Near-critical bipartite configuration models and their associated intersection graphs

TL;DR

This paper analyzes the behavior of bipartite configuration models and their intersection graphs near criticality, revealing that component sizes converge to excursion lengths of a thinned Lévy process and deriving asymptotic triangle counts.

Contribution

It extends the analysis of bipartite configuration models at criticality, establishing convergence of component sizes to Lévy process excursions and calculating asymptotic triangle counts in the associated intersection graphs.

Findings

Component sizes converge to Lévy process excursions at criticality.

Asymptotic triangle counts in the intersection graph are derived.

Results align with recent findings for bipartite Erdős-Rényi graphs.

Abstract

Recently, van der Hofstad, Komj\'{a}thy, and Vadon (2022) identified the critical point for the emergence of a giant connected component for the bipartite configuration model (BCM) and used this to analyze its associated random intersection graph (RIG) (2021). We extend some of this analysis to understand the graph at, and near, criticality. In particular, we show that under certain moment conditions on the empirical degree distributions, the number of vertices in each connected component listed in decreasing order of their size converges, after appropriate re-normalization, to the excursion lengths of a certain thinned L\'{e}vy process. Our approach allows us to obtain the asymptotic triangle counts in the RIG built from the BCM. Our limits agree with the limits recently identified by Wang (2023) for the RIG built from the bipartite Erd\H{o}s-R\'{e}nyi random graph.