Rank-k random graphs and finite type branching processes

Abstract

In this note, we investigate fundamental relations between exploration processes in random graphs, and branching processes. We formulate a class of models that we call {\em rank-$k$ random graphs}, and that are special in that their neighborhood explorations can be obtained by a {\em thinning} of multi-type branching processes. We show that any rank-2 random graph can be described in terms of thinning of a 2-type branching process, while for higher rank, it is not clear how many types are needed.