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On the Asymptotics of the Connectivity Probability of Random Bipartite Graphs

Boris Chinyaev

TL;DR

This work addresses the exact asymptotics of the connectivity probability $P_{n,m}(p)$ for the bipartite random graph $G(n,m,p)$, allowing $p$ to scale with $(n,m)$. It adapts the inhomogeneous random-walk framework from $G(n,p)$ to the bipartite setting via a nonasymptotic exploration process, establishing a representation that reduces the problem to a conditioned nonnegative trajectory of a two-walk system built from Poisson increments. The main result characterizes $P_{n,m}(p)$ across several regimes $p(n,m)$, including cases where the base edge probability vanishes slowly or rapidly, by providing explicit asymptotic formulas and monotonicity properties with respect to $p$. This framework lays the groundwork for rigorous asymptotics and practical sampling methods for connected bipartite graphs, with potential extensions to broader random-graph models.

Abstract

In this paper, we analyze the exact asymptotic behavior of the connectivity probability in a random binomial bipartite graph $G(n,m,p)$ under various regimes of the edge probability $p=p(n)$. To determine this probability, a method based on the analysis of inhomogeneous random walks is proposed.

On the Asymptotics of the Connectivity Probability of Random Bipartite Graphs

TL;DR

This work addresses the exact asymptotics of the connectivity probability for the bipartite random graph , allowing to scale with . It adapts the inhomogeneous random-walk framework from to the bipartite setting via a nonasymptotic exploration process, establishing a representation that reduces the problem to a conditioned nonnegative trajectory of a two-walk system built from Poisson increments. The main result characterizes across several regimes , including cases where the base edge probability vanishes slowly or rapidly, by providing explicit asymptotic formulas and monotonicity properties with respect to . This framework lays the groundwork for rigorous asymptotics and practical sampling methods for connected bipartite graphs, with potential extensions to broader random-graph models.

Abstract

In this paper, we analyze the exact asymptotic behavior of the connectivity probability in a random binomial bipartite graph under various regimes of the edge probability . To determine this probability, a method based on the analysis of inhomogeneous random walks is proposed.

Paper Structure

This paper contains 6 sections, 4 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exploration process in the Bipartite Graph
  4. Connectivity Probability and Random Walks
  5. Formulation of the Main Theorem
  6. Conclusion

Key Result

Lemma 2.1

Let $G(n,p)$ be an Erdős–Rényi graph. Then the connectivity probability is given by where $S_k = \sum_{i=1}^{k} X_i$, and $X_i$ are independent random variables such that $X_i + 1\sim Poiss\left(\lambda_i\right)$, with

Figures (2)

  • Figure 1: Plot of the expected value and sample realizations of $S_k$.
  • Figure 2: Plot of sample realizations of $B_k$ and $V_{k}^{A}.$

Theorems & Definitions (6)

  • Lemma 2.1: chinyaev2024er_eng
  • Lemma 2.2: Connectivity Probability of $G(n,m,p)$
  • proof
  • Lemma 2.3: Monotonicity
  • proof
  • Theorem 2.1: On the Connectivity Probability of $G(n,m,p)$