I addressed this question somewhat in a previous post here , but here is a relevant part to your question:
Which brings us to Catholic tradition, which attempts to create an unbroken line of Bishops of Rome to give the Pope the apostolic authority that justifies his position in the church today (and the episcopal ecclesiology that the Catholic Church structures itself on). Apostolic authority is essential to the Catholic religion because it gives the Pope authority from Jesus himself, who allegedly installed Peter as the first Bishop of Rome. Contrary to (perhaps) popular belief, the New Testament (or other 1st and 2nd century texts, for that matter) does not explicitly (or, in the mind of many, implicitly) outline this installment. Most sources indicate that Peter visited Rome, but whether he led the church there is matter for much debate (in fact, if his ministry mirrored his peers and we consider the political situation in Rome at the time, he probably did not “settle down” to lead the church in Rome as its bishop). Still, as it is reflected in the Second Vatican Council, it is important for Catholics to see Peter as the “First among the Apostles,” as recognized by Jesus himself, so that the Pope reflects this style of leadership as the First Priest of the Church.
Immediately after Peter, our sources for apostolic succession get extremely muddy. Forgetting the 3rd-century texts such as the Liberian Catalogue and Liber Pontificalis—2nd century “lists” of the Bishop of Rome are also contradictory and contain numerous anomalies! Some protestant and secular scholars have proposed that the early Church in Rome operated in some sort of “collegiate episcopacy,” a theory obviously inconsistent with Catholic views on succession. Some lists claim Clement as Peter’s successor, while Eusebius tells us of Clement succeeded Linus and Anencletus after Peter’s death. Regardless, many of these very-early “Bishops of Rome” (to whatever extent they were) were well-known figures in early Christianity. While Eusebius didn’t offer references for much of his early list, it is safe to assume that these leaders existed and were influential in the early church in some capacity. Clement, in particular, is famous for the letter commonly attributed to him that failed to make New Testament canon at Nicaea.
The list of Bishops of Rome that we have today, in their varying forms, mostly consists of historical church leaders who were recognized and their feats recorded by (mostly) Christian historians and church leaders of diverse ethnicities. While biographical details are usually hard to come by for many of these figures, their theological, doctrinal, and canonical contributions in the so-called “Apostolic Age” are documented. Whether or not these leaders functioned in any role recognizable as the one that Pope Francis fills is debatable, if not downright unlikely. Thus, there isn’t much reason to believe that these “popes” didn’t exist, and, as you say, held “some position of authority” within the early church.
I am not a historian, just very interested in the early church period, I hope I don't mistep on the rules it seems you can answer even though you are not a historian, but as long as you provide sources.
The early papacy is murky, specifically because it was an underground organisation that was persecuted and at a certain times they were forced to hand over writings to the Roman authorities, coincidently that is where we get the word traditores or traitors, "those who handed over". The Latin word for traitor was Proditor.
So it gets difficult to even have a lot of writing until the religion is legalized, or there are lulls in persecution.
There are sources for Linus as the succesor for Peter. Linus and the next 4 popes are mentioned in the bible as disciples, Linus specifically in the Second Epistle to Timothy and Paul mentions him keeping him company in Rome.
The first mention preserved is by Irenaeus at 180 AD. He was a Greek Bishop working in what would later become France, he also confirms Anacletus and Clement.
Hegesippus a contemporary of Irenaeus operating in Palestine also confirmed the 3 first popes or bishops of Rome to be more accurate. It should be noted that some later lists, list Cletus instead of Anacletus, some even go more of kilter by listing them as seperate people. However the earliest sources I mentioned have a consensus that goes. Peter - Linus - (Ana) cletus
Things get muddy after Clement in regards to reigns, my personal interpretation is that it coincides with the persecution by Trajan, which would have disrupted a lot of the organisation efforts by the early Christians. The names are similar in Irenaeus and Hegesippus, and their lists go up to Pope Eleutherius.
Then there is an unknown author of the "Poem against Marcion" that has a similar list but uses the names Anacletus and Cletus interchangeably. The Poem is form around 200 AD.
After Pope Victor things really get muddy, and all we really know for sure are names of at least some of the Popes, because the lists we have vary depending on location and time. So much so that we even end up having an anti-pope commemorated as Popes in some Eastern Calendars like the Copts and Armenians.
By the 4th century, things are so muddy, that when the Church is legalized you have several lists going about, and Eusebius is one of the few that tries to source his info instead of just relying on what was handed down orally. He uses Iraneus and Hegesippus, unfortunatley only few fragments of Hegesippus texts have been preserved.
It is only after the Edict of Serdica and the Edit of Milan that you start to have Christians and others seriously ponder the origins of the Church and its history.
The most Reliable efforts are from Eusebius, in his Chronica and his histories.
Then there is the Catalogue of Liberius from the Chronography of 354. Which save for a few errors in assuming Anacletus and Cletus were different people, it lines up Iranaeus and Hegesippus lists. An important note on the Catalogue is that it seems to source its information from two chronicles from the 3rd century, namely Julius Sextus Africanus, and Hippolytus of Rome.
Book "Popes and the Tale of Their Names" by Anura Guruge. for info on Anacletus/Cletus mixup
synaxarion of the Coptic church with Hippolytus as Pope
Eusebius Church History (Book V)