What kind of ecclesiastical structure did Ignatius of Antioch discuss in his epistles? There are approximately seven "genuine" epistles attributed to Ignatius of Antioch. I'm curious to know what kind of ecclesiastical structure he mentions in his epistles. If he discussed it, in what ways was it similar to and different from the ecclesiastical structure of today's Roman Catholic Church?
Francis A. Sullivan S.J devotes some 23 pages (pages 103-125) in From Apostles to Bishops to the letters of Ignatius.  Five of the seven letter credited to Ignatius were written to churches in the Roman province of Asia, and each of these communities had a bishop. On page 105, Sullivan says that the Letter to Polycarp, bishop of Smyrna, was the only letter that Ignatius actually addressed to a bishop. This is clear evidence that by the time of Ignatius (ca. 115 CE), bishops were being appointed in at least some cities in the eastern Mediterranean. There is a clear hierarchy in these letters, with deacons subject to presbyters and to the bishop, presbyters subject to the bishop, and the bishop receiving his authority from the Father, the "Bishop of all" (page 109).

Francis A. Sullivan S.J says, in page 107, that in Ephesians 3.2, Ignatius refers to "bishops appointed throughout the world," but says that most scholars now doubt whether in the time of Ignatius, bishops like him had been appointed "throughout the world." 

Ignatius makes no mention of a bishop or presbyters in his Letter to the Romans, with his appeal directed to the whole community. On page 111, Sullivan says we can draw no conclusions from the absence of any mention of a bishop of Rome in this letter, although Sullivan elsewhere discusses the broader evidence that bishops had not yet been appointed in Rome. Ignatius was himself a bishop, as he refers to himself as the "bishop of Syria" in the Letter to the Romans. It is unclear from this whether other cities in Syria had yet begun to appoint their own bishops.

To provide context for the church hierarchy of which Ignatius was writing, Sullivan says (page 14) that a bishop is a residential pastor who presides in a stable manner over the church in a city and its environs. He says there is no evidence, nor is it at all likely, that any one of the apostles ever took up permanent residence in a particular church as its bishop. Sullivan goes on to say (page 15) the letters of Ignatius are the first Christian documents that witness to the presence of a bishop who is clearly distinct from the presbyterate, but this evidence is certain only for Antioch and several churches of western Asia Minor.

In the salutation of the Letter to the Romans, Ignatius refers to the Roman church "presides in the place of the district of the Romans ... and presiding over love." Sullivan says older Catholic scholars have seen here a recognition of the primacy of the church of Rome over the church throughout the Roman world, but says the word translated
"district" (chorion) means a limited area such as a city or town.