Securing Cooperative Sensing in UAV Swarms Against Conformity-Driven Byzantine Attacks

Motivation

• Imitation-based conformity lets normal UAVs switch behaviors using neighbors’ fitness, unlike unbiased quantized consensus.The paper focuses on imitation-based conformity; aggregation-weight robustness is outside its scope.Source: §I · p. 1

Threat model

• Under conformity, malicious UAVs can lure normal nodes into wrong strategies over repeated rounds, making effective errors dynamic and much larger than sensing errors.This creates model mismatch for fusion methods assuming constant, independent errors.Source: §I · pp. 1–2, §III · p. 3

Contributions

• The paper uses graph evolutionary game theory to derive misinformation dynamics, the ESS, and the majority-flip threshold under independent fixed-intensity attacks.The framework models bounded-rational, local strategy adaptation under death-birth updating.Source: §I · p. 2
• With heterogeneous sensing errors, the zeroth-order ESS depends on the error distribution only through its mean.This supports mean-field fusion design under realistic channel conditions.Source: §I · p. 2
• A closed-form first-order correction and an exact arbitrary-selection fixed point extend the evolutionary analysis beyond the zeroth-order approximation.The first-order correction and exact mean-field result quantify selection effects on misinformation.Source: §I · p. 2

Fusion framework

• The proposed per-node MAP fusion tracks evolving normal-report errors and achieves nearly 100% accuracy across conditions, remaining above 99% under ±20% parameter mismatch.Evaluated conditions include different topologies, attack intensities, network scales, and sensing-error distributions.Source: §I · p. 2, §III · p. 3

Transition construction

• The exact transition probability is obtained by averaging over central and neighbor errors before applying the weak-selection approximation.The construction explicitly incorporates heterogeneous central and neighbor error roles in the exact formulation.Source: A. State Transition Probability · p. 4

State Transition Probability

• A UAV’s transition probability accounts separately for correct and erroneous central observations and averages over neighbor report configurations.Fitness functions are specified for both truthful and misleading strategies in each observation condition.Source: A. State Transition Probability · p. 4
• The model derives the probability that a normal UAV switches from the truthful strategy to the misleading strategy under death-birth updating.The probability compares the fitness of neighbors judged opposite to the true state with total neighbor fitness.Source: §IV · p. 3, A. State Transition Probability · p. 4

Weak-selection expansion

• The derivation substitutes payoff-based fitness expressions into the death-birth transition rule and separates baseline and conformity-dependent terms.The weak-selection coefficient α isolates the first-order payoff contribution.Source: A. State Transition Probability · p. 4
• The resulting expansion retains an O(1) zeroth-order term and a closed-form first-order term in α.The zeroth-order component is used for population evolution, while the first-order form is developed later.Source: A. State Transition Probability · pp. 4–5

Population Evolution Equation

• The expected misleading-neighbor fraction is ko(pm + βPa), combining normal misinformation pm with malicious attack probability Pa.Here β is the malicious-to-normal UAV ratio and ko is the number of ordinary neighbors.Source: B. Population Evolution Equation · p. 5
• The expected total neighborhood size is ko(1 + β), which normalizes the misleading-neighbor contribution in the population dynamics.The denominator includes ordinary and malicious neighbors.Source: B. Population Evolution Equation · p. 5

Zeroth-order approximation

• Under weak selection, the analysis drops the higher-order αE[ϖ(km)] term and retains the zeroth-order evolution equation.This approximation treats α as very small and the payoff-dependent expectation as bounded.Source: B. Population Evolution Equation · p. 5
• The resulting population equation is a linear differential equation in pm, rewritten around the ESS using the error variable ϱ = pm − pESS.This form isolates deviations from equilibrium.Source: B. Population Evolution Equation · p. 5

Stability

• The equilibrium is asymptotically stable because A < 0 for 0 < ε ≤ 0.5 and β > 0, causing deviations to decay exponentially.The solution is ϱ(t) = ϱ(0)e^At, so ϱ(t) approaches zero.Source: B. Population Evolution Equation · p. 5

• Pa > 1/2 is the majority-flip threshold, independent of both ε and β.Source: C. Majority-Flip Threshold · p. 6
• When Pa > 1/2, aggregate swarm reports are systematically inverted and majority-trusting defenses fail.Source: C. Majority-Flip Threshold · p. 6
• When Pa < 1/2, the majority remains trustworthy and classical defenses survive.Source: C. Majority-Flip Threshold · p. 6

• The exact transition probability retains the full fitness and makes no weak-selection truncation.Source: D. Exact Mean-Field Transition… · p. 6
• At α = 0, the exact transition probability reduces to the zeroth-order approximation, while its Taylor coefficient is the closed-form correction G.Source: D. Exact Mean-Field Transition… · p. 6
• The exact mean-field ESS is computed as the solution of a fixed-point equation by simple iteration.Source: D. Exact Mean-Field Transition… · p. 6

• The analysis examines ESS dependence on heterogeneous sensing-error distributions.Source: §V · p. 6
• It derives a first-order closed-form correction under weak selection.Source: §V · p. 6
• It discusses graph-structural effects beyond the mean-field approximation.Source: §V · p. 6

A. Heterogeneous Sensing Errors

• The result assumes local misinformation ratios ρi are uncorrelated with sensing errors εi.Source: A. Heterogeneous Sensing Errors · pp. 6–7
• Theorem 1 makes heterogeneous zeroth-order dynamics identical to homogeneous dynamics with ε = ¯ε.Source: A. Heterogeneous Sensing Errors · p. 6
• The zeroth-order ESS depends on the sensing-error distribution only through its mean ¯ε.Source: A. Heterogeneous Sensing Errors · p. 6
• Per-node errors εi remain relevant to the fusion likelihood and are used by the per-node MAP rule.Source: A. Heterogeneous Sensing Errors · p. 7
• Strong coupling between channel conditions and network topology requires correction and is left for future work.Source: A. Heterogeneous Sensing Errors · p. 7

B. First-Order Weak-Selection Closed-Form Correction

• The first-order population correction is expressed as a payoff-weighted linear combination G(e, ρ) = gns uns + gnd und + gms ums + gmd umd.Source: B. First-Order Weak-Selection… · p. 7
• The first-order ESS shift is obtained by applying the implicit function theorem around the zeroth-order ESS.Source: B. First-Order Weak-Selection… · p. 7
• Mean-field analysis replaces local misinformation ratios with the population-average ρs, while finite-graph fluctuations can cause structural amplification and a slightly steeper descent.Source: B. First-Order Weak-Selection… · p. 7

• The MAP estimator infers the true state sequence Θ from UAV reports under a uniform prior.Source: §VI · pp. 7–8

Report models

• The fusion likelihood marginalizes over malicious and normal node types using the malicious ratio β and conditional independence.Source: §VI · p. 7
• Malicious UAV reports are modeled using attack probability Pa together with sensing error, yielding a report error probability.Source: §VI · p. 7
• Normal UAV report errors evolve after the first slot according to the misinformation ratio generated by conformity dynamics.Source: §VI · p. 7

Online procedure

• Algorithm 1 propagates per-slot error probabilities and accumulates log-likelihoods to implement the fusion procedure.Source: §VI · p. 8

• Theorem 1 requires only the population mean sensing error ¯ε to predict misinformation evolution, while individual εi improve the fusion likelihood.Source: A. Per-Node Heterogeneous Fusion · p. 8
• Each UAV feeds back its sensing-SINR estimate so the fusion center can obtain individual εi for per-node error probabilities.Source: A. Per-Node Heterogeneous Fusion · p. 8
• The per-slot MAP decision uses conditional probabilities evaluated with malicious-node and normal-node error parameters.Source: A. Per-Node Heterogeneous Fusion · p. 8
• Combining Equation (65) with evolutionary tracking of the misinformation ratio defines EGTM fusion, including homogeneous and per-node variants.Source: A. Per-Node Heterogeneous Fusion · p. 8

B. Online Implementation

• The misinformation ratio is propagated online using model parameters and the payoff matrix, with a sliding window reducing enumeration complexity from 2^T to 2^τ.Source: B. Online Implementation · p. 8
• τ = 1 achieves full accuracy with the lowest fusion latency because the normal-node error probability is predicted slot by slot.Source: B. Online Implementation · p. 8

C. Robustness to Heterogeneous Sensing Errors

• Across regular, BA, and ER networks, evolution curves nearly coincide with the theoretical ODE, indicating topology-independent dynamics under the tested settings.Source: A. Evolution Dynamics · p. 8, Figure 2 · p. 9
• The steady-state misinformation ratio decreases with ε, increases with β and Pa, and reaches 0.5 at ε = 0.5 when neighbor influence vanishes.Source: A. Evolution Dynamics · p. 8

B. First-Order Weak-Selection Effect

• The zeroth-order ESS is 0.6860 and decreases to 0.6781 at α = 0.5, while the first-order approximation deviates by less than 3 × 10^-5 at α = 0.05.Source: B. First-Order Weak-Selection Effect · p. 8, Figure 3 · p. 9

C. Robustness to Heterogeneous Sensing Errors

• With mean ¯ε = 0.1, four markedly different sensing-error distributions produce cross-distribution ESS dispersion below 0.002.Source: C. Robustness to Heterogeneous Sensing… · pp. 8–9, Figure 4 · p. 9

D. Fusion Performance

• For weak attacks, all fusion methods exceed 0.99 accuracy; after the strong-attack ESS crosses 1/2, majority voting, reputation methods, and independent fusion collapse.Source: D. Fusion Performance · p. 9, Table I · p. 10
• Under ε = 0.1, β = 0.5, and Pa = 0.8, misinformation crosses 0.5 within about two slots, while the proposed method maintains accuracy close to 1.0.Source: D. Fusion Performance · pp. 9–10, Figure 6 · p. 10

Malicious ratio and heterogeneity

• 0.83–1.0 accuracy is maintained by EGTM-pernode across β ∈[0.05, 0.5], reaching full accuracy from β = 0.2 under homogeneous errors and β = 0.3 under SNR errors.IFM and majority voting fall to 0–0.25 over the same β range, while EGTM-homog is slightly worse under SNR heterogeneity.Source: E. Fusion under Heterogeneous Errors · p. 10
• 0.83–0.85 accuracy is achieved at β = 0 despite conformity driving the misinformation ESS to 1/2.Majority voting and independent fusion degenerate to approximately 0.5, whereas the proposed method extracts residual statistical asymmetry.Source: E. Fusion under Heterogeneous Errors · p. 10

Robustness and parameter sensitivity

• Nearly 1.0 accuracy is sustained across small to large swarm sizes and improves slightly with N, with reduced variance.The observed stabilization is attributed to the law of large numbers.Source: F. Robustness Analysis · p. 10
• Above 0.99 accuracy is retained under ±20% parameter mismatch, compared with 0.9992 from ODE prediction and 0.9995 using the true simulated error.The result indicates reliance on modeled error dynamics rather than exact parameter knowledge.Source: F. Robustness Analysis · p. 10
• 100% accuracy is approached above about 0 dB across all β, while accuracy improves with SINR and remains nearly insensitive to window length.At τ = 1, accuracy is already ≥0.99 for β ∈{0.5, 1.0}; latency grows linearly with τ.Source: F. Robustness Analysis · p. 10