Information Dissemination via Random Walks in d-Dimensional Space

TL;DR

This paper analyzes how quickly information spreads among mobile agents performing random walks in three-dimensional space, revealing a phase transition phenomenon that differs from lower-dimensional cases.

Contribution

It extends previous work to three dimensions, providing tight bounds on dissemination time and uncovering a phase transition in behavior for d=3.

Findings

Dissemination time bounds are established for d=3.

A phase transition in spreading behavior occurs in 3D space.

Differences in dissemination dynamics between 2D and 3D spaces.

Abstract

We study a natural information dissemination problem for multiple mobile agents in a bounded Euclidean space. Agents are placed uniformly at random in the $d$-dimensional space $\{-n, ..., n\}^d$ at time zero, and one of the agents holds a piece of information to be disseminated. All the agents then perform independent random walks over the space, and the information is transmitted from one agent to another if the two agents are sufficiently close. We wish to bound the total time before all agents receive the information (with high probability). Our work extends Pettarin et al.'s work (Infectious random walks, arXiv:1007.1604v2, 2011), which solved the problem for $d \leq 2$. We present tight bounds up to polylogarithmic factors for the case $d = 3$. (While our results extend to higher dimensions, for space and readability considerations we provide only the case $d=3$ here.) Our results show the behavior when $d \geq 3$ is qualitatively different from the case $d \leq 2$. In particular, as the ratio between the volume of the space and the number of agents varies, we show an interesting phase transition for three dimensions that does not occur in one or two dimensions.