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

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.

💡 Research Summary

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The paper tackles the problem of evaluating physical‑layer security in non‑terrestrial networks (NTNs) that span multiple layers—space, air, ground, and sea—and employ multi‑hop decode‑and‑forward relaying. Existing works on secure connection probability (SCP) assume Rayleigh fading for all links, which is unrealistic for heterogeneous NTN environments where each layer exhibits distinct line‑of‑sight (LOS) conditions and therefore different Rician K‑factors. Moreover, prior studies rely on heuristic posterior calibration of model coefficients without theoretical justification.

The authors first model the locations of passive eavesdroppers (Eves) in each layer as independent homogeneous Poisson point processes (HPPPs). For the i‑th hop, the legitimate SNR is defined as
( \text{SNR}{s,i}= \frac{P_i |h{s,i}|^2}{n_0 d_{s,i}^{\alpha_i}} )
and the eavesdropping SNR for Eve m as
( \text{SNR}{e}(i,m)= \frac{P_i |h{e}(i,m)|^2}{n_0 d_{e}(i,m)^{\alpha_i}} ).
Both channel gains follow Rician fading with possibly different K‑factors (K_i).

The secrecy capacity of layer (l) is defined as the difference between the minimum legitimate link rate and the maximum eavesdropping link rate across the multi‑hop path. An end‑to‑end SCP is the probability that every layer’s secrecy capacity is positive, which factorizes as (P = \prod_{l=1}^{L} P_l) with
( P_l = \Pr\bigl(\min_i \text{SNR}{s,i} > \max_m \text{SNR}{e,m}\bigr) ).

To obtain a tractable expression for (P_l), the authors first average over the random locations of Eves using the probability generating functional of the HPPP. This yields an exponential term involving the density (\lambda_l) and an integral over the plane. The remaining difficulty lies in the distribution of the sum of non‑central chi‑squared variables (\sum_i |h_{e}(i,m)|^2) with different scaling factors. The authors approximate this sum by a Gamma distribution via moment matching, defining shape (m_l) and scale (\theta_l).

The CDF of the minimum legitimate SNR, (X = \min_i \text{SNR}_{s,i}), is expressed as a product of first‑order Marcum Q‑functions. The product is approximated by a single equivalent Marcum Q‑function (Q_1(\hat a_l, \hat b_l \sqrt{x})). The parameter (\hat b_l) is set to the root‑sum‑square of the individual (b_i) terms, while (\hat a_l) is obtained by curve fitting to minimize the deviation from the exact product. Numerical validation shows that the mean absolute error stays below 0.06 across the whole range of interest, confirming the adequacy of the approximation.

Using a spatial integral lemma, the authors simplify the exponential term to a closed‑form constant (\kappa_l) that depends on (\lambda_l), the Gamma parameters, and the fitted (\hat a_l). Substituting the approximated CDF of (X) and evaluating the remaining integral yields the compact expression

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