Is there a folk theorem for repeated games on networks?
Yes, there are folk theorems for games on networks, depending on information structure and possible communication. Here are some of the most relevant papers:


Ben-Porath, E., &amp; Kahneman, M. (1996). Communication in repeated
games with private monitoring. Journal of Economic Theory, 70(2),
281-297. - public communication, private monitoring
Renault, J., &amp; Tomala, T. (1998). Repeated proximity games.
International Journal of Game Theory, 27(4), 539-559. - complete information, imperfect monitoring
Tomala, T. (2011). Fault reporting in partially known networks and
folk theorems. Operations research, 59(3), 754-763. - partial knowledge of the network and restricted communication
Laclau, M. (2012). A folk theorem for repeated games played on a
network. Games and Economic Behavior, 76(2), 711-737. - private communication


Thanks @Ubiquitous for basically providing the answer.