Moderate deviations in random graphs and Bernoulli random matrices

TL;DR

This paper establishes a moderate deviation principle for subgraph counts in Erdős-Rényi graphs and traces of Bernoulli matrices, providing bounds on tail probabilities and extending to symmetric statistics.

Contribution

It introduces a new moderate deviation principle for subgraph counts and matrix traces, using log-Laplace estimates and the Gärtner-Ellis theorem, with applications to symmetric statistics.

Findings

Moderate deviation principle for subgraph counts in Erdős-Rényi graphs.

Upper bounds on tail probabilities for small subgraph occurrences.

Extension of results to symmetric statistics including U-statistics.

Abstract

We prove a moderate deviation principle for subgraph count statistics of Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gaertner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries.