Local Conformity-Based Evolutionary Game Modeling of UAV Swarm Under Byzantine Attack

He Fang; Junhui Zhao; Ruixing Ren

arxiv: 2606.21206 · v1 · pith:JZLYDLEYnew · submitted 2026-06-19 · 📡 eess.SY · cs.SY

Local Conformity-Based Evolutionary Game Modeling of UAV Swarm Under Byzantine Attack

Ruixing Ren , Junhui Zhao , He Fang This is my paper

Pith reviewed 2026-06-26 13:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords UAV swarmByzantine attackevolutionary game theorylocal conformitydeath-birth updatingdeceptive strategiesnetwork topologytopological robustness

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The pith

Local conformity spreads deceptive strategies in UAV swarms under Byzantine attacks via evolutionary game dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper models UAV swarms facing Byzantine attacks as an evolutionary game in which each UAV adopts strategies based on local conformity with its network neighbors. Using death-birth updating rules on the communication graph, the authors derive a macroscopic dynamic equation for the fraction of deceptive strategies and obtain analytical expressions for its evolutionary stable states. Simulations show that observation errors reduce the spread of deception while larger fractions of malicious nodes and higher attack intensity increase it, with the overall dynamics remaining consistent across regular, scale-free, and random network topologies.

Core claim

The paper establishes that local conformity rules, implemented through graph evolutionary game theory with death-birth updating, govern the propagation of deceptive strategies in UAV swarms under Byzantine attacks. The macroscopic dynamic equation for the fraction of deceptive strategies admits analytical evolutionary stable states. Observation errors weaken malicious induction, whereas higher proportions of malicious nodes and greater attack intensity amplify attack impacts, and the model exhibits strong topological robustness across regular, scale-free, and random networks.

What carries the argument

death-birth updating rules in a graph evolutionary game on the UAV communication network

If this is right

The fraction of deceptive strategies obeys a differential equation whose equilibria depend explicitly on the fraction of malicious nodes and attack intensity.
Observation errors reduce the equilibrium fraction of deceptive strategies, thereby limiting the effectiveness of Byzantine attacks.
Higher malicious-node density and stronger attacks drive the system toward higher stable fractions of deception.
The location and stability of the equilibria remain unchanged when the underlying interaction network is switched among regular, scale-free, and random topologies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The analytical stable-state expressions could be used to set intervention thresholds for UAV operators when the deceptive fraction approaches critical values.
Because the dynamics prove insensitive to network type, mitigation techniques need not be redesigned for each possible swarm topology.
Allowing conformity strength to vary with mission phase might reveal parameter regimes in which attacks fail to reach high stable fractions.

Load-bearing premise

Local conformity rules, implemented through graph evolutionary game theory and death-birth updating, accurately capture how UAVs adopt deceptive strategies under Byzantine attacks.

What would settle it

Simulating the death-birth process on a small network with known attack parameters and checking whether the measured fraction of deceptive strategies follows the derived dynamic equation and reaches the predicted stable states would test the model; systematic mismatch would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.21206 by He Fang, Junhui Zhao, Ruixing Ren.

**Figure 3.** Figure 3: Observation error ε vs p ESS m under different β, Pa = 1 proportion p ESS m decreases monotonically as ε grows. The underlying reason is that when the observation error is small, legitimate UAVs can perceive the true system state relatively accurately. In this case, the deceptive strategies induced by malicious nodes can spread effectively, keeping the steadystate proportion of deceptive strategies at a h… view at source ↗

**Figure 4.** Figure 4: Theoretical ESS Heatmap 0 25 50 75 100 125 150 175 200 Time step 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 p m(t) Pa=0.3 Theory Pa=0.3 Pa=0.7 Theory Pa=0.7 Pa=1.0 Theory Pa=1.0 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Impact of Attack Probability Pa, ε = 0.1, β = 0.5. steady-state value is the lowest at a smaller Pa. This further illustrates that a stronger attack intensity of malicious nodes produces a more prominent interference effect on the decisionmaking of legitimate UAVs, inducing more normal UAVs to adopt deceptive strategies. V. CONCLUSION Aiming at UAV swarm under Byzantine attack, this paper took local confo… view at source ↗

read the original abstract

Leveraging their flexible and efficient deployment capabilities, unmanned aerial vehicle (UAV) swarms have been widely applied in various mission scenarios. However, the open communication environment also exposes them to the threat of Byzantine attacks. Most existing studies assume independent decision-making by each UAV, neglecting that local conformity amplifies false information propagation. This paper constructs an evolutionary game model for UAV swarm under malicious attacks based on graph evolutionary game theory, revealing how local conformity rules govern the spread of deceptive strategies. Using death-birth updating rules, we derive the macroscopic dynamic equation for the fraction of deceptive strategies and the analytical solutions to its evolutionary stable states. Sim ulations reveal observation errors weaken malicious induction, while higher proportions of malicious nodes and greater attack intensity drastically amplify attack impacts. Moreover, the model exhibits strong topological robustness across regular, scale-free and random networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a graph evolutionary game model with local conformity and death-birth updates for deceptive strategies in UAV swarms under Byzantine attack, but the claimed closed-form stable states likely rest on mean-field approximations that weaken on scale-free networks.

read the letter

The new element here is folding local conformity into the evolutionary game setup for this specific attack scenario on UAV swarms. Prior work apparently treated UAV decisions as independent, so adding conformity via graph structure and death-birth rules is a straightforward extension that targets a practical gap.

The simulations across regular, scale-free, and random networks are the part that lands cleanly. They show observation errors dampening attack effects while higher malicious fractions and attack strength increase impact, and they report topological robustness. That gives a usable picture for the subfield even if the numbers are illustrative.

The soft spot is the derivation step. The abstract says they obtain a macroscopic dynamic equation and analytical evolutionary stable states from death-birth updating. Standard death-birth on heterogeneous graphs usually requires pair or mean-field closure, and those closures lose accuracy when degree variance is high. The stress-test concern is on point: if the equation replaces local averages with global fractions without correction terms, the closed-form solutions become approximate, and the scale-free results may not match the analytics exactly. The paper would need to show the explicit steps and any error bounds to clear this.

This is for researchers working on UAV network security or evolutionary games on graphs. A reader in either area could extract the model and simulation protocol for their own work. It deserves peer review because the modeling direction is relevant and the simulation evidence is concrete, even though the analytical claims need tighter verification on the heterogeneous cases.

Referee Report

2 major / 1 minor

Summary. The paper constructs an evolutionary game model of UAV swarms under Byzantine attacks using graph evolutionary game theory. It incorporates local conformity and death-birth updating to derive a macroscopic dynamic equation governing the fraction of deceptive strategies together with closed-form solutions for the evolutionary stable states. Simulations are used to examine the effects of observation errors, malicious-node fraction, attack intensity, and network topology (regular, scale-free, random) on strategy spread.

Significance. If the derivation is exact, the work supplies an analytically tractable model linking local conformity rules to macroscopic deception dynamics in attacked UAV networks, with potential value for security analysis of multi-agent systems on graphs.

major comments (2)

[derivation of macroscopic dynamic equation] The central derivation replaces local neighborhood averages with global fractions to obtain a closed macroscopic equation from death-birth updating. On scale-free networks this step is not shown to remain exact once degree heterogeneity is present; standard pair or mean-field closures introduce higher-order correction terms that are omitted here. Consequently the reported analytical stable states and the claim of strong topological robustness on scale-free graphs rest on an unverified approximation (see the derivation following the statement of the death-birth rule and the subsequent stability analysis).
[simulation results] The abstract asserts that simulations confirm the analytical solutions across topologies, yet no parameter values, network-generation details, or quantitative comparison between the closed-form equilibria and the simulated trajectories are supplied. Without these, it is impossible to assess whether the reported robustness holds beyond the mean-field regime.

minor comments (1)

[abstract] Abstract contains the typographical error "Sim ulations".

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's comments. We address each major point below and propose revisions where appropriate to strengthen the manuscript.

read point-by-point responses

Referee: [derivation of macroscopic dynamic equation] The central derivation replaces local neighborhood averages with global fractions to obtain a closed macroscopic equation from death-birth updating. On scale-free networks this step is not shown to remain exact once degree heterogeneity is present; standard pair or mean-field closures introduce higher-order correction terms that are omitted here. Consequently the reported analytical stable states and the claim of strong topological robustness on scale-free graphs rest on an unverified approximation (see the derivation following the statement of the death-birth rule and the subsequent stability analysis).

Authors: The derivation follows the standard mean-field approach commonly used in evolutionary game theory on graphs to close the system by approximating local frequencies with global ones. This yields the macroscopic equation and closed-form ESS. While degree heterogeneity on scale-free networks can introduce corrections, our simulations demonstrate that the analytical predictions match well even on scale-free topologies, supporting the robustness claim. We will add a paragraph discussing the approximation's validity and limitations in the revised manuscript. revision: partial
Referee: [simulation results] The abstract asserts that simulations confirm the analytical solutions across topologies, yet no parameter values, network-generation details, or quantitative comparison between the closed-form equilibria and the simulated trajectories are supplied. Without these, it is impossible to assess whether the reported robustness holds beyond the mean-field regime.

Authors: We will revise the manuscript to include detailed simulation parameters (e.g., network sizes, generation algorithms such as Watts-Strogatz for regular, Barabási–Albert for scale-free with specific parameters, Erdős–Rényi for random), initial conditions, and quantitative metrics (e.g., mean squared error between analytical equilibria and averaged simulation results over multiple runs) to allow verification of the agreement across topologies. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from death-birth rules is independent of target results

full rationale

The abstract and provided text describe a standard construction of an evolutionary game model on graphs using death-birth updating to obtain a macroscopic dynamic equation and its stable states. No quoted step reduces the claimed predictions or stable-state solutions to fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation is presented as following directly from the updating rule and graph structure; simulations are reported separately as validation. This is self-contained against external benchmarks in evolutionary game theory and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from evolutionary game theory applied to UAV decision-making; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)

domain assumption Graph evolutionary game theory with death-birth updating accurately models UAV strategy adoption under attacks
Invoked to derive the macroscopic dynamic equation and stable states.
domain assumption Local conformity governs the spread of deceptive strategies in the swarm
Core modeling choice stated as the key difference from prior independent-decision studies.

pith-pipeline@v0.9.1-grok · 5671 in / 1337 out tokens · 31109 ms · 2026-06-26T13:41:24.316287+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references

[1]

R. Ren, J. Zhao and Q. Zhang, ”UA V-Assisted Collaborative Sensing Task Offloading and Resource Allocation in IoV ,” IEEE Transactions on Vehicular Technology, vol. 75, no. 4, pp. 6806-6815, Apr. 2026

2026
[2]

J. Zhao, F. Huang, L. Liao and Q. Zhang, ”Blockchain-Based Trust Management Model for Vehicular Ad Hoc Networks,” IEEE Internet of Things Journal, vol. 11, no. 5, pp. 8118-8132, Mar. 2024

2024
[3]

X. Wang, Z. Zhao, et al. ”A survey on security of UA V swarm networks: attacks and countermeasures,” ACM Computing Surveys, vol. 57, no. 3, pp. 1-37, Nov. 2024

2024
[4]

R. Ren, J. Zhao and Q. Zhang, ”Integrated sensing and communication for low-altitude security,” Physical Communication, vol. 76, pp. 103117, Apr. 2026

2026
[5]

S. Hu, Z. Zhou, P. Wang and Y . Zhang, ”A Hierarchical Cooperative Au- thentication Protocol for Attack-Resilient UA V Swarms With Ultra-Low Overhead,” IEEE Transactions on Information Forensics and Security, vol. 20, pp. 12652-12664, Nov. 2025

2025
[6]

Q. Wu, Y . Zhang, Z. Yang and M. R. Shikh-Bahaei, ”Deep Learning for Secure UA V Swarm Communication Under Malicious Attacks,” IEEE Transactions on Wireless Communications, vol. 23, no. 10, pp. 14879- 14894, Oct. 2024

2024
[7]

Y . Wu, X. Guan, W. Yang and Q. Wu, ”UA V Swarm Communication Under Malicious Jamming: Joint Trajectory and Clustering Design,” IEEE Wireless Communications Letters, vol. 10, no. 10, pp. 2264-2268, Oct. 2021

2021
[8]

C. Zhu, B. Jiang, et al., ”Privacy-Preserving Average Consensus for Swarm Systems Subject to DoS Attacks,” IEEE Internet of Things Journal, vol. 12, no. 24, pp. 53659-53669, 15 Dec. 2025

2025
[9]

Gong, et al., ”Resilient Human-in-the-Loop Formation-Tracking of Multi-UA V Systems Against Byzantine Attacks,” IEEE Transactions on Automation Science and Engineering, vol

X. Gong, et al., ”Resilient Human-in-the-Loop Formation-Tracking of Multi-UA V Systems Against Byzantine Attacks,” IEEE Transactions on Automation Science and Engineering, vol. 22, pp. 3797-3809, 2025

2025
[10]

J. Zhao, R. Ren, Y . Nie and D. Wang, ”Decision Intelligence Em- powering Resource Management for 6G-V2X Communications,” IEEE Internet of Things Magazine, vol. 8, no. 6, pp. 88-96, Nov. 2025

2025
[11]

Zhao, et al., ”A UA V-Aided Vehicular Integrated Platooning Network for Heterogeneous Resource Management,” IEEE Transactions on Green Communications and Networking, vol

J. Zhao, et al., ”A UA V-Aided Vehicular Integrated Platooning Network for Heterogeneous Resource Management,” IEEE Transactions on Green Communications and Networking, vol. 7, no. 1, pp. 512-521, Mar. 2023

2023
[12]

Huang, B

R. Huang, B. Li, et al. ”Dual-feature-driven nonlinear evolution of social misinformation,” International Journal of Swarm Intelligence Research, vol. 16, no.1, Feb. 2025

2025
[13]

C. Wu, H. Deng, et al. ”Enhancing resilience of unmanned autonomous swarms through game theory-based cooperative reconfiguration”, Reli- ability Engineering & System Safety, vol. 260, pp. 110951, Aug. 2025

2025
[14]

The replicator equation on graphs

Ohtsuki H, Nowak M A. The replicator equation on graphs. Journal of Theoretical Biology, 2006, 243(1): 86–97

2006
[15]

Lin, ”Analysis of public opinion security based on bounded rational behaviors in social network,” Ph.D

Y . Lin, ”Analysis of public opinion security based on bounded rational behaviors in social network,” Ph.D. dissertation, Tsinghua University, Beijing, China, 2025

2025

[1] [1]

R. Ren, J. Zhao and Q. Zhang, ”UA V-Assisted Collaborative Sensing Task Offloading and Resource Allocation in IoV ,” IEEE Transactions on Vehicular Technology, vol. 75, no. 4, pp. 6806-6815, Apr. 2026

2026

[2] [2]

J. Zhao, F. Huang, L. Liao and Q. Zhang, ”Blockchain-Based Trust Management Model for Vehicular Ad Hoc Networks,” IEEE Internet of Things Journal, vol. 11, no. 5, pp. 8118-8132, Mar. 2024

2024

[3] [3]

X. Wang, Z. Zhao, et al. ”A survey on security of UA V swarm networks: attacks and countermeasures,” ACM Computing Surveys, vol. 57, no. 3, pp. 1-37, Nov. 2024

2024

[4] [4]

R. Ren, J. Zhao and Q. Zhang, ”Integrated sensing and communication for low-altitude security,” Physical Communication, vol. 76, pp. 103117, Apr. 2026

2026

[5] [5]

S. Hu, Z. Zhou, P. Wang and Y . Zhang, ”A Hierarchical Cooperative Au- thentication Protocol for Attack-Resilient UA V Swarms With Ultra-Low Overhead,” IEEE Transactions on Information Forensics and Security, vol. 20, pp. 12652-12664, Nov. 2025

2025

[6] [6]

Q. Wu, Y . Zhang, Z. Yang and M. R. Shikh-Bahaei, ”Deep Learning for Secure UA V Swarm Communication Under Malicious Attacks,” IEEE Transactions on Wireless Communications, vol. 23, no. 10, pp. 14879- 14894, Oct. 2024

2024

[7] [7]

Y . Wu, X. Guan, W. Yang and Q. Wu, ”UA V Swarm Communication Under Malicious Jamming: Joint Trajectory and Clustering Design,” IEEE Wireless Communications Letters, vol. 10, no. 10, pp. 2264-2268, Oct. 2021

2021

[8] [8]

C. Zhu, B. Jiang, et al., ”Privacy-Preserving Average Consensus for Swarm Systems Subject to DoS Attacks,” IEEE Internet of Things Journal, vol. 12, no. 24, pp. 53659-53669, 15 Dec. 2025

2025

[9] [9]

Gong, et al., ”Resilient Human-in-the-Loop Formation-Tracking of Multi-UA V Systems Against Byzantine Attacks,” IEEE Transactions on Automation Science and Engineering, vol

X. Gong, et al., ”Resilient Human-in-the-Loop Formation-Tracking of Multi-UA V Systems Against Byzantine Attacks,” IEEE Transactions on Automation Science and Engineering, vol. 22, pp. 3797-3809, 2025

2025

[10] [10]

J. Zhao, R. Ren, Y . Nie and D. Wang, ”Decision Intelligence Em- powering Resource Management for 6G-V2X Communications,” IEEE Internet of Things Magazine, vol. 8, no. 6, pp. 88-96, Nov. 2025

2025

[11] [11]

Zhao, et al., ”A UA V-Aided Vehicular Integrated Platooning Network for Heterogeneous Resource Management,” IEEE Transactions on Green Communications and Networking, vol

J. Zhao, et al., ”A UA V-Aided Vehicular Integrated Platooning Network for Heterogeneous Resource Management,” IEEE Transactions on Green Communications and Networking, vol. 7, no. 1, pp. 512-521, Mar. 2023

2023

[12] [12]

Huang, B

R. Huang, B. Li, et al. ”Dual-feature-driven nonlinear evolution of social misinformation,” International Journal of Swarm Intelligence Research, vol. 16, no.1, Feb. 2025

2025

[13] [13]

C. Wu, H. Deng, et al. ”Enhancing resilience of unmanned autonomous swarms through game theory-based cooperative reconfiguration”, Reli- ability Engineering & System Safety, vol. 260, pp. 110951, Aug. 2025

2025

[14] [14]

The replicator equation on graphs

Ohtsuki H, Nowak M A. The replicator equation on graphs. Journal of Theoretical Biology, 2006, 243(1): 86–97

2006

[15] [15]

Lin, ”Analysis of public opinion security based on bounded rational behaviors in social network,” Ph.D

Y . Lin, ”Analysis of public opinion security based on bounded rational behaviors in social network,” Ph.D. dissertation, Tsinghua University, Beijing, China, 2025

2025