Concepts of succession become more prominent in second-century Christianity. But we can't assume that what one source wrote on the subject represents what every other Christian believed or even what most others believed. Robert Lee Williams comments:
"It is the particular nature of the ecclesiastical struggle in each case that shapes the content of the doctrine of apostolic succession at that point. The content of apostolic succession, linchpin though it is in defense of orthodoxy, changes in content and emphasis from writer to writer...The bishops in apostolic succession are the legitimate leaders of the churches not in every way, with a carte blanche of authority, but in specific ways that emerge at the particular historical junctures at which bishop lists are cited. Political needs changed theological emphases." (Bishop Lists [Piscataway, New Jersey: Gorgias Press, 2005], pp. 8-9)
I've already discussed two early patristic sources, Clement of Rome and Papias.
Regarding another, Williams writes, "Ignatius says nothing of apostolic succession." (p. 68) Ignatius does say a lot about church government, and he sometimes tells his audience to obey church leaders as they would obey the apostles. But he also compares church leaders to the Father and Christ, for example. Such language is commonly used without any sort of succession or infallibility in view (Ephesians 6:5, Philemon 17). Allen Brent, a scholar who has specialized in the study of Ignatius, similarly concludes that there's no relevant concept of apostolic succession in Ignatius (Ignatius Of Antioch [New York, New York: T & T Clark International, 2009], pp. 86-87, 122-129). As Brent notes, Ignatius mostly parallels presbyters, not bishops, with the apostles, and he never refers to himself as a successor of the apostles or as having the authority of Peter or the other apostles. (Ignatius' church, the church of Antioch, was apostolic.) He may have not referred to a monarchical bishop when writing to Rome because that form of church government hadn't developed in Rome yet, as other sources from the same time period also suggest. It should be kept in mind that Ignatius puts a lot of emphasis on issues of church government, so the absence of Dave's concept of apostolic succession in Ignatius' writings (as well as the absence of a papacy) is accordingly significant.
Justin Martyr, like other early patristic sources, makes many explicit references to the authority of the Father, Christ, the Holy Spirit, and scripture, and he appeals to many extra-Biblical sources (government records, written or oral traditions about Jesus, etc.). But he says nothing of apostolic succession (or an infallible church), and his Jewish and Gentile opponents raise no objections to Christianity that assume Christian belief in such concepts.
Much the same can be said of Celsus, who wrote against Christianity late in the second century. However, we do begin to see some concepts of apostolic succession that are closer to Dave's view around the time when Celsus wrote. The first relevant source is Hegesippus, and I'll discuss him in my next post.
However, before I move on to Hegesippus, it should be noted that many other Christian sources from the late second century onward continue to write at length, including in many contexts relevant to apostolic succession as Dave defines it, without mentioning the concept. We shouldn't assume that the concept was universally accepted, or even accepted by a majority, once a Christian somewhere advocates it. The appearance of concepts of apostolic succession in Hegesippus, Irenaeus, and other sources is significant evidence of the popularity of such ideas. But the earlier absence of those ideas and their ongoing absence or lesser emphasis in other sources is significant as well. Both lines of evidence have to be addressed.