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Near-critical bipartite configuration models and their associated intersection graphs

David Clancy

Abstract

Recently, van der Hofstad, Komjáthy, 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 Lévy 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 Erdős-Rényi random graph.

Near-critical bipartite configuration models and their associated intersection graphs

Abstract

Recently, van der Hofstad, Komjáthy, 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 Lévy 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 Erdős-Rényi random graph.

Paper Structure

This paper contains 31 sections, 50 theorems, 219 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results for the configuration model
  3. Bipartite Erdős-Rényi Model
  4. Main Results for the Random Intersection Graph
  5. Finite Third Moment Case
  6. Connection to random hypergraphs
  7. Infinite Third Moment Case
  8. Overview
  9. Exploration and Weak Convergence
  10. Weak convergence of $Y_n^{\mathfrak{l}}$ and $Y_n^{\mathfrak{r}}$
  11. Weak Convergence of ${Z}_n^{(r_n)}$
  12. Preliminary Tightness Results
  13. Number of Cycles Constructed
  14. Finishing up
  15. Basic Scaling Relationships
  16. ...and 16 more sections

Key Result

Theorem 1.1

Suppose both $D_n^{\mathfrak{l}}\to D_\infty^{\mathfrak{l}}$ and $D_n^{\mathfrak{l}}\to D_\infty^{\mathfrak{r}}$ in $\mathscr{W}_1$ and that $\mathbb{P}(D_\infty^{\mathfrak{l}}=2)+\mathbb{P}(D_\infty^{\mathfrak{r}}=2)<2$. Let ${\mathcal{C}}_{1}^{(n)}$ denote the largest connected component of $\math Moreover, $\xi>0$ if and only if

Theorems & Definitions (76)

  • Theorem 1.1: van der Hofstad, Komjáthy, Vadon vdHKV.22
  • Theorem 1.2: Dhara et al. DvdHvLS.17DvdHvLS.20
  • Theorem 1.3: Wang Wang.23
  • Proposition 1.4: Wang Wang.23
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.8
  • ...and 66 more