Multiple Random Walks in Random Regular Graphs

We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random $r$-regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for $r\geq3$ the cover time is asymptotic to $\theta_r n\ln n$, where $\theta_r=(r-1)/(r-2)$. In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph $G$. For $k$ independent walks on $G$, the cover time $C_G(k)$ is asymptotic to $C_G/k$, where $C_G$ is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is $\theta_r n/\binom{k}{2}$. If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for $k$ walks to broadcast a single piece of information to each other is asymptotic to $\frac{2\ln k}{k}\theta_r n$ as $k,n\rightarrow\infty$. We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of $k$ explosive particles ($k$ even) tends to $(2\ln2)\theta_r n$ as $k\rightarrow\infty$. The case of $n$ coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to $2\theta_r n$. Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with $k$ predators and $\ell$ prey the expected detection time is $\theta_r H_{\ell}n/k$, where $H_{\ell}$ is the $\ell$th harmonic number.