Scaling Limits and Generic Bounds for Exploration Processes

TL;DR

This paper analyzes exploration algorithms on random graphs and geometric graphs, deriving scaling limits, central limit theorems, and generic bounds for exploration processes, with applications to random sequential adsorption.

Contribution

It provides exact scaling limits and a CLT for homogeneous graphs, and introduces dimension-independent bounds for exploration on geometric graphs.

Findings

Exact limits for homogeneous graphs

Central limit theorem for the jamming constant

Dimension-independent bounds for geometric graph exploration

Abstract

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.