Component sizes of rank-2 multiplicative random graphs

TL;DR

This paper investigates the sizes of connected components in rank-2 multiplicative random graphs across three critical regimes, showing convergence to Lévy process excursions and extending prior results in related models.

Contribution

It introduces a unified analysis of component sizes in rank-2 multiplicative graphs across multiple regimes, connecting to Lévy processes and extending existing theoretical results.

Findings

Component sizes converge to Lévy process excursions

Results apply to three critical regimes including bipartite and intersection graph regimes

Extends prior work on stochastic blockmodels and intersection graphs

Abstract

We show that in three different critical regimes, the masses of the connected components of rank-2 multiplicative random graph converge to lengths of excursions of a thinned L\'{e}vy process, perhaps with random coefficients. The three critical regimes are those identified by Bollob\'{a}s, Janson and Riordan (2007), the interacting regime identified by Konarovskyi and Limic (2021), and what we call the nearly bipartite regime which has recently gained interest for its connection to random intersection graphs. Our results are able to extend some of the results by Baslingker et al. (2023) on component sizes of the stochastic blockmodel with two types and those of Federico (2019) and Wang (2023) on the sizes of the connected components of random intersection graphs.