Dispersion processes

TL;DR

This paper analyzes a synchronous dispersion process on various graphs, showing phase transitions in the time to complete based on the number of particles relative to the graph size, with implications for different graph structures.

Contribution

It provides new bounds and phase transition results for the dispersion process on complete graphs, star graphs, and other structures, including lazy variants and diverse graph classes.

Findings

Fast dispersion in complete and star graphs for low particle counts

Exponential time needed for high particle counts in certain graphs

Bounds on dispersion time and maximum distance for paths, trees, and hypercubes

Abstract

We study a synchronous dispersion process in which $M$ particles are initially placed at a distinguished origin vertex of a graph $G$. At each time step, at each vertex $v$ occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbour of $v$ chosen independently and uniformly at random. The dispersion process ends once the particles have all stopped moving, i.e. at the first step at which each vertex is occupied by at most one particle. For the complete graph $K_n$ and star graph $S_n$, we show that for any constant $\delta>1$, with high probability, if $M \le n/2(1-\delta)$, then the process finishes in $O(\log n)$ steps, whereas if $M \ge n/2(1+\delta)$, then the process needs $e^{\Omega(n)}$ steps to complete (if ever). We also show that an analogous lazy variant of the process exhibits the same behaviour but for higher thresholds, allowing faster dispersion of more particles. For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes (in terms of $M$) we give bounds on the time to finish and the maximum distance traveled from the origin as a function of the number of particles $M$.