Dispersion processes

12/09/2017
by   Colin Cooper, et al.
0

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 δ>1, with high probability, if M < n/2(1-δ), then the process finishes in O( n) steps, whereas if M > n/2(1+δ), then the process needs e^Ω(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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/04/2023

Dispersion on the Complete Graph

We consider a synchronous process of particles moving on the vertices of...
research
08/28/2018

The dispersion time of random walks on finite graphs

We study two random processes on an n-vertex graph inspired by the inter...
research
03/22/2019

Best-of-Three Voting on Dense Graphs

Given a graph G of n vertices, where each vertex is initially attached a...
research
02/15/2019

The excluded area of two-dimensional hard particles

The excluded area between a pair of two-dimensional hard particles with ...
research
08/03/2023

Interactive High-Resolution Simulation of Granular Material

We introduce a particle-based simulation method for granular material in...
research
07/26/2017

Pileup Mitigation with Machine Learning (PUMML)

Pileup involves the contamination of the energy distribution arising fro...
research
08/16/2021

Smoluchowski processes and nonparametric estimation of functionals of particle displacement distributions from count data

Suppose that particles are randomly distributed in ^d, and they are subj...

Please sign up or login with your details

Forgot password? Click here to reset