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End-to-End Secure Connection Probability in MultiLayer Networks with Heterogeneous Rician Fading

Hyeonsu Lyu, Yumin Kim, Hyun Jong Yang

TL;DR

This work addresses end-to-end secure connection probability (SCP) for multi-hop routes in space–air–ground–sea NTNs under heterogeneous Rician fading. It develops a tractable closed-form SCP expression by gamma-approximating the sum of eavesdropper channel gains and simplifying the Marcum Q-function product, yielding a Rayleigh-like form with layer-specific coefficients. The analytic SCP is validated against Monte-Carlo simulations and real SAGIN data, achieving high accuracy (mean absolute error ≈ 0.034 and <1%p gap in high-SCP regimes). The results provide a theoretical basis for prior posterior calibrations and offer a practical tool for secure NTN design under unknown eavesdroppers.

Abstract

Ensuring physical-layer security in non-terrestrial networks (NTNs) is challenging due to their global coverage and multi-hop relaying across heterogeneous network layers, where the locations and channels of potential eavesdroppers are typically unknown. In this work, we derive a tractable closedform expression of the end-to-end secure connection probability (SCP) of multi-hop relay routes under heterogeneous Rician fading. The resulting formula shares the same functional form as prior Rayleigh-based approximations but for the coefficients, thereby providing analytical support for the effectiveness of heuristic posterior coefficient calibration adopted in prior work. Numerical experiments under various conditions show that the proposed scheme estimates the SCP with an 1%p error in most cases; and doubles the accuracy compared with the conventional scheme even in the worst case. As a case study, we apply the proposed framework to real-world space-air-groundsea integrated network dataset, showing that the derived SCP accurately captures observed security trends in practical settings.

End-to-End Secure Connection Probability in MultiLayer Networks with Heterogeneous Rician Fading

TL;DR

This work addresses end-to-end secure connection probability (SCP) for multi-hop routes in space–air–ground–sea NTNs under heterogeneous Rician fading. It develops a tractable closed-form SCP expression by gamma-approximating the sum of eavesdropper channel gains and simplifying the Marcum Q-function product, yielding a Rayleigh-like form with layer-specific coefficients. The analytic SCP is validated against Monte-Carlo simulations and real SAGIN data, achieving high accuracy (mean absolute error ≈ 0.034 and <1%p gap in high-SCP regimes). The results provide a theoretical basis for prior posterior calibrations and offer a practical tool for secure NTN design under unknown eavesdroppers.

Abstract

Ensuring physical-layer security in non-terrestrial networks (NTNs) is challenging due to their global coverage and multi-hop relaying across heterogeneous network layers, where the locations and channels of potential eavesdroppers are typically unknown. In this work, we derive a tractable closedform expression of the end-to-end secure connection probability (SCP) of multi-hop relay routes under heterogeneous Rician fading. The resulting formula shares the same functional form as prior Rayleigh-based approximations but for the coefficients, thereby providing analytical support for the effectiveness of heuristic posterior coefficient calibration adopted in prior work. Numerical experiments under various conditions show that the proposed scheme estimates the SCP with an 1%p error in most cases; and doubles the accuracy compared with the conventional scheme even in the worst case. As a case study, we apply the proposed framework to real-world space-air-groundsea integrated network dataset, showing that the derived SCP accurately captures observed security trends in practical settings.
Paper Structure (12 sections, 5 theorems, 22 equations, 6 figures, 3 tables)

This paper contains 12 sections, 5 theorems, 22 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. System Model and Formulation
  3. System Model
  4. Closed-form Derivation of SCP
  5. Numerical Experiments
  6. Simulation Settings
  7. Validation of the Marcum Q-function Approximation
  8. Performance Evaluation of the SCP
  9. Conclusion and Discussion
  10. Visualization results with a 99% SCP threshold
  11. Proof of SCP for Rician fading
  12. Proof of SCP for Rayleigh fading

Key Result

Lemma 1

(Spatial Integral) The following equality holds for the integration over the two-dimensional Euclidean plane:

Figures (6)

  • Figure 1: Visualization of the proposed scheme with baselines in the North America space-air-ground integrated network testbed.
  • Figure 2: Comparison of the exact product of Marcum Q-functions and the proposed approximation for LEO and ground layers.
  • Figure 3: Impact of Rician $K$-factor on SCP in terrestrial and LEO layers. The red, green, and blue curves correspond to Eve densities of $10^{-5}$, $3 \times 10^{-5}$, and $10^{-4}$ (terrestrial), and $10^{-8}$, $3 \times 10^{-8}$, and $10^{-7}$ (LEO).
  • Figure 4: SCP of {3,5,7} hops over various average link distances. The red, green, and blue curves correspond to the {3,5,7} hop scenarios, respectively.
  • Figure 5: SCP under various Eve densities.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof