Local weak limits for collapsed branching processes with random out-degrees

TL;DR

This paper establishes local weak limits for collapsed branching processes with random out-degrees, revealing their structure as stopped continuous-time branching processes and exploring implications for network degree distributions.

Contribution

It introduces a novel framework for analyzing local weak limits of collapsed branching processes with random out-degrees, including applications to network models.

Findings

Local weak limits are continuous-time branching processes stopped at exponential times.

In-components of multiple vertices converge to i.i.d. copies of the limit.

Results provide bounds and phase transition insights for in-degree distributions.

Abstract

We obtain local weak limits in probability for Collapsed Branching Processes (CBP), which are directed random networks obtained by collapsing random-sized families of individuals in a general continuous-time branching process. The local weak limit of a given CBP, as the network grows, is shown to be a related continuous-time branching process stopped at an independent exponential time. The proof involves the construction of an explicit coupling of the in-components of vertices with the limiting object. We also show that the in-components of a finite collection of uniformly chosen vertices locally weakly converge (in probability) to i.i.d. copies of the above limit, reminiscent of propagation of chaos in interacting particle systems. We obtain as special cases novel descriptions of the local weak limits of directed preferential and uniform attachment models. We also outline some applications of our results for analyzing the limiting in-degree and PageRank distributions. In particular, upper and lower bounds on the tail of the in-degree distribution are obtained and a phase transition is detected in terms of the growth rate of the attachment function governing reproduction rates in the branching process.