2017/12/09 by Colin Cooper, Cooper, Colin, Andrew McDowell +8
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #Complex Network Analysis Techniques #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.1712.03389
arxiv created 2017/12/09 · openalex publication_date 2017/12/09 · arxiv updated 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Kn and star graph Sn, we show that for any constant δ>1, with high probability, if M ≤ n/2(1-δ), then the process finishes in O(log n) steps, whereas if M ≥ n/2(1+δ), then the process needs eΩ(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.