Functional Central limit theorems for microscopic and macroscopic functionals of inhomogeneous random graphs
Shankar Bhamidi, Amarjit Budhiraja, Akshay Sakanaveeti
TL;DR
This work analyzes inhomogeneous random graphs with a finite type space through a dynamic graph-valued process, proving functional central limit theorems for microscopic type-densities as infinite-dimensional conditionally Gaussian processes in a Banach space and establishing joint functional CLTs for macroscopic giant-component observables in the supercritical regime. It then derives a central limit theorem for the weight of the minimum spanning tree on dense graph sequences driven by a finite-type graphon, tying MST fluctuations to the integrated fluctuations of the microscopic densities. The results encompass Erdős-Rényi as a special case and extend to graphon-modulated dense graphs, providing a comprehensive path from microscopic fluctuation analysis to macroscopic giant-component behavior and MST fluctuations. The methodology centers on infinite-dimensional SDEs driven by cylindrical Brownian motion, with careful handling of the critical window and tail contributions to ensure tractable limiting objects. These contributions deepen our understanding of second-order fluctuations in coagulation-like random graph dynamics and offer tools for broader graphon-based models and percolation phenomena.
Abstract
We study inhomogeneous random graphs with a finite type space. For a natural generalization of the model as a dynamic network-valued process, the paper establishes the following results: (a) Functional central limit theorems for the infinite vector of microscopic type-densities and characterizations of the limits as infinite-dimensional conditionally Gaussian processes in a certain Banach space. (b) Functional (joint) central limit theorems for macroscopic observables of the giant component in the supercritical regime including size, surplus and number of vertices of various types in the giant component. As a corollary this provides central limit theorems for the size of the largest connected component, its surplus, and its type vector, for percolation on dense graphs obtained from a finite type Graphon. (c) Central limit theorem for the weight of the minimum spanning tree with random i.i.d. Exponential edge weights on dense graph sequences driven by an underlying finite type graphon.