A Central Limit Theorem for Diffusion in Sparse Random Graphs

TL;DR

This paper establishes a central limit theorem demonstrating that, in sparse random graphs with fixed degrees, the final count of activated nodes in a diffusion process exhibits Gaussian fluctuations under certain conditions.

Contribution

It introduces a central limit theorem for the final activated node count in bootstrap percolation on sparse random graphs with fixed degrees.

Findings

Final activated nodes follow an asymptotically Gaussian distribution.

The theorem applies under specific assumptions on degree and threshold distributions.

Provides a probabilistic understanding of diffusion outcomes in sparse networks.

Abstract

We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of nodes with threshold zero which consists of initially activated nodes, whereas every other node is inactive. Subsequently, in each round, if an inactive node with threshold $\theta$ has at least $\theta$ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more nodes become activated. The main result of this paper provides a central limit theorem for the final size of activated nodes. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated nodes has asymptotically Gaussian fluctuations.