So where does Apostolic succession come into play? I've searched, but have been unable to find any records where Peter himself granted his authority to Linus, and then to Cletus, and then to Clement I , and even Linus and Cletus only seem to head the church in Rome, not the entirety of Christendom. The earliest evidence that I can find on the subject is a letter from Linus to the church at Corinth where he asserts a global command to head off a schism.
Check out Isaiah 22. The chapter records the succession in office of the vice-king of the House of David. The office (and the key) is passed from Shebna to Eliakim. The key represents an office. Office involves succession.
In Matthew 16, we see Jesus establishing His Church, and giving the keys to the kingdom (the Church) to Peter. The keys represent the office of the vice-regency of the eternal House of David, the Church.
Additionally, in Acts we see the succession in office of Matthias, who takes the office of Judas. (In the KJV, the "bishopric").
This thread is moving so fast that by the time I'd found anything on Linus, I'd lost your post! Anyway, here is a link: http://www.newadvent.org/cathen/09272b.htm
Maybe it's the news, but it's loading very slowly.