Rank-k random graphs and finite type branching processes

TL;DR

This paper introduces rank-k random graphs, linking their exploration processes to thinned multi-type branching processes, and explores how the number of types relates to the rank.

Contribution

It formulates rank-k random graphs and establishes their connection to thinned multi-type branching processes, especially detailing the rank-2 case.

Findings

Rank-2 random graphs correspond to thinning of 2-type branching processes.

Higher rank graphs' relation to branching processes remains uncertain.

Provides a framework for analyzing exploration processes in complex networks.

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.