An essay on the subject : http://www.myriobibl.../milton1_4.html
The above essay gives some historical information on this issue. But it doesn't go into depth. In 382 a purported council in Rome under Pope Damasus invented the petrine theory. That Rome, Alexandria and Antioch were connected as these three Sees all were associated with Peter. The title of "Apostolic See" dates back to Pope Damasus and with reason. The entire reason for this newly formed petrine theory and the new title is that the the 2nd Ecumenical Council raised Constantinople to the second place eclipsing Alexandria and Antioch in the Nicene rankings.
Byzantium was originally a small greek colony and was not known as having been found by apostles. In the East it did not matter as the pentarchy was never based on apostolic foundations, it was simply based on the importance of the city to the empire.
The truth is the concept of an "Apostolic See' is a 4th century innovation itself, of the approximate 42 apostolic churches, Byzantium was not on that list. So what better way to lessen its importance than to always remind everyone it was not apostolic?
The 'Decree of Damasus' demonstrates a newly-created rivalry between Constantinople and Rome, Elder Rome and New Rome created a sibling rivalry:
1. After the prophets and the evangelical and apostolic scriptures which we discussed above, on which the catholic church is founded by the grace of God, we also have thought necessary to say what, although the catholic churches (note the plural) diffused throughout the world is the single bride of Christ, however the holy Roman church is given first place by the rest of the churches without [the need for] a synodical decision, but from the voice of the Lord our saviour in the gospel obtained primacy: 'You are Peter,' he said, 'and upon this rock I shall build my church and the gates of hell shall not prevail against it; and to you I give the keys of the kingdom of heaven, and whatever you shall bind upon Earth shall be bound also in heaven and whatever you release upon Earth shall also be released in heaven'.
2. In addition there is also the presence of the blessed apostle Paul, 'the chosen vessel', who not in opposition, as the heresies jabber, but on the same date and the same day was crowned in glorious death with Peter in the city of Rome suffering under Nero Caesar; and equally they made the above-mentioned holy Roman church special in Christ the Lord and gave preference in their presence and veneration-worthy triumph before all other cities in the whole world.
3. Therefore first is the seat at the Roman church of the apostle Peter 'having no spot or wrinkle or any other [defect]'.
However the second place was given in the name of blessed Peter to Mark his disciple and gospel-writer at Alexandria, and who himself wrote down the word of truthdirected by Peter the apostle in Egypt and gloriously consummated [his life] in martyrdom.
Indeed the third place is held at Antioch of the most blessed and honourable apostle Peter, who lived there before he came to Roma and where first the name of the new race of the Christians was heard.
Historically. Peter himself was never known to have been the head bishop ruling from Rome, only that he visited in the latter end where he was eventually martyred. Infact we know exactly where Peter made his residence and that was in Corinth. Not only does Paul's epistle to the Corinthians spells this out, its confirmed by Dionysios of Corinth in 180 AD ,when he wrote his epistle to the Roman Church explaining their common heritage.
From this ridiculous Petrine Theory, even nestorian- minded bishops from Antioch attempted to argue their superior theology compared to the monophysite theology of Alexandria!:
"But this man (Patriarch Dioscorus of Alexandria) will not abide by the decrees (of Nicaea), but brings forward at every turn that his is the ****Throne of Mark****; and yet he knows well that the great city of Antioch has the Throne of Peter, who was both teacher of Mark, and the first and the leader (coryphaeus) of the choir of the Apostles." (Theodoret, T. iv. Ep. lxxxvi.).
Regardless, the early christians of the Asia Minor region (and neighboring areas) have always considered John, Andrew and Philip as the apostles that laid the foundations of the Church in their region. Infact the early christians of Asia Minor also held the daughters of Philip in the highest esteem as they were long-remembered and venerated as prophetesses in the region. Byzantium would have been within that hegemony. Hence the Gospel of John gives the story of how greeks sought out Christ by turning to Philip who in turn told Andrew and both brought the message to Christ. Christ responded by saying, 'the hour has come for the Son of Man to be glorified' (Jn 12.20-36). Also unique in John's gospel is that Andrew's name is placed before Peter's name which is quite telling. Its the only place in the gospel that this occurs and gives us historical insight that in Ephesus Andrew held a special place in their hearts:
The following day Jesus wanted to go to Galilee, and He found Philip and said to him, “Follow Me.” 44 Now Philip was from Bethsaida, the city of Andrew and Peter. (Jn 1.43)