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Testing Properties of Edge Distributions

Yumou Fei

Abstract

We initiate the study of distribution testing for probability distributions over the edges of a graph, motivated by the closely related question of ``edge-distribution-free'' graph property testing. The main results of this paper are nearly-tight bounds on testing bipartiteness, triangle-freeness and square-freeness of edge distributions, whose sample complexities are shown to scale as $Θ(n)$, $n^{4/3\pm o(1)}$ and $n^{9/8\pm o(1)}$, respectively. The technical core of our paper lies in the proof of the upper bound for testing square-freeness, wherein we develop new techniques based on certain birthday-paradox-type lemmas that may be of independent interest. We will discuss how our techniques fit into the general framework of distribution-free property testing. We will also discuss how our results are conceptually connected with Turán problems and subgraph removal lemmas in extremal combinatorics.

Testing Properties of Edge Distributions

Abstract

We initiate the study of distribution testing for probability distributions over the edges of a graph, motivated by the closely related question of ``edge-distribution-free'' graph property testing. The main results of this paper are nearly-tight bounds on testing bipartiteness, triangle-freeness and square-freeness of edge distributions, whose sample complexities are shown to scale as , and , respectively. The technical core of our paper lies in the proof of the upper bound for testing square-freeness, wherein we develop new techniques based on certain birthday-paradox-type lemmas that may be of independent interest. We will discuss how our techniques fit into the general framework of distribution-free property testing. We will also discuss how our results are conceptually connected with Turán problems and subgraph removal lemmas in extremal combinatorics.
Paper Structure (59 sections, 47 theorems, 183 equations, 2 figures)

This paper contains 59 sections, 47 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Distribution-Free Testing of Functions
  3. Additional Results
  4. Related Work
  5. Further Motivation for Edge-Distribution-Free Testing
  6. Technical Overview
  7. General Framework
  8. Applying the Framework to Graph Problems
  9. Subgraph-Removal for Sparse Graphs
  10. Preliminaries
  11. General Notations
  12. Sets.
  13. Probability.
  14. Sampling.
  15. Empirical vectors.
  16. ...and 44 more sections

Key Result

Theorem 1.1

The sample complexities of testing bipartiteness, triangle-freeness and square-freeness of edge distributions on $n$ vertices are $\Theta(n)$, $n^{4/3\pm o(1)}$ and $n^{9/8\pm o(1)}$, respectively.

Figures (2)

  • Figure 1: Relations between functions in Case 1
  • Figure 2: Relations between functions in Case 2

Theorems & Definitions (111)

  • Theorem 1.1: Informal
  • Definition 1.2
  • Definition 1.3: goldreich1998property
  • Proposition 1.4
  • Theorem 1.5: Formal version of Theorem \ref{['thm:main_informal']}
  • Remark 1
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9: Implicit in chen2024distribution
  • ...and 101 more