Computer science fields that deal with peer-to-peer systems
Distributed systems, algorithms and even graph theory are fields that are likely to be relevant.

But, in a sense, "fields" aren't particularly relevant. You don't get a PhD "in a field" but, rather, by finding an advisor who wants to do a research project with you. That person would probably be in one of the fields I've suggested but being in a field isn't a formal status.