On the moderate deviation principles in the sparse multi-type Erd\H{o}s R\'enyi random graph

TL;DR

This paper establishes moderate deviation principles for key properties of sparse multi-type Erdős-Rényi random graphs, providing explicit rate functions and a law of large numbers for the total number of components.

Contribution

It introduces the first moderate deviation principles for the size and count of connected components in multi-type sparse Erdős-Rényi graphs, with explicit rate functions and a novel proof approach.

Findings

Explicit rate functions for deviations in component sizes and counts

Law of large numbers for total number of components

Representation via multi-dimensional compound Poisson processes

Abstract

This paper investigate the sparse multi-type Erd\H{o}s R\'enyi random graphs studied in S\"{o}derberg~\cite{soderberg2002general} and also Bollob\'as et al.~\cite{bollobas2007phase}. Although the corresponding central limit results are currently unknown, we establish moderate deviation principles for the size of the largest connected component, the number of specific types of connected components, and the total number of connected components. The associated rate functions are provided explicitly. As a byproduct of this work, we present the law of large numbers for the total number of connected components. Our proof methodology relies on representing the multi-type random graph using a conditional multi-dimensional compound Poisson process. We also discuss the properties of related multi-type branching processes and the properties of the matrices in the rate functions.