Impact of location heterogeneity on random walk mobility models

Abstract

This paper investigates random walk mobility models with location heterogeneity, where different locations may have different neighboring regions.With the assumption of totally n locations, we consider 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 for moving 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 Θ(n) to Θ(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) 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 as we vary 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 (∝ r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−2</sup> and ∝ r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> , respectively). Furthermore, extensive simulation is conducted to verify our observations and enhance the understanding on the effect of network parameters.