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Component sizes of rank-2 multiplicative random graphs

David Clancy

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 Lévy process, perhaps with random coefficients. The three critical regimes are those identified by Bollobás, 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.

Component sizes of rank-2 multiplicative random 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 Lévy process, perhaps with random coefficients. The three critical regimes are those identified by Bollobás, 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.

Paper Structure

This paper contains 31 sections, 52 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Component sizes of rank-1 models
  4. Rank-2 model
  5. Classic regime: $\kappa_{ij}>0$
  6. Interacting regime: $\kappa_{12} = 0$
  7. Nearly bipartite regime: $\kappa_{ii} = 0$
  8. Connections with Related Results
  9. Classic Regime
  10. Bipartite Regime
  11. Exploring the graph
  12. First hitting times of fields
  13. Fields and Random Graphs
  14. Bipartite Analog
  15. Some Results in the $M_1$ topology
  16. ...and 16 more sections

Key Result

Theorem 2.1

Let $({\mathcal{C}}_j^{(n)};j\ge 1)$ be the connected components of ${\mathcal{G}}({\mathbf{z}}^{(n)},q)$ listed in decreasing order of mass. If ${\mathbf{z}}^{(n)}\rightsquigarrow (\beta,\boldsymbol{\theta})\in \mathcal{I}$ and eqn:qrange holds then

Theorems & Definitions (76)

  • Theorem 2.1: Aldous and Limic AL.98
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Theorem 3.1: Baslingker et al. BBBSW.23
  • ...and 66 more