Mobile Geometric Graphs, and Detection and Communication Problems in Mobile Wireless Networks

Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks, in which the nodes move over time; moreover, these results often make unrealistic assumptions about node mobility such as the ability to make very large jumps. In this paper we consider a realistic model for mobile wireless networks which we call mobile geometric graphs, and which is a natural extension of the random geometric graph model. We study two fundamental questions in this model: detection (the time until a given “target” point - which may be either fixed or moving - is detected by the network), and percolation (the time until a given node is able to communicate with the giant component of the network). For detection, we show that the probability that the detection time exceeds t is \exp(-\Theta(t/\log t)) in two dimensions, and \exp(-\Theta(t)) in three or more dimensions, under reasonable assumptions about the motion of the target. For percolation, we show that the probability that the percolation time exceeds t is \exp(-\Omega(t^\frac{d}{d+2})) in all dimensions d\geq 2. We also give a sample application of this result by showing that the time required to broadcast a message through a mobile network with n nodes above the threshold density for existence of a giant component is O(\log^{1+2/d} n) with high probability.

💡 Research Summary

The paper introduces a mathematically rigorous model for mobile wireless networks called Mobile Geometric Graphs (MGG), which extends the classic Random Geometric Graph (RGG) by allowing nodes to move over time. In an MGG, nodes are initially placed according to a Poisson point process of intensity λ in ℝⁿ (n ≥ 2). At each discrete time step each node independently chooses a random direction and moves a fixed distance (or follows a continuous‑time Brownian motion), preserving the same speed for all nodes. At time t a graph G(t) is formed by connecting any two nodes whose Euclidean distance does not exceed a fixed communication radius r. This construction captures the essential features of realistic ad‑hoc or sensor networks where devices are mobile but their motion is bounded and isotropic.

The authors focus on two fundamental performance metrics: (i) Detection time – the first time τ_det at which a designated “target” point (which may be static or itself moving according to a similar random walk) falls within distance r of at least one node, and (ii) Percolation time – the first time τ_perc at which a given node becomes connected to the giant component C_∞(t) that exists when λ exceeds the percolation threshold λ_c. Both metrics are expressed as tail probabilities, i.e., the probability that the respective time exceeds a given t.

Detection results.
For a static or randomly moving target the authors prove dimension‑dependent exponential tails. In two dimensions the probability that τ_det > t decays as
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