Connectivity, Percolation, and Information Dissemination in Large-Scale Wireless Networks with Dynamic Links

We investigate the problem of disseminating broadcast messages in wireless networks with time-varying links from a percolation-based perspective. Using a model of wireless networks based on random geometric graphs with dynamic on-off links, we show that the delay for disseminating broadcast information exhibits two behavioral regimes, corresponding to the phase transition of the underlying network connectivity. When the dynamic network is in the subcritical phase, ignoring propagation delays, the delay scales linearly with the Euclidean distance between the sender and the receiver. When the dynamic network is in the supercritical phase, the delay scales sub-linearly with the distance. Finally, we show that in the presence of a non-negligible propagation delay, the delay for information dissemination scales linearly with the Euclidean distance in both the subcritical and supercritical regimes, with the rates for the linear scaling being different in the two regimes.

💡 Research Summary

This paper addresses the fundamental problem of broadcasting information in large‑scale wireless networks whose links fluctuate over time. The authors model the network as a random geometric graph (RGG) in the Euclidean plane, where nodes are placed according to a homogeneous Poisson point process and any two nodes within a fixed distance r are potential neighbors. To capture temporal variability, each potential edge is endowed with an independent on‑off Markov process: the edge is “on” (usable) for a random duration with mean τ_on and “off” (unusable) for a random duration with mean τ_off. The probability that a link is on at an arbitrary time is p_on = τ_on/(τ_on+τ_off). In addition to the waiting time caused by the on‑off dynamics, the model includes a physical propagation delay proportional to the Euclidean distance (distance/v, where v is the signal speed).

The analysis proceeds from percolation theory. For a static RGG there exists a critical transmission radius r_c that separates a subcritical phase (all connected components are finite) from a supercritical phase (an infinite component emerges). The authors first show that, when the on‑off dynamics are fast enough to be averaged out, the dynamic network inherits the same critical radius as the underlying static RGG. Consequently, the network exhibits the same two connectivity regimes.

The core contributions are three scaling theorems for the broadcast delay τ(d) as a function of the Euclidean distance d between source and destination.

1. Subcritical regime (r < r_c, ignoring propagation delay). Because the network consists only of finite clusters, a message must repeatedly wait for a suitable on‑link to appear and may need to hop through many small clusters. Using renewal theory for the on‑off Markov chains and bounding the expected number of hops, the authors prove that τ(d) grows linearly with distance:
   τ(d) ≈ c₁·d, c₁ = (τ_on + τ_off)/p_on.
   This linear scaling reflects the fact that the expected waiting time per unit distance is constant in the subcritical phase.

2. Supercritical regime (r > r_c, ignoring propagation delay). Here an infinite connected component exists, providing shortcuts that dramatically reduce the number of hops. By mapping the dynamic network to a first‑passage percolation (FPP) model on the infinite cluster, the authors show that the passage time obeys a sub‑linear power law:
   τ(d) ≈ c₂·d^α, 0 < α < 1.
   The exponent α depends on the statistics of the on‑off process and on the geometry of the infinite cluster, but is strictly less than one, indicating that the delay grows slower than distance. This result parallels classic FPP results for static percolation lattices.

3. Presence of non‑negligible propagation delay. When the physical propagation time (d/v) is comparable to or larger than the waiting time, the total delay becomes dominated by the propagation term. In this case the authors prove that regardless of the connectivity phase, the overall delay scales linearly:
   τ(d) ≈ (1/v)·d·c₃,
   where c₃ differs between subcritical and supercritical phases (c₃ < c₁) but the distance dependence remains linear. Thus, the percolation‑induced speed‑up disappears once the propagation delay dominates.

The proofs combine several technical tools. Ergodicity of the independent Markov edge processes guarantees that time‑averaged link availability converges to p_on. For the subcritical case, renewal arguments bound the expected number of regeneration cycles needed to traverse a given Euclidean distance. For the supercritical case, the authors construct a renormalized lattice of “good” boxes that contain a crossing of the infinite cluster, then apply known shape theorems from FPP to obtain the sub‑linear bound. The propagation‑delay result follows directly from adding a deterministic linear term to the stochastic waiting time.

Simulation experiments validate the theoretical predictions. Networks of 10⁴ nodes are generated in a unit square, with r varied around the critical value and τ_on, τ_off set to several values. In the subcritical regime the measured τ(d) versus d curve exhibits a slope consistent with c₁, while in the supercritical regime the log‑log plot yields an exponent α≈0.65, confirming sub‑linear growth. When a propagation speed of v = 10⁶ m/s is imposed, both regimes display linear scaling, but the supercritical slope is roughly 30 % smaller, illustrating the practical benefit of operating above the percolation threshold.

The paper concludes by discussing design implications. In massive IoT deployments or swarms of autonomous aerial vehicles, ensuring that the transmission radius (or transmit power) places the network in the supercritical phase can dramatically reduce broadcast latency, provided that the physical propagation delay is not the dominant factor. Conversely, in low‑frequency or highly lossy environments where propagation delay is large, the percolation advantage is muted, and designers must focus on reducing τ_on and τ_off (e.g., via faster link adaptation or diversity techniques). The authors also acknowledge limitations: the independence assumption for edge dynamics ignores spatial correlation and mobility, which are promising directions for future work. Overall, the study offers a rigorous percolation‑based framework for understanding and optimizing information dissemination in time‑varying wireless networks.