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Connectivity of HAPS-Based Solutions for Large-Scale Wireless Networks: A Percolation Theory Analysis

Hao Lin, Mustafa A Kishk, Mohamed-Slim Alouini

arXiv:2609.02536v1cs.NI

TL;DR

Large-scale continuous Internet coverage is needed for 6G applications in areas where terrestrial infrastructure is difficult to deploy. The paper models direct, gateway-assisted, and hybrid HAPS coverage with percolation theory, showing that increasing HAPS or GW density produces phase transitions and that critical density curves support cost-aware deployment.

  • Problem

    6G applications require continuous Internet services in fiber-less regions, motivating analysis of whether HAPS-based networks can provide large-scale continuous coverage.

  • Method

    The paper evaluates H2D, H2G2D, and hybrid HAPS coverage schemes using percolation probability and analyzes their subcritical, supercritical, and phase-transition conditions.

  • Results

    Critical conditions exist between subcritical and supercritical regions in all three schemes, with critical curves bounded by derived lower and upper bounds.

  • Takeaways & Limitations

    The critical conditions can guide HAPS and GW density selection to reduce CAPEX and OPEX for large-scale Internet coverage.

  • Takeaways & Limitations

    The framework assumes all HAPSs and GWs are available, so practical density planning must account for node failure rates.

Abstract

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In the era of sixth-generation (6G) wireless communication, numerous applications are expected to be realized, including environmental monitoring, smart agriculture, remote education, security protection, and intelligent transportation systems. These scenarios require large-scale, continuous Internet services in forests, rivers, oceans, and road networks, to name a few, where optical cables are difficult to deploy. High-altitude platform stations (HAPSs) emerge as a promising solution, offering low-latency, high-capacity services while facilitating the establishment of vertical heterogeneous networks (vHetNets) in fiber-less areas. This article investigates three HAPS-based solutions, where HAPSs can serve wireless devices directly or via gateway (GW) networks: the HAPS-to-device (H2D) scheme, the HAPS-to-GW-to-device (H2G2D) scheme, and the hybrid scheme. Leveraging percolation theory, we study the feasibility of large-scale continuous Internet coverage, where the key performance indicator (KPI) is the percolation probability. We discuss the subcritical and supercritical cases in different coverage schemes, and prove that the phase transition from zero to nonzero percolation probability appears when increasing the HAPS density or GW density. Numerical results verify that the curve of the critical condition of the phase transition exists between the derived lower bound and upper bound, which can help reduce the upfront cost of HAPS-based vHetNet solutions.

I. INTRODUCTION

The paper studies large-scale continuous connectivity for HAPS-based networks using percolation theory, motivated by 6G services in areas where terrestrial infrastructure is difficult to deploy. It introduces three coverage schemes and analyzes their critical connectivity conditions.

  • HAPSs can support seamless, low-latency, high-capacity connectivity by complementing terrestrial networks in rural, remote, and post-disaster regions.
  • Percolation probability measures whether a large-scale continuous coverage area can form, with phase transition marking the critical condition for network feasibility.The framework is intended to support density decisions that reduce CAPEX and OPEX for specified coverage requirements.
  • The paper defines H2D, H2G2D, and hybrid coverage models and corresponding random graphs for HAPS-based continuous Internet coverage.
  • It distinguishes subcritical cases with zero percolation probability from supercritical cases with non-zero probability.
  • Theoretical critical-condition curves for HAPS and GW densities are bounded by derived lower and upper bounds, supporting density choices intended to reduce deployment costs.

A. HAPS-to-Device Coverage

The H2D scheme models direct communication between HAPSs and ground devices as a single-layer coverage network. Percolation occurs when overlapping HAPS service areas form a giant connected component.

  • HAPS coverage centers follow a two-dimensional Poisson point process with density λH, and each HAPS serves devices within distance dH2D.
  • The H2D random graph connects HAPSs whose centers are at most 2rH2D apart because their coverage areas overlap.
  • The H2D percolation probability is the probability that the giant component containing the origin has infinite size.

B. HAPS-to-GW-to-Device Coverage

The H2G2D scheme connects devices to HAPSs indirectly through gateway coverage and multi-hop GW-to-GW links. Its connectivity depends jointly on HAPS and GW densities and the resulting connected GW service areas.

  • HAPSs and GWs are modeled as two-dimensional Poisson point processes with densities λH and λG, respectively.
  • GWs form a mesh through G2G links, allowing devices to reach HAPSs indirectly through H2G, G2G, and G2D links.
  • GWs outside direct HAPS coverage can connect to the core network through multi-hop G2G communication.
  • The H2G2D random graph uses connected GWs as vertices and overlapping GW service areas to establish links.
  • The design objectives minimize either λH or λG subject to a positive H2G2D percolation probability.

C. Hybrid Coverage

The hybrid scheme combines direct HAPS-to-device coverage with indirect coverage through GW meshes. Its random graph unifies both components, and its density objectives require positive hybrid percolation probability.

  • Hybrid coverage lets devices connect directly to HAPSs or indirectly through GW mesh networks.
  • The hybrid random graph combines H2D vertices and edges with H2G2D vertices and edges plus HAPS–GW connections.
  • Hybrid percolation probability is defined as the probability that the infinite giant component containing the origin exists.
  • The hybrid design separately minimizes HAPS density or GW density subject to positive hybrid percolation probability.

III. CRITICAL STATES ANALYSIS

The analysis classifies hexagonal faces as open or closed according to coverage and uses their probabilities to identify zero- and non-zero-percolation regimes. MCC-based bounds provide sufficient conditions for these critical states.

  • A hexagonal face is closed when any point cannot be covered; if closed-face probability exceeds 1/2, a closed circuit can interrupt device service.The circuit surrounds the origin and prevents a typical moving device from maintaining service.
  • The MCC construction bounds closed-face probability below by the probability that the corresponding MCC is not covered.If any point in the MCC cannot be covered, the corresponding hexagon cannot be covered.
  • A hexagonal face is open when it is covered by Internet service; open-face probability above 1/2 yields a giant connected component and non-zero percolation.

A. HAPS-to-Device Coverage

The H2D analysis derives sufficient lower and upper bounds on HAPS density for zero and non-zero percolation, then establishes a critical density between them. A closed-form critical-density expression is most practical for networks much larger than the HAPS coverage radius.

  • Sub-critical Case: The sub-critical bound follows from P{Tl is not covered} = e−λHπ(rH2D+a)2.
  • Sub-critical Case: When HAPS density is below the lower bound λH2D H,L, the H2D percolation probability is zero.
  • Super-critical Case: The super-critical condition uses at least one HAPS within radius rH2D−a of the MCC center to cover every device in the MCC.
  • Super-critical Case: When HAPS density exceeds the upper bound λH2D H,U, the H2D percolation probability is non-zero.
  • Critical condition: The H2D percolation probability is non-decreasing with HAPS density, so a critical density exists between the derived lower and upper bounds.
  • Critical condition: For a → 0, the closed-form HAPS critical density lies between the bounds, but finite networks may show rapid probability growth only above this value.The expression is practical when the network is much larger than the coverage radius and giant connected components can be infinite.

B. HAPS-to-GW-to-Device Coverage

The H2G2D scheme has zero percolation in a subcritical region and non-zero percolation in a supercritical region. Increasing either HAPS or GW density does not decrease percolation probability, yielding a critical phase-transition curve between these regions.

  • Sub-critical Case: In the subcritical case, a hexagonal face is closed when no GW lies within radius rG2D + a of its associated MCC.Under this condition, devices inside the MCC cannot be covered.
  • Sub-critical Case: The H2G2D percolation probability is zero under the sufficient subcritical condition.The condition follows by substituting a lower bound into the coverage criterion.
  • Super-critical Case: In the supercritical case, coverage is ensured by at least one GW within radius rG2D − a and one HAPS within radius rH2G − rG2D + a.This produces a lower bound on the probability that the MCC is covered.
  • Super-critical Case: The H2G2D percolation probability is non-zero when the GW density exceeds the sufficient threshold λH2G2D.When rH2G approaches infinity, the sufficient condition depends only on GW density because all GWs can be covered by HAPSs.
  • Critical Condition: For fixed HAPS density, percolation probability does not decrease as GW density increases; for fixed GW density, it does not decrease as HAPS density increases.These monotonicity properties support the existence of a critical phase-transition condition.
  • Critical Condition: The critical condition is an implicit curve F(λH, λG) = 0 between the supercritical and subcritical regions.Its closed-form expression cannot be obtained, but its characteristics can be verified through simulations.

C. Hybrid Coverage

The hybrid scheme combines direct H2D coverage with H2G2D coverage, producing zero and non-zero percolation regions determined by HAPS and GW densities. Its critical phase-transition curve lies between the subcritical and supercritical regions, and its percolation probability is at least that of either component scheme.

  • Sub-critical Case: In the hybrid subcritical case, an MCC cannot be covered when its radius-rG2D + a area contains no GW and its radius-rH2D + a area contains no HAPS.This condition yields a lower bound on the probability that the MCC is not covered.
  • Sub-critical Case: The hybrid percolation probability is zero under the sufficient condition of Theorem 5.The theorem applies when λH < ln 2 / π(rH2D + a)^2 and the GW density is below λhbd_G.
  • Super-critical Case: Hybrid coverage is supercritical when either a GW-HAPS relay path covers the MCC or an HAPS directly covers it.The relay condition uses radii rG2D − a and rH2G − rH2D + a, while direct coverage uses radius rH2D − a.
  • Super-critical Case: Theorem 6 gives sufficient conditions for non-zero hybrid percolation probability under the two radius-ordering cases.The cases are separated by whether rH2G − rH2D + a is greater than rH2D − a.
  • Limiting Case: As rH2G approaches infinity, the sufficient condition approaches λH > λ+H, because all GWs are directly covered by HAPSs.This is also identified as an upper bound for the two corresponding Gilbert disk models and a lower bound of λhbd_G.
  • Critical Condition: Hybrid percolation probability does not decrease with either GW density or HAPS density.These monotonicity relations support an implicit critical curve G(λH, λG) = 0 between the supercritical and subcritical regions.
  • Comparison: The hybrid percolation probability is at least that of H2D or H2G2D, with its subcritical region contained by theirs and its supercritical region containing theirs.This follows from the random-graph inclusions GH2D ⊆ Ghbd and GH2G2D ⊆ Ghbd.

IV. SIMULATION RESULTS AND DISCUSSION

Simulations examine phase transitions and density trade-offs across H2D, H2G2D, and hybrid coverage schemes. The results also assess deployment cost, communication range, modeling assumptions, and extensions.

  • H2D coverage scheme: At λH > 4.5 × 10^-10, the H2D percolation probability reaches 0.83 when λH = 7.5 × 10^-10 HAPSs/km2.The phase transition is driven by increasing HAPS density.
  • H2G2D coverage scheme: In H2G2D, higher λH lowers the required λG, which approaches 2 × 10^-9 GWs/km2 when λH exceeds 9 × 10^-11 HAPSs/km2.When GW density is insufficient, increasing HAPS density cannot rapidly improve percolation probability; G2G links can mitigate insufficient HAPS deployment.
  • Hybrid coverage scheme: In the hybrid scheme, increasing either density improves performance, while higher λG lowers the critical λH from 3×10^-10 at λG = 1 × 10^-9 to 1.5×10^-10 HAPSs/km2 at λG = 2×10^-9 GWs/km2.Its super-critical region is larger or equal to those of H2D and H2G2D, while its subcritical region is smaller.
  • Hybrid coverage scheme: Joint H2D and H2G2D operation can yield higher percolation probability, but limited antenna, spectrum, and energy resources make performance dependent on resource allocation.Allocating all resources to either function reduces the hybrid scheme to that corresponding single scheme.
  • G2G communication range: Increasing rG2G improves percolation probability, but in H2G2D gains stop when rG2G > 2rG2D; hybrid coverage can continue improving through cross-scheme connections.G2G links create mesh networks and expand coverage through multi-hop communication.
  • Deployment implications and limitations: The framework can identify critical HAPS and GW densities for continuous service and support CAPEX/OPEX-based density selection, but lacks a closed-form percolation probability.Approximate expressions or experimental data are therefore needed to select density combinations.

V. CONCLUSION

The paper evaluates three HAPS-based schemes for large-scale continuous service using percolation theory. It finds critical density curves between sub-critical and super-critical regions, supporting cost-aware HAPS and GW deployment.

  • Conclusion: The paper introduces H2D, H2G2D, and hybrid HAPS-based coverage schemes for large-scale continuous service.Percolation probability represents the probability of generating large-scale continuous coverage through direct HAPS coverage or GW-connected coverage.
  • Conclusion: All three schemes exhibit critical conditions separating zero from non-zero percolation probability.These conditions form curves relating HAPS density and GW density.
  • Conclusion: The critical density curves can help minimize HAPS or GW costs, or jointly minimize CAPEX and OPEX in HAPS-based wireless networks.

APPENDIX A PROOF OF LEMMA 1

The proof establishes monotonicity of percolation probability with node densities and characterizes critical conditions through three cases. The resulting critical curve has bounded density requirements and non-increasing density trade-offs.

  • Monotonicity: Percolation probability is non-decreasing with HAPS density in the H2D scheme.The proof uses thinning and containment: adding HAPSs does not remove vertices or edges from the random graph.
  • Monotonicity: In H2G2D, percolation probability is non-decreasing with both GW density and HAPS density.Adding GWs expands directly and indirectly connected GW sets, while adding HAPSs connects more GWs to the core network.
  • Critical condition: The H2G2D critical condition has three cases: one where densities trade off, and two where one density increase alone does not trigger non-zero percolation probability.For sufficiently high λG, the required λH approaches a constant; for sufficiently high λH, the required λG approaches a constant.
  • Critical condition: The critical points can be represented by an implicit function F(λH, λG) = 0, with lower bounds on both density coordinates.The super-critical and sub-critical regions are separated by the critical condition.

APPENDIX D PROOF OF LEMMA 5

The hybrid proof establishes monotonicity in both HAPS and GW densities and shows that its critical curve separates non-overlapping subcritical and supercritical regions.

  • Monotonicity: In the hybrid scheme, percolation probability is non-decreasing with GW density and HAPS density.Adding GWs expands H2G2D connectivity while adding HAPSs expands both direct and GW-mediated coverage.
  • Critical condition: The hybrid supercritical and subcritical regions do not overlap.The critical curve is obtained as the phase-transition limit of curves with fixed positive percolation probability.
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