Functional Central Limit Theorems for Inhomogeneous Random Graphs

Inhomogeneous random graphs (IRGs) are generalisations of the Erdos-Renyi random graph. Several random graph models can be viewed as inhomogeneous random graphs with an appropriate type space. Although tools to understand the law of large numbers limits and phase transitions are well established, there has been little progress on central limit theorems for IRGs. The main reason for this is that combinatorial techniques, which are the primary tools used for Erdős–Rényi random graphs, fail in this setting. In this work, we study the dynamic version of IRG and establish 

1. Functional central limit theorems for the infinite vector of microscopic type-densities and characterizations of the limits as infinite-dimensional Gaussian processes in a certain Banach space in both subcritical and supercritical regimes.
2. Functional (joint) central limit theorems for macroscopic observables of the giant component in the supercritical regime, including size, surplus, and its type composition. 
3. Leveraging these results, we also establish the central limit theorem for the weight of the minimum spanning tree with iid Exponential edge weights on dense graph sequences driven by an underlying finite type graphon.