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Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs

Sayak Chatterjee, Anirban Chatterjee, Abhinav Chakraborty, Bhaswar B. Bhattacharya

Abstract

In this paper, we derive the asymptotic distribution of the number of copies of a fixed graph $H$ in a random graph $G_n$ sampled from a sparse graphon model. Specifically, we provide a refined analysis that separates the contributions of edge randomness and vertex-label randomness, allowing us to identify distinct sparsity regimes in which each component dominates or both contribute jointly to the fluctuations. As a result, we establish asymptotic normality for the count of any fixed graph $H$ in $G_n$ across the entire range of sparsity (above the containment threshold for $H$ in $G_n$). These results provide a complete description of subgraph count fluctuations in sparse inhomogeneous networks, closing several gaps in the existing literature that were limited to specific motifs or suboptimal sparsity assumptions.

Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs

Abstract

In this paper, we derive the asymptotic distribution of the number of copies of a fixed graph in a random graph sampled from a sparse graphon model. Specifically, we provide a refined analysis that separates the contributions of edge randomness and vertex-label randomness, allowing us to identify distinct sparsity regimes in which each component dominates or both contribute jointly to the fluctuations. As a result, we establish asymptotic normality for the count of any fixed graph in across the entire range of sparsity (above the containment threshold for in ). These results provide a complete description of subgraph count fluctuations in sparse inhomogeneous networks, closing several gaps in the existing literature that were limited to specific motifs or suboptimal sparsity assumptions.

Paper Structure

This paper contains 21 sections, 17 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fluctuations of Subgraph Counts in Sparse $W$-Random Graphs
  3. Preliminaries
  4. Statements of the Results
  5. A General Conditional CLT and Proof Outline
  6. Prior Work
  7. Erdős-Rényi Model
  8. Sparse Graphon Model
  9. Dense Graphon Model
  10. Proof of Proposition \ref{['prop:threshold']}
  11. Proof of Lemma \ref{['lem:meanvar']}
  12. Proof of Proposition \ref{['prop:main']}
  13. Proof of Theorem \ref{['thm:regular']}
  14. Proof of Theorem \ref{['thm:irregular']}
  15. Conclusions
  16. ...and 6 more sections

Key Result

Proposition 2.2

Fix a graph $H = (V(H), E(H))$ with $|E(H)| \geq 1$ and a graphon $W$ such that $t(H, W) > 0$. Then for $G_n \sim G(n, \rho_n, W)$ the following holds:

Figures (3)

  • Figure 1: (a) A graph which is balanced, but not strictly balanced; (b) an unbalanced graph.
  • Figure 2: (a) A strictly balanced graph that is not strongly balanced; and (b) a balanced (but not strictly balanced) graph, which is not strongly balanced.
  • Figure 3: The $(a, b)$- vertex join of the 2 copies of $H$.

Theorems & Definitions (36)

  • Definition 1.1: Sparse graphon model
  • Definition 2.1: Balanced graph
  • Proposition 2.2
  • Definition 2.4: $H$-regularity
  • Remark 2.5
  • Theorem 2.6
  • Definition 2.7: Strongly balanced graph
  • Remark 2.8
  • Proposition 2.9
  • Remark 2.10
  • ...and 26 more