Bounds on the mixing time and partial cover of ad-hoc and sensor networks

In [Avin, C et al., (2004)], the authors proposed the partial cover of a random walk on a broadcast network to be used to gather information and supported their proposal with experimental results. In this paper, we demonstrate analytically that for sufficiently large broadcast radius r, the partial cover of a random walk on a random broadcast network is in fact efficient and generates a good distribution of the visited nodes. Our result is based on bounding the conductance, which intuitively measures the amount of bottlenecks in a graph. We show that the conductance of a random broadcast network in a unit square is ⊗(r) with high probability, and this bound allows us to analyze properties of the random walk such as mixing time and load balancing. We find that for the random walk to be both efficient and have a high quality cover and partial cover (i.e. rapid mixing), radius at least r_rapid = ⊗(1/poly (log n)) is sufficient and necessary. Experimental results on the random geometric graphs, namely graphs that represent broadcast networks, that resemble the conductance of the 3-dimensional grid indicate that the analytical bounds on efficiency, namely cover time and partial cover time, are not tight. In particular, r = ⊗(1/n¹3/) is sufficient radius to obtain optimal cover time and partial cover time.