Fluctuations of subgraph counts in exponential random graphs

Abstract

In this paper, we study fluctuations of fixed subgraph counts in dense exponential random graphs on n vertices. We compare each normalized subgraph count V with the normalized edge count W: their difference V−W is O(n^−1/2) in L², whereas 𝔼[(V−W)f(W)]=O(n⁻¹) for every $1$-Lipschitz function f. We also obtain O(n⁻¹) Kolmogorov and Wasserstein bounds, optimal for the edge count, with relative variance error of the same order. These estimates hold uniformly on compact subsets of the ferromagnetic subcritical and absolute-coefficient Dobrushin regions, and under small fixed signed perturbations of the former. Using this uniform control, we derive a local Gaussian likelihood expansion with rank-one leading Fisher information and two-sided moderate deviations with an explicit cubic correction. The proofs combine two applications of the normal Stein equation, covariance estimates from Glauber dynamics and exponential tilting.

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BibTeX RIS

BibTeX

@misc{liu2026fluctuations,
  title = {{Fluctuations of subgraph counts in exponential random graphs}},
  author = {Hui-Lin Liu and Song-Hao Liu and Dawei Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28078},
  eprint = {2610.28078},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28078},
}

RIS

TY  - UNPB
TI  - Fluctuations of subgraph counts in exponential random graphs
AU  - Hui-Lin Liu
AU  - Song-Hao Liu
AU  - Dawei Lu
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28078
UR  - https://arxiv.org/abs/2610.28078
AB  - In this paper, we study fluctuations of fixed subgraph counts in dense exponential random graphs on n vertices. We compare each normalized subgraph count V with the normalized edge count W: their difference V−W is O(n^−1/2) in L^2, whereas 𝔼[(V−W)f(W)]=O(n^−1) for every $1$-Lipschitz function f. We also obtain O(n^−1) Kolmogorov and Wasserstein bounds, optimal for the edge count, with relative variance error of the same order. These estimates hold uniformly on compact subsets of the ferromagnetic subcritical and absolute-coefficient Dobrushin regions, and under small fixed signed perturbations of the former. Using this uniform control, we derive a local Gaussian likelihood expansion with rank-one leading Fisher information and two-sided moderate deviations with an explicit cubic correction. The proofs combine two applications of the normal Stein equation, covariance estimates from Glauber dynamics and exponential tilting.
ER  -

Abstract

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