CSoS-STRE: A combat system-of-system space-time resilience enhancement framework

Abstract

A combat system-of-systems (CSoS) is a network of independent entities that interact to provide overall operational capabilities. Enhancing the resilience of CSoS is garnering increasing attention due to its practical value in optimizing network architectures, improving network security and refining operational planning. Accordingly, we present a unified framework called CSoS space-time resilience enhancement (CSoS-STRE) to enhance the resilience of CSoS. Specifically, we develop a spatial combat network model and a space-time resilience optimization model that captures the complex spatial relationships between entities and reformulates the resilience enhancement problem as a linear optimization model with spatial features. Moreover, we extend the model to include obstacles. Next, a resilience-oriented recovery optimization method based on the improved non-dominated sorting genetic algorithm II (R-INSGA) is proposed to determine the optimal recovery sequence for the damaged entities. This method incorporates spatial features while providing the optimal travel paths for multiple recovery teams. Finally, the feasibility, effectiveness, and superiority of the CSoS-STRE are demonstrated through a case study, providing valuable insights for guiding recovery and developing more resilient CSoS.

Graphical abstract

Keywords

combat system-of-systems / space-time resilience enhancement / resilience optimization model / recovery optimization method / no-obstacles and obstacles

References

[1] Artime O, Grassia M, De Domenico M, Gleeson J P, Makse H A, Mangioni G, Perc M, Radicchi F, (2024). Robustness and resilience of complex networks. Nature Reviews. Physics, 6( 2): 114–131

[2] Aslani B, Mohebbi S, Oughton E, (2024). A systematic review of optimization methods for recovery planning in cyber–physical infrastruc-ture networks: Current state and future trends. Computers & Industrial Engineering, 192: 110224

[3] Chatterjee A, Helbig C, Malak R, Layton A, (2023). A comparison of graph-theoretic approaches for resilient system of systems design. Journal of Computing and Information Science in Engineering, 23( 3): 030906

[4] Chatterjee A, Malak R, Layton A, (2022). Ecology-inspired resilient and affordable system of systems using degree of system order. Systems Engineering, 25( 1): 3–18

[5] Chen W, Li W, Zhang T, (2024a). Complex network-based resilience capability assessment for a combat system of systems. Systems, 12( 1): 31

[6] Chen Z, Hong D, Cui W, Xue W, Wang Y, Zhong J, (2023a). Resilience evaluation and optimal design for weapon system of systems with dynamic reconfiguration. Reliability Engineering & System Safety, 237: 109409

[7] ChenZYinSLiLCuiWHongD (2024b). Resilience metric and dynamic assessment of unmanned system-of-systems considering cooperative reconfiguration strategies. IEEE Transactions on Reliability. doi:10.1109/TR.2024.3438810

[8] Chen Z, Zhou Z, Zhang L, Cui C, Zhong J, (2023b). Mission reliability modeling and evaluation for reconfigurable unmanned weapon system-of-systems based on effective operation loop. Journal of Systems Engineering and Electronics, 34( 3): 588–597

[9] Cimellaro G P, Reinhorn A M, Bruneau M, (2010). Framework for analytical quantification of disaster resilience. Engineering Structures, 32( 11): 3639–3649

[10] Danziger M M, Barabási A L, (2022). Recovery coupling in multilayer networks. Nature Communications, 13( 1): 955

[11] De Domenico M, (2023). More is different in real-world multilayer networks. Nature Physics, 19( 9): 1247–1262

[12] Deb K, Pratap A, Agarwal S, Meyarivan T, (2002). A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Transactions on Evolutionary Computation, 6( 2): 182–197

[13] DellerSBowlingS RRabadiG ATolkABellM I (2009). Applying the information age combat model: quantitative analysis of network centric operations. The international C2 Journal, 3(1)

[14] DeptulaD APenneyH RStutzriemL AGunzingerM (2019). Restoring america’s military competitiveness: Mosaic warfare. Mitchell Institute for Airpower Studies

[15] Du C, Ouyang M, Zhang H, Wang B, Wang N, (2023). Resilience patterns of urban road networks under the worst-case localized disruptions. Risk Analysis, 44( 10): 2333–2347

[16] Dui H, Zhu Y, Tao J, (2024). Multi-phased resilience methodology of urban sewage treatment network based on the phase and node recovery importance in IoT. Reliability Engineering & System Safety, 247: 110130

[17] Edwards C M, Nilchiani R R, Miller I M, (2024). Impact of graph energy on a measurement of resilience for tipping points in complex systems. Systems Engineering, 27( 4): 745–758

[18] Fang Z, Wu S, Zhang X, Sun Y, (2021). ADC-GERT network parameter estimation model for mission effectiveness of joint operation system. Journal of Systems Engineering and Electronics, 32( 6): 1394–1406

[19] Filippini R, Silva A, (2014). A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies. Reliability Engineering & System Safety, 125: 82–91

[20] Gao F, He W, Bi W, (2024). An intuitionistic fuzzy weighted influence non-linear gauge system for equipment evaluation under system-of-systems warfare environment. Expert Systems with Applications, 238: 122187

[21] Gao Y, Cheng J, Tian Y, Liu H, (2023). Machine learning-based evaluation of the contribution effectiveness in SoS missions. IEEE Systems Journal, 17( 4): 5877–5888

[22] Hafsa M, Wattebled P, Jacques J, Jourdan L, (2025). Solving a multiobjective professional timetabling problem using evolutionary algorithms at mandarine academy. International Transactions in Operational Research, 32( 1): 244–269

[23] Han Q, Pang B, Li S, Li N, Guo P S, Fan C L, Li W M, (2023). Evaluation method and optimization strategies of resilience for air and space defense system of systems based on kill network theory and improved self-information quantity. Defence Technology, 21: 219–239

[24] Holling C S, (1973). Resilience and stability of ecological systems. Annual Review of Ecology and Systematics, 4( 1): 1–23

[25] Huang Y, Luo A, Chen T, Zhang M, Ren B, Song Y, (2023). When architecture meets RL+EA: A hybrid intelligent optimization approach for selecting combat system-of-systems architecture. Advanced Engineering Informatics, 58: 102209

[26] Khasawneh M T, Shearer N, Rabadi G, Bowling S, (2017). The information age combat model: A vision for a discrete event simulation approach. International Journal of Simulation and Process Modelling, 12( 5): 429–445

[27] KoehlerM TBricio-NetoJ LPageE HTolkA (2024). Applying complex adaptive systems research results to combat simulations of the generation-after-next. The Journal of Defense Modeling and Simulation, 15485129241233608

[28] LanierBPetngaL (2019). Spatial functions for modeling and analysis of safety-critical systems of systems. 2019 14th Annual Conference System of Systems Engineering, SoSE 2019, page 352–357

[29] Li J, Jiang J, Yang K, Chen Y, (2019). Research on functional robustness of heterogeneous combat networks. IEEE Systems Journal, 13( 2): 1487–1495

[30] Li J, Tan Y, Yang K, Zhang X, Ge B, (2017). Structural robustness of combat networks of weapon system-of-systems based on the operation loop. International Journal of Systems Science, 48( 3): 659–674

[31] Li J, Zhao D, Jiang J, Yang K, Chen Y, (2021). Capability oriented equipment contribution analysis in temporal combat networks. IEEE Transactions on Systems, Man, and Cybernetics. Systems, 51( 2): 696–704

[32] Li R, Yuan H, Ren B, Zhang X, Chen T, Luo X, (2024). Optimal unmanned combat system-of-systems reconstruction strategy with heterogeneous cost via deep reinforcement learning. Mathematics, 12( 10): 1476

[33] Liu J, Xu R, Li J, Yang K, Lou Z, (2024). Enhancing the resilience of combat system-of-systems under continuous attacks: Novel index and reinforcement learning-based protection optimization. Expert Systems with Applications, 251: 123912

[34] Liu M, Li J, Yang Z, Yang K, (2023). Higher-order functional structure exploration in heterogeneous combat network based on operational motif spectral clustering. IEEE Systems Journal, 17( 3): 4279–4290

[35] Liu X, Li D, Ma M, Szymanski B K, Stanley H E, Gao J, (2022). Network resilience. Physics Reports, 971: 1–108

[36] Lozano-Osorio I, Sánchez-Oro J, López-Sánchez A, Duarte A, (2025). A variable neighborhood search for the median location problem with interconnected facilities. International Transactions in Operational Research, 32( 1): 69–89

[37] Markina-Khusid A, Jacobs R B, Antul L, Cho L, Tran H T, (2022). A complex network framework for validated assessments of systems of systems robustness. IEEE Systems Journal, 16( 1): 1092–1102

[38] Miller K, Bordetsky A, Mun J, Maule R, Pollman A, (2021). Merging future knowledgebase system of systems with artificial intelli-gence/machine learning engines to maximize reliability and availability for decision support. Military Operations Research, 26( 4): 77–93

[39] Qi X, Mei G, (2024). Network resilience: Definitions, approaches, and applications. Journal of King Saud University – Computer and Information Sciences, 36( 1): 101882

[40] Rescia A, Gómez Menéndez A, González Lodares C, Ortega M, (2023). Quantification of the spatial resilience of the spanish rural landscape to fire occurrence analysed using the sispares network of plots. Landscape Ecology, 38( 12): 3621–3636

[41] Sinex C H, Lilly T C, Harlow M A, (2000). Using the war room process to explore network-centric warfare. Johns Hopkins APL Technical Digest, 21( 3): 368–377

[42] Sun L, Zhou Y, Zhu C, Zhang W, (2024a). A new important nodes identification method in multi-layer heterogeneous combat network with meta-path centrality. Journal of Complex Networks, 12( 3): cnae021

[43] Sun Q, Li H, Wang Y, Zhang Y, (2022). Multi-swarm-based cooperative reconfiguration model for resilient unmanned weapon system-of-systems. Reliability Engineering & System Safety, 222: 108426

[44] Sun Q, Li H, Zhong Y, Ren K, Zhang Y, (2024b). Deep reinforcement learning-based resilience enhancement strategy of unmanned weapon system-of-systems under inevitable interferences. Reliability Engineering & System Safety, 242: 109749

[45] Sun Y, Zhang T, (2023). Research on autonomous reconstruction method for dependent combat networks. IEEE Systems Journal, 17( 4): 6104–6113

[46] Tran H T, Domerçant J C, Mavris D N, (2016). A network-based cost comparison of resilient and robust system-of-systems. Procedia Computer Science, 95: 126–133

[47] Uday P, Chandrahasa R, Marais K, (2019). System importance measures: Definitions and application to system-of-systems analysis. Reliability Engineering & System Safety, 191: 106582

[48] Uday P, Marais K, (2015). Designing resilient systems-of-systems: A survey of metrics, methods, and challenges. Systems Engineering, 18( 5): 491–510

[49] Wang H, (2023). A new algorithm for euclidean shortest paths in the plane. Journal of the Association for Computing Machinery, 70( 2): 11

[50] Wang L, Jin J G, Sun L, Lee D H, (2024a). Urban rail transit disruption management: Research progress and future directions. Frontiers of Engineering Management, 11( 1): 79–91

[51] Wang Z, Su Z, Deng Y, Kurths J, Wu J, (2024b). Spatial network disintegration based on kernel density estimation. Reliability Engineering & System Safety, 245: 110005

[52] Watson B, Chowdhry A, Weissburg M, Bras B, (2022). A new resilience metric to compare system of systems architecture. IEEE Systems Journal, 16( 2): 2056–2067

[53] Watson B C, Morris Z B, Weissburg M, Bras B, (2023). System of system design-for-resilience heuristics derived from forestry case study variants. Reliability Engineering & System Safety, 229: 108807

[54] Pan X, Wang H, Yang Y, Zhang G, (2019). Resilience based importance measure analysis for SoS. Journal of Systems Engineering and Electronics, 30( 5): 920–930

[55] Xu R, Gong L, Xie J, Liu X, Yang K, (2023a). Operation network link importance evaluation and recovery strategy based on equipment system-of-systems resilience. Journal of Systems Engineering and Electronics, 45( 1): 139–147

[56] Xu R, Liu J, Li J, Yang K, Zio E, (2024). TSoSRA: A task-oriented resilience assessment framework for system-of-systems. Reliability Engineering & System Safety, 248: 110186

[57] Xu R, Liu X, Cui D, Xie J, Gong L, (2023b). An evaluation method of contribution rate based on fuzzy bayesian networks for equipment system-of-systems architecture. Journal of Systems Engineering and Electronics, 34( 3): 574–587

[58] Yabe T, Rao S C, Ukkusuri S V, Cutter S L, (2022). Toward data-driven, dynamical complex systems approaches to disaster resilience. Proceedings of the National Academy of Sciences of the United States of America, 119( 8): e2111997119

[59] Yang D, Li Q, Zhu F, Cui H, Yi W, Qin J, (2023). Parallel emergency management of incidents by integrating ooda and prea loops: The c2 mechanism and modes. IEEE Transactions on Systems, Man, and Cybernetics. Systems, 53( 4): 2160–2172

[60] Zhang J, Fang Z, Dong W, Zhang L, (2024). Portfolios selection decision model for equipment system of systems considering development costs. Expert Systems with Applications, 246: 123235

[61] Zhong Y, Li H, Sun Q, Huang Z, Zhang Y, (2024). A kill chain optimization method for improving the resilience of unmanned combat system-of-systems. Chaos, Solitons, and Fractals, 181: 114685

[62] Zobel C W, (2011). Representing perceived tradeoffs in defining disaster resilience. Decision Support Systems, 50( 2): 394–403

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