A large-deviations principle for all the components in a sparse inhomogeneous random graph

TL;DR

This paper establishes a large-deviations principle for the entire component structure of an inhomogeneous sparse random graph, providing detailed asymptotics and phase transition criteria.

Contribution

It introduces a comprehensive large-deviations framework for all components in inhomogeneous sparse graphs, including explicit rate functions and minimizers.

Findings

Explicit large-deviations rate function for component statistics

Logarithmic asymptotics for connectivity probability

Criteria for phase transition in the graph

Abstract

We study an inhomogeneous sparse random graph on [N] = {1, . . . , N } as introduced in a seminal paper by Bollobas, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N to infinity, we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size proportional to N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of the graph. In particular, we recover the criterion for the existence of the phase transition given in [5].