From trees to graphs: collapsing continuous-time branching processes

TL;DR

This paper introduces collapsed branching processes (CBPs) derived from continuous-time branching processes (CTBPs) to model multigraphs with fixed out-degree, extending the analysis of preferential attachment models beyond trees.

Contribution

It develops a novel collapsing procedure on CTBPs to create multigraph models and analyzes their degree distribution, linking it to the original CTBP distribution, especially in preferential attachment contexts.

Findings

Degree distribution of CBPs closely related to original CTBP distribution

First application of CTBPs to study degree distribution in PAMs beyond trees

Extension of branching process models to multigraphs with fixed out-degree

Abstract

Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks, since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree $m\geq 2$. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.