Analysis of Random Walk Mobility Models with Location Heterogeneity

Abstract

This paper investigates random walk mobility models with location heterogeneity, where different locations may have different neighboring regions. We consider $n$ locations in a one-dimension network and investigate two cases, i.e., full-range locations where nodes situated have the capability to shuffle throughout the network and long-range locations where nodes are allowed to move to positions nearby within a certain range. In the former situation, with the exact expressions derived, we find location heterogeneity has a critical impact on the first hitting time of random walk, varying from $\Theta (n)$ to $\Theta \left(n^3\right)$ according to different extent of heterogeneity. The result covers, as two special cases, both the classic independent and identically distributed (i.i.d) mobility and traditional random walk when varying the number of full-range locations. In the latter one, our asymptotic results on both the first crossing time and cover time suggest that they are inversely proportional to the range of neighboring region $r$ ( $\propto r^{-2}$ and $\propto r^{-1}$ , respectively). Furthermore, with multiple concurrent random walks introduced, the first hitting time can be drastically decreased and the effect is strengthened if combined with location heterogeneity. In addition, our investigation into the stationary distribution of nodes indicates that the uniformity no longer holds due to different transition probabilities, as a result of location heterogeneity. We also conduct extensive simulation results to verify our observations and enhance the understanding on the impact of network parameters. Based on the insights obtained, we move forward to investigate the impact of location heterogeneity in two-dimension networks.