. Of course there are exceptions, but the predominant movers are from the Eastern Empires territories: North Africa, Greecian lands, or the Middle East (by today's geography).

Thus, for Rome to assert dominance was, frankly, ludicrous from an intellectual standpoint - the main claim was its identification with the Roman empire, and, aftwr Constantine and his heirs attempted to move the focal point of the Empire to the East, even that argument was weaker.

It really didnt make much sense for Rome's bishop to be seen as the leader of the global church at all, by amy reckoning. Even after the collapse of Rome's political situation by Odoacer etc., the Latin church only ever really controlled Latin-speaking Christian territories. And even then its control was loosely exerted byond the Alps.

A pretty good resource is 'The History of Global Christianity Vol. 1' by Scott Sunquist. He avoids much of the nitty gritty philosophical details whilst prioritizing the larger, global, picture. I especially liked his insistence on covering the Persian church too - something typically overlooked in Church History classes. That being said, there will likely be things you aren't as convinced about, but on the whole it would be my first resource to suggest.

Let me expand my comment on /u/Otherwise_Cap_9073's answer to a full answer: The earliest assertion of the Bishop of Rome's special position in the church hierarchy is a text called Adversus Haereses (Against Heresies) by St Irenaeus. St Irenaeus lived in the 2nd century (the exact birth and death dates are not know, but he lived roughly around 120 AD to 200AD). Irenaeus was a disciple of St Polycarp, who was a disciple of John (the disciple of Jesus who is traditionally held to have written the gospel of John). St Irenaeus was Greek, from the city of Smyrna, but he became bishop of Lugdunum (Lyon in today's France).

Adversus Haereses was a text arguing against may different Christian heresies. St Irenaeus argues that orthodox doctrine is derived from apostolic tradition, and asserts the authority of the Bishop of Rome due to its association with Saints Peter and Paul.

The relevant portions of the text can be found below. This translation is from the Ante-Nicene Fathers (Buffalo, NY: Christian Literature Company, 1885).https://www.ewtn.com/catholicism/library/st-irenaeus-on-the-roman-see-22975

III, 3.2 Since, however, it would be very tedious, in such a volume as this, toreckon up the successions of all the Churches, we do put to confusionall those who, in whatever manner, whether by an evil self-pleasing, byvainglory, or by blindness and perverse opinion, assemble inunauthorized meetings; [we do this, I say,] by indicating that traditionderived from the apostles, of the very great, the very ancient, anduniversally known Church founded and organized at Rome by the two mostglorious apostles, Peter and Paul; as also [by pointing out] the faithpreached to men, which comes down to our time by means of thesuccessions of the bishops. For it is a matter of necessity that everyChurch should agree with this Church, on account of its pre-eminentauthority, that is, the faithful everywhere, inasmuch as the apostolicaltradition has been preserved continuously by those [faithful men] whoexist everywhere.

III, 3.3 The blessed apostles, then, having founded andbuilt up the Church, committed into the hands of Linus the office of theepiscopate. Of this Linus, Paul makes mention in the Epistles toTimothy. To him succeeded Anacletus; and after him, in the third placefrom the apostles, Clement was allotted the bishopric. This man, as hehad seen the blessed apostles, and had been conversant with them, mightbe said to have the preaching of the apostles still echoing [in hisears], and their traditions before his eyes. Nor was he alone [in this],for there were many still remaining who had received instructions fromthe apostles. In the time of this Clement, no small dissension havingoccurred among the brethren at Corinth, the Church in Rome despatched amost powerful letter to the Corinthians, exhorting them to peace,renewing their faith, and declaring the tradition which it had latelyreceived from the apostles, proclaiming the one God, omnipotent, theMaker of heaven and earth, the Creator of man, who brought on thedeluge, and called Abraham, who led the people from the land of Egypt,spake with Moses, set forth the law, sent the prophets, and who hasprepared fire for the devil and his angels. From this document,whosoever chooses to do so, may learn that He, the Father of our LordJesus Christ, was preached by the Churches, and may also understand theapostolical tradition of the Church, since this Epistle is of older datethan these men who are now propagating falsehood, and who conjure intoexistence another god beyond the Creator and the Maker of all existingthings. To this Clement there succeeded Evaristus. Alexander followedEvaristus; then, sixth from the apostles, Sixtus was appointed; afterhim, Telephorus, who was gloriously martyred; then Hyginus; after him,Pius; then after him, Anicetus. Soter having succeeded Anicetus,Eleutherius does now, in the twelfth place from the apostles, hold theinheritance of the episcopate. In this order, and by this succession,the ecclesiastical tradition from the apostles, and the preaching of thetruth, have come down to us. And this is most abundant proof that thereis one and the same vivifying faith, which has been preserved in theChurch from the apostles until now, and handed down in truth.