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A Method for Generating Connected Erdos-Renyi Random Graphs

Boris Chinyaev

TL;DR

This paper addresses sampling connected graphs from the sparse Erdős–Rényi model $G(n,p)$ where connectivity conditioning is exponentially unlikely. It introduces an exact sampler that couples the graph exploration process to an inhomogeneous Poisson random walk, generating the exploration tree and then completing the graph by independently sampling the remaining edges with probability $p$, achieving $O(n)$ expected time for $p=c/n$. The method extends to $G(n,M)$ via an acceptance-rejection step on the number of edges, with a parameter tuning based on the target $M$. Empirical results validate distributional correctness, symmetry of vertex degrees, and the predicted degree behavior, and the authors provide open-source code for implementation.

Abstract

We propose a novel exact algorithm for generating connected Erdos-Renyi random graphs $G(n,p)$. The method couples the graph exploration process to an inhomogeneous Poisson random walk, which yields an exact sampler that runs in $O(n)$ time in the sparse regime $p=c/n$. We also show how the method extends to the $G(n,M)$ model via an additional acceptance-rejection step.

A Method for Generating Connected Erdos-Renyi Random Graphs

TL;DR

This paper addresses sampling connected graphs from the sparse Erdős–Rényi model where connectivity conditioning is exponentially unlikely. It introduces an exact sampler that couples the graph exploration process to an inhomogeneous Poisson random walk, generating the exploration tree and then completing the graph by independently sampling the remaining edges with probability , achieving expected time for . The method extends to via an acceptance-rejection step on the number of edges, with a parameter tuning based on the target . Empirical results validate distributional correctness, symmetry of vertex degrees, and the predicted degree behavior, and the authors provide open-source code for implementation.

Abstract

We propose a novel exact algorithm for generating connected Erdos-Renyi random graphs . The method couples the graph exploration process to an inhomogeneous Poisson random walk, which yields an exact sampler that runs in time in the sparse regime . We also show how the method extends to the model via an additional acceptance-rejection step.

Paper Structure

This paper contains 20 sections, 2 theorems, 29 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary Information
  3. Basic Definitions
  4. Exploration Process of $G(n,p)$
  5. Connection with an Inhomogeneous Poisson Random Walk
  6. Generation of Connected $G(n,p)$
  7. Generation of the Exploration
  8. Generation of a Graph for a Fixed Exploration
  9. Reconstruction of the Tree from the Trajectory.
  10. Generation of the Remaining Edges.
  11. Explanation of the Method's Correctness
  12. Complexity Analysis
  13. Experimental Results
  14. Generation of Connected Graphs $G(n,M)$
  15. Generation Algorithm idea
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $G \sim G(n,p)$ be an Erdős–Rényi graph. Then the connectivity probability of $G$ is given by where $S_k = \sum_{i=1}^{k} (X_i - 1)$, and the $X_i$ are independent random variables $X_i \sim \mathrm{Poiss}(\lambda_i)$. Here $\lambda_i$ are defined in (lambda_i).

Figures (5)

  • Figure 1: Plot of the expected value and examples of realizations of $S_k$ for $n=100$, $p = 3/n$.
  • Figure 2: Visualization of random graphs for $n=30$ and different values of $c$.
  • Figure 3: Empirical vertex degree distribution for $c = 2$, $n = 100$.
  • Figure 4: Average vertex degree as a function of parameter $c$ for $n = 300$.
  • Figure 5: Distribution of the number of edges when generating $G(n,M)$ via $G(n,p)$

Theorems & Definitions (4)

  • Lemma 2.1: chinyaev2024er_eng
  • Corollary 2.1
  • Remark 3.1
  • Remark 3.2