Asymptotic Normality of Subgraph Counts in Sparse Inhomogeneous Random Graphs

TL;DR

This paper characterizes the asymptotic distribution of subgraph counts in sparse inhomogeneous random graphs, revealing how different sources of randomness influence fluctuations across various sparsity regimes.

Contribution

It provides a comprehensive analysis of subgraph count fluctuations in sparse inhomogeneous graphs, identifying regimes where edge or vertex randomness dominate, and establishes normality across all relevant sparsity levels.

Findings

Asymptotic normality of subgraph counts in all sparsity regimes

Separation of edge and vertex randomness contributions

Complete description of fluctuations in sparse inhomogeneous networks

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.