Covert Multi-Hop Communications for Heterogeneous Networks With Multiple Wardens

Abstract: This paper investigates covert multi-hop communications in heterogeneous wireless networks monitored by multiple passive wardens. To maximize network-wide covertness while satisfying a strict end-to-end rate requirement, we jointly optimize routing, modality selection, and transmit power. Under a simultaneous multi-hop transmission scheme, we analyze the detection capabilities of two distinct warden models: colluding wardens employing a central fusion center, and non-colluding wardens operating independently. For both models, we derive optimal detectors and exact expressions for the detection error probability (DEP). In addition, to reduce the complexity of evaluating the DEP, we develop highly accurate closed-form approximations based on gamma moment matching and establish rigorous DEP lower bounds using Kullback-Leibler (KL) divergence. Building on this theoretical foundation, we propose an efficient two-stage optimization algorithm that decouples link-level resource allocation from network-level path selection. By translating the KL divergence bounds into a novel, low-complexity routing metric, which universally simplifies to a linear summation of signal-to-noise ratios, we substantially reduce the computational overhead compared to conventional per-hop detection-based metrics. Finally, numerical simulations validate the theoretical analysis and demonstrate the near-optimal performance of the proposed framework.

• The paper develops exact detection-error probability analysis for simultaneous multi-hop transmissions observed by colluding and non-colluding wardens, validated by Monte Carlo simulations and supported by tight gamma approximations and KL-divergence bounds.
• The proposed two-stage method sets each link’s transmit power to meet its finite-blocklength rate constraint, then uses total warden-observed SNR as a Dijkstra-compatible routing and modality-selection metric with substantially lower complexity than exhaustive search.
• Results show that warden collusion sharply reduces covertness, per-hop DEP products underestimate the true network DEP, and heterogeneous frequency modalities improve covert performance by enabling routes that avoid unfavorable propagation conditions.

Overview

This paper studies covert multi-hop communications in heterogeneous wireless networks where each node can transmit over one of $M$ modalities (distinct center frequencies), and the network is monitored by multiple passive wardens. The central contributions are threefold: exact detection error probability (DEP) analysis under a worst-case simultaneous multi-hop transmission model, tractable approximations and bounds of that DEP, and a low-complexity joint optimization of routing, modality selection, and transmit power that provably reduces to Dijkstra's algorithm with an SNR-summation link metric (2606.18435).

Two warden threat models are considered: colluding wardens, which forward raw observations to a central fusion center performing joint hypothesis testing, and non-colluding wardens, each running an independent binary hypothesis test, with network covertness dictated by the most capable warden. The paper's positioning relative to prior work is notable on two counts. First, prior multi-warden routing frameworks assume wardens process hops independently, whereas this work models wardens observing the coherent superposition of all concurrent transmissions — a strictly harder adversarial setting. Second, the authors observe that no prior covert routing study, even for a single warden, provides exact DEP expressions; earlier schemes rely on per-hop DEP products or bounds.

Network and Detection Model

The network routes traffic from source $S$ to destination $D$ through relays, each equipped with $M$ modalities operating at distinct center frequencies. Under $\mathcal{H}_1$, all transmitters along the active route transmit concurrently; inter-hop interference at legitimate receivers is handled by orthogonal code-division multiplexing, and self-interference is assumed suppressed. Each warden observes, per modality, either noise alone ($\mathcal{H}_0$) or the coherent superposition of signals from all transmitters using that modality plus noise ($\mathcal{H}_1$).

The analysis adopts a deliberately pessimistic information assumption: wardens know the route, channel state information, powers, bandwidths, and modality assignments, so the derived DEP constitutes a robust lower bound on achievable covertness. Link rates use a finite-blocklength normal approximation with decoding error probability $\epsilon$, and the end-to-end rate is the minimum link rate along the route. The optimization problem maximizes the network-wide DEP subject to per-node power limits and an end-to-end rate requirement $R_{\text{req}}$.

Exact DEP Analysis for Colluding Wardens

For colluding wardens, the fusion center observes a $J$-dimensional Gaussian vector per modality. After whitening by the noise covariance and projecting onto the eigenbasis of the whitened SNR matrix $S$0, the optimal Neyman–Pearson detector under equal priors reduces to a weighted energy test:

$S$1

where $S$2 depends on the eigenvalues of $S$3 and $S$4 accumulates squared magnitudes over the blocklength $S$5. Because each $S$6 is gamma-distributed, the test statistic is a weighted sum of independent gamma variables whose characteristic function admits a closed product form; the exact DEP then follows via the Gil-Pelaez inversion formula. A useful structural remark is that this detector generalizes the single-warden case by replacing scalar aggregate SNRs with matrix eigenvalues, recovering the single-warden detector when $S$7.

Two computationally lighter alternatives are derived. A gamma moment-matching approximation equates the first two moments of the test statistic to a single gamma distribution, yielding a closed-form DEP involving incomplete gamma functions. Separately, Pinsker's inequality gives a rigorous lower bound on the DEP expressed through the KL divergence between the hypothesis distributions, which decomposes additively across modalities, eigenmodes, and channel uses. The paper also formulates a conventional baseline in which wardens detect each hop in isolation with maximal ratio combining and the end-to-end DEP is the product of per-hop DEPs; the authors show analytically that this product formulation systematically underestimates the true DEP because it decays geometrically with hop count, a point confirmed numerically.

Exact DEP Analysis for Non-Colluding Wardens

Each non-colluding warden runs the single-warden optimal weighted-energy detector on its own observations, with weights determined by the aggregate SNR $S$8 summed over all transmitters sharing modality $S$9. Per-warden false alarm and missed detection probabilities are again obtained exactly via characteristic-function inversion, and the network DEP is the minimum over wardens. Moment-matching approximations and KL-divergence lower bounds follow analogously. Numerically, the DEP curves for multiple non-colluding wardens closely track the single-warden curve with a near-uniform downward shift, indicating that covertness against independent wardens is dominated by the single warden with the strongest physical channel — a practically meaningful simplification for threat assessment.

A direct comparison shows colluding wardens achieve markedly lower DEP than non-colluding ones for any $D$0, with the gap widening as $D$1 grows. The implication is that maintaining a target covertness level requires substantially reducing transmit power as surveillance density increases, particularly against coordinated adversaries exploiting coherent integration and spatial correlation.

Joint Route and Resource Optimization

Solving the mixed-integer joint problem directly is intractable due to coupling between physical-layer signal superposition at the wardens and network-layer path constraints. The proposed two-stage decomposition separates link-level resource allocation from network-level path selection.

Link level: the paper proves the finite-blocklength link rate is strictly increasing in transmit power whenever the rate is positive (shown by establishing that the positive-rate condition implies the monotonicity condition via the inequality $D$2). Since all candidate link metrics are also strictly increasing in power, the optimal power for any link-modality pair satisfies the rate constraint with equality, uniquely determining $D$3 and reducing the search to modality selection only.

Network level: starting from the KL divergence upper bound chain, the authors derive a Dijkstra-compatible link weight

$D$4

i.e., total SNR leakage at all wardens. Two claims deserve emphasis. First, this weight is identical for colluding and non-colluding warden models — the min-max structure of the non-colluding case is handled via the relaxation $D$5 — so a single set of link weights serves both threat models without separate optimization pipelines. Second, the metric requires only SNR summation, avoiding the incomplete-gamma evaluations needed by the conventional per-hop DEP weights; the complexity analysis quantifies a per-link speedup factor of $D$6 relative to the per-hop DEP-based Dijkstra, while exhaustive search incurs combinatorial $D$7 cost.

It should be noted that the KL-based routing metric optimizes a bound rather than the exact DEP, so its near-optimality is an empirical finding rather than a proven guarantee; the simulations below address this gap.

Numerical Validation

Simulations use a 36-node topology with three modalities at 600 MHz, 900 MHz, and 1200 MHz, blocklength $D$8, and results averaged over $D$9 random realizations of warden locations and fading. Key findings include:

Finding	Detail
Exactness	Analytical DEPs match Monte Carlo simulations perfectly
Approximation quality	Gamma moment matching is tight across the entire power regime
Baseline behavior	Product-of-per-hop DEPs consistently underestimates; KL lower bound is loose/conservative
Hop-count effect	Fewer simultaneous transmitters yield higher DEP (3-hop route outperforms 5-hop)
Collusion penalty	DEP degrades sharply with $M$0 under collusion; steepest drops occur at low power

For the optimized system, the proposed algorithm achieves near-optimal DEP versus exhaustive search under both warden models across all tested rate requirements. Expanding the modality set from one band to all three substantially improves covertness by enabling routes that bypass wardens via favorable propagation characteristics. Behavioral analysis shows average hop count increases with $M$1 but decreases with $M$2 (fewer transmission events reduce aggregate exposure), and modality selection shifts toward the lowest-frequency modality $M$3 at stringent rates due to its superior channel gains, with slightly more distributed selections as $M$4 grows.

Limitations and Open Questions

Several assumptions constrain the applicability of the results. The worst-case analysis grants wardens full knowledge of routes, channels, and resource allocations; the authors themselves identify warden CSI uncertainty as requiring robust or uncertainty-aware allocation schemes, which remains unaddressed. The near-optimality of the KL-based metric is demonstrated numerically rather than established theoretically, leaving open whether the bound-driven metric can incur non-negligible optimality gaps in regimes outside those simulated. The framework also assumes synchronized transmission states, orthogonal code-division multiplexing suppressing inter-hop interference, effective self-interference cancellation, and static per-block channels; temporal dynamics are not modeled, and the authors note that integrating queuing theory to expose latency-covertness trade-offs is open. Finally, the algorithm is centralized; extending to decentralized architectures supporting multiple concurrent flows via distributed learning or game-theoretic methods is explicitly left as an unresolved challenge.

Conclusion

This paper delivers the first exact DEP characterization for covert multi-hop communications under simultaneous transmission observed by multiple wardens, covering both colluding and non-colluding adversarial models, together with accurate closed-form approximations and rigorous KL-divergence bounds. Its principal practical contribution is a universal, provably additive routing metric — total SNR leakage across wardens — that collapses joint routing, modality selection, and power control into a polynomial-time two-stage algorithm achieving near-optimal covertness at a fraction of the cost of exhaustive search. The results quantify the substantial covertness penalty imposed by warden collusion and demonstrate that spectral diversity across heterogeneous modalities is a reliable lever for evading distributed surveillance.