Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Asymptotics of the Connectivity Probability of Erdos-Renyi Graphs

B. Chinyaev, A. Shklyaev

TL;DR

This work addresses the sparse-regime connectivity problem for $G(n,p)$ by introducing an inhomogeneous Poisson increment random-walk representation that conditions on a bridge to analyze connectivity. The authors derive a non-asymptotic formulation $P_n(p) = \left(1-(1-p)^n\right)^{n-1} \mathbf{P}(S_k \ge 0, 0<k<n \mid S_n=-1)$ with $X_i+1\sim \mathrm{Poiss}(\lambda_i)$ and provide a rigorous asymptotic classification across four regimes of $p=c_n/n$, including new and known results. The core contribution is a set of auxiliary lemmas on homogeneous meanders, Poisson-bridge comparisons, and bounds for hitting $-1$ that enable precise asymptotics via a three-interval decomposition of the walk. The framework offers a non-combinatorial, broadly applicable method for sparse random-graph connectivity and suggests potential for efficient generation of conditioned connected graphs and extensions to other random-graph models.

Abstract

In this paper, we investigate the exact asymptotic behavior of the connectivity probability in the Erdos-Renyi graph G(n,p), under different asymptotic assumptions on the edge probability p=p(n). We propose a novel approach based on the analysis of inhomogeneous random walks to derive this probability. We show that the problem of graph connectivity can be reduced to determining the probability that an inhomogeneous random walk with Poisson-distributed increments, conditioned to form a bridge, is actually an excursion

On the Asymptotics of the Connectivity Probability of Erdos-Renyi Graphs

TL;DR

This work addresses the sparse-regime connectivity problem for by introducing an inhomogeneous Poisson increment random-walk representation that conditions on a bridge to analyze connectivity. The authors derive a non-asymptotic formulation with and provide a rigorous asymptotic classification across four regimes of , including new and known results. The core contribution is a set of auxiliary lemmas on homogeneous meanders, Poisson-bridge comparisons, and bounds for hitting that enable precise asymptotics via a three-interval decomposition of the walk. The framework offers a non-combinatorial, broadly applicable method for sparse random-graph connectivity and suggests potential for efficient generation of conditioned connected graphs and extensions to other random-graph models.

Abstract

In this paper, we investigate the exact asymptotic behavior of the connectivity probability in the Erdos-Renyi graph G(n,p), under different asymptotic assumptions on the edge probability p=p(n). We propose a novel approach based on the analysis of inhomogeneous random walks to derive this probability. We show that the problem of graph connectivity can be reduced to determining the probability that an inhomogeneous random walk with Poisson-distributed increments, conditioned to form a bridge, is actually an excursion

Paper Structure

This paper contains 11 sections, 8 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Exploration as a Random Walk
  4. Expression of Graph Connectivity via an Inhomogeneous Random Walk
  5. Main Result
  6. Auxiliary Results
  7. Probability That a Homogeneous Walk Forms a Meander
  8. Lemma on the Comparison of Poisson Bridges
  9. Inequality for the Probability of the Inhomogeneous Random Walk Hitting -1
  10. Proof of the Theorem
  11. 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 the $X_i$ are independent random variables such that $X_i + 1\sim Poiss\left(\lambda_i\right)$, with

Figures (1)

  • Figure 1: Graph of the mathematical expectation and sample realizations of $S_k$ for $n=100$, $p = 3/n$.

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Theorem 2.1: On the Connectivity Probability of a Graph
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.3
  • ...and 7 more