High-altitude platform stations (HAPSs) can extend wireless service across regions where fiber and dense terrestrial infrastructure are difficult to deploy. Coverage footprints alone, however, do not prove that a large random deployment forms a continuously connected service region. A collection of isolated coverage islands can still leave long paths broken.
A new public preprint studies this system-level transition using percolation theory. Rather than asking whether one user is covered, it asks when an unbounded network model develops a large connected component.
Three architectures distribute the relay burden differently
The paper compares direct HAPS-to-device service, a HAPS-to-gateway-to-device architecture, and a hybrid of the two. In the direct scheme, aerial density and air-to-ground coverage dominate. The gateway-assisted design also depends on the spatial availability of terrestrial gateways. The hybrid can use either path, expanding the possible connectivity graph.
These are not merely three link budgets. They create different random network topologies and therefore different thresholds for the emergence of continuous connectivity.
Connectivity changes abruptly around a critical density
Percolation probability is the primary metric. The analysis distinguishes subcritical regimes, where only finite connected clusters exist, from supercritical regimes, where a network-spanning component appears with nonzero probability.
The authors show that increasing HAPS density or gateway density produces a phase transition from zero to nonzero percolation probability. Numerical results place the critical-condition curve between derived lower and upper bounds for the evaluated coverage models.
Bounds can guide deployment before exact design is known
A bounded critical region can help planners reject densities that are clearly insufficient and avoid treating every denser deployment as equally necessary. The public abstract suggests this can reduce upfront cost when selecting among direct, gateway-assisted, and hybrid architectures.
That conclusion remains model-dependent. Percolation in an infinite random geometry is a planning abstraction, not a site plan. Finite boundaries, terrain, atmospheric loss, backhaul capacity, energy availability, failures, regulation, and correlated placement can shift the practical threshold.
Research notes
Connectivity of HAPS-Based Solutions for Large-Scale Wireless Networks: A Percolation Theory Analysis
Authors: Hao Lin, Mustafa A Kishk, and Mohamed-Slim Alouini.
Status: Public arXiv record dated 2 September 2026.
What the public evidence establishes: The work applies percolation theory to direct, gateway-assisted, and hybrid HAPS networks and derives bounds around density-driven connectivity phase transitions.
Limits: Infinite-network and stochastic-geometry results do not by themselves establish finite-site coverage, cost, reliability, or behavior under correlated infrastructure and real propagation.