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MCCARTNEY M . The solution of Lanchester’s equations with inter-battle reinforcement strategies [J ] . Physica A:Statistical Mechanics and its Applications , 2022 , 586 : 126477 . DOI: 10.1016/j.physa.2021.126477 http://doi.org/10.1016/j.physa.2021.126477 https://linkinghub.elsevier.com/retrieve/pii/S0378437121007500 https://linkinghub.elsevier.com/retrieve/pii/S0378437121007500

KRESS M , CAULKINS J P , FEICHTINGER G , et al . Lanchester model for three-way combat [J ] . European Journal of Operational Research , 2018 , 264 ( 1 ): 46 - 54 . DOI: 10.1016/j.ejor.2017.07.026 http://doi.org/10.1016/j.ejor.2017.07.026 https://linkinghub.elsevier.com/retrieve/pii/S0377221717306537 https://linkinghub.elsevier.com/retrieve/pii/S0377221717306537

赵丹玲 , 谭跃进 , 李际超 , 等 . 基于异质网络的武器装备体系结构抗毁性研究 [J ] . 系统工程理论与实践 , 2019 , 39 ( 12 ): 3197 - 3207 . DOI: 10.12011/1000-6788-2018-1302-11 http://doi.org/10.12011/1000-6788-2018-1302-11 考虑到武器装备体系中装备节点的复杂性和异质性，以及装备之间关系的多样性，本文首先提出了基于异质网络的武器装备体系建模方法.其次通过借鉴OODA作战循环理论的概念，结合异质网络中元路径的定义，提出评估指标——杀伤链数对武器装备体系结构抗毁性进行评估.然后将杀伤链数和常用的抗毁性评估指标——自然连通度进行比较，分析随机攻击和选择性攻击策略下武器装备体系结构抗毁性的变化，发现具备实际语义信息的杀伤链数对于评估武器装备体系结构抗毁性更加有效.最后，通过想定背景下的武器装备作战体系实例，分析该实例下的武器装备体系结构抗毁性，为进攻方首要攻击对象和防守方首要保护对象提供辅助决策.

ZHAO D L , TAN Y J , LI J C , et al . Research on structural robustness of weapon system-of-systems based on heterogeneous network [J ] . Systems Engineering Theory & Practice , 2019 , 39 ( 12 ): 3197 - 3207 . (in Chinese)

李尔玉 , 龚建兴 , 黄健 . 基于功能链的融合网络功能抗毁性评估 [J ] . 兵工学报 , 2019 , 40 ( 7 ): 1450 - 1459 . DOI: 10.3969/j.issn.1000-1093.2019.07.015 http://doi.org/10.3969/j.issn.1000-1093.2019.07.015 为解决现有作战网络模型抽象度过高、网络功能复杂性研究不够充分的问题，提出了一种 针对融合网络的功能抗毁性分析方法。建立作战体系融合复杂网络模型，在单层网路中处理异质、融合的节点；提出新的功能链概念，并基于功能链定义了节点的功能参与度和业务强度；在此基础上，定义功能完整度为网络功能抗毁性测度，对融合网络模型进行抗毁性评估。在抗毁性实验中，采用3类攻击方式对网络进行打击：网络信息完全未知情况下的随机攻击;网络信息不完全已知情况下的蓄意攻击——按节点度攻击;网络信息完全已知情况下的蓄意攻击——按功能重要度攻击和按贪婪规则攻击。实验结果显示：网络对抗随机攻击能力较强，对抗信息不完全已知的蓄意攻击能力稍弱，对抗信息已知的蓄意攻击能力较弱，验证了所提算法合理性。

LI E Y , GONG J X , HUANG J . Analysis about functional invulnerability of convergent network based on function chain [J ] . Acta Armamentarii , 2019 , 40 ( 7 ): 1450 - 1459 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2019.07.015 http://doi.org/10.3969/j.issn.1000-1093.2019.07.015 As the existing combat network model is abstracted and the study on network functional complexity is not sufficient enough, an analysis method for functional invulnerability of convergent networks is presented. A complex convergent network model of combat system is established, which deals with hetero- geneous and convergent nodes in a single-layer network, and a new concept of function chain is proposed. The functional participation and functional importance of nodes are defined by function chain. On this basis, the functional integrity is defined as the measure of network functional invulnerability. Three attack modes are adopted in experiment: the first is random attack when network information is unknown, the second is intentional attack as network information is known incompletely, including attacking according to node degree, and the third is intentional attack within network information is known, including attacking according to functional importance and greedy algorithm. Experimental results show that the network has a higher robustness against random attack; it has less robustness against intentional attack in the case of incompletely known information ; and it has more less robustness against intentional attack in the case of known information.Key

王运明 , 陈思 , 陈波 , 等 . 基于作战链路效率的指挥与控制网络抗毁测度 [J ] . 指挥与控制学报 , 2017 , 3 ( 1 ): 61 - 66 .

WANG Y M , CHEN S , CHEN B , et al . Measure of invulnerability for command and control network based on combat link effciency [J ] . Journal of Command and Control , 2017 , 3 ( 1 ): 61 - 66 . (in Chinese)

郭晓成 , 马润年 , 陈彤睿 , 等 . 跨层协同指挥控制网络抗毁性研究 [J ] . 火力与指挥控制 , 2019 , 44 ( 6 ): 36 - 40 .

GUO X C , MA R N , CHEN T R , et al . Study on invulnerability of cross-layer cooperative C2 network [J ] . Fire Control & Command Control , 2019 , 44 ( 6 ): 36 - 40 . (in Chinese)

LI J C , JIANG J , YANG K W , et al . Research on functional robustness of heterogeneous combat networks [J ] . IEEE Systems Journal , 2019 , 13 ( 2 ): 1487 - 1495 . DOI: 10.1109/JSYST.2018.2828779 http://doi.org/10.1109/JSYST.2018.2828779 The combat system-of-systems in high-tech informative warfare consists of different types of combat systems connected via multiple interconnections, which can be abstracted as types of complex heterogeneous networks. Research on the functional robustness of heterogeneous combat networks (HCNs) is of significant military value for optimizing operation process planning and for improving network survivability in complex electromagnetic battle fields. This study presents an integrated framework, known as the HCN functional robustness (HCNFR) framework to measure the functional robustness of HCNs under various attacking strategies. Specifically, a heterogeneous network model is introduced to characterize HCNs considering different types of functional entities and information flows. Based on this setup, the rescaled operational capability index (ROCI) is first proposed to evaluate the functional robustness of HCNs, and the criterion for functional disintegration of HCNs is established by summing the values of ROCI during the attacking process. Finally, the reliability and effectiveness of HCNFR are illustrated with an empirical case study. The results provide useful insights for operational guidance and design of a more functional and robust combat system-of-systems.

陈登 , 陈楚湘 , 周春华 . 基于“OODA环”的杀伤网节点重要性评估 [J/OL ] . 兵工学报 , 2022 (2022-08-25)[2022-11-10 ] . https://doi.org/10.12382/bgxb.2022.0623 https://dx.doi.org/10.12382/bgxb.2022.0623 .

CHEN D , CHEN C X , ZHOU C H . Importance evaluation of kill network nodes based on “OODA loop” [J/OL ] . Acta Armamentarii , 2022 (2022-08-25) [2022-11-10 ] . https://doi.org/10.12382/bgxb.2022.0623 https://dx.doi.org/10.12382/bgxb.2022.0623 . (in Chinese)

王超 , 郭基联 , 符凌云 . 基于拓扑势的作战体系网络节点重要度评估方法 [J ] . 兵工学报 , 2020 , 41 ( 8 ): 1658 - 1664 . DOI: 10.3969/j.issn.1000-1093.2020.08.020 http://doi.org/10.3969/j.issn.1000-1093.2020.08.020 针对传统重要度排序算法无法客观全面评估作战体系中各装备实体重要度的不足，运用复杂网络理论，结合数学中的拓扑学和物理学中的场论思想，提出基于拓扑势的重要度排序方法。通过定义序列距离，引入覆盖率指标，对拓扑势排序结果与常规方法排序结果的相似度进行对比分析。设计一个作战想定示例，对基于拓扑势的重要度排序算法的有效性进行分析验证。结果表明，该算法可以精确客观地量化分析体系中不同装备对体系结构构建以及任务完成能力的支撑作用，为战时兵力部署、指挥体系构建、战场抢修决策提供重要依据。

WANG C , GUO J L , FU L Y . Research on evaluation method of node importance of combat system-of-systems network based on topological potential [J ] . Acta Armamentarii , 2020 , 41 ( 8 ): 1658 - 1664 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2020.08.020 http://doi.org/10.3969/j.issn.1000-1093.2020.08.020 The traditional importance ranking algorithm can not be used to objectively and comprehensively evaluate the importance of each equipment entity in the combat system-of-systems network. An importance ranking method based on topological potential is proposed based on complex network theory combining with the topology theory in mathematics and field theory in physics. A coverage index is introduced to compare and analyze the similarity between the topological potential ranking results and the conventional method ranking results by defing a sequenced distance. An operational scenario is designed to analyze and verify the effectiveness of the importance ranking algorithm based on topological potential. The results show that the proposed algorithm can accurately and objectively quantify the supporting roles of different equipment in the system-of-systems to the construction of the structure and the ability to complete the task, and provide an important basis for the wartime deployment of forces, the construction of command system-of-systems, and the battlefield repair decisions.

李尔玉 , 龚建兴 , 黄健 , 等 . 基于功能链的作战体系复杂网络节点重要性评价方法 [J ] . 指挥与控制学报 , 2018 , 4 ( 1 ): 42 - 49 .

LI E Y , GONG J X , HUANG J , et al . Node importance analysis of complex networks for combat systems based on function chain [J ] . Journal of Command and Control , 2018 , 4 ( 1 ): 42 - 49 . (in Chinese)

周琛 , 宋笔锋 , 尚柏林 , 等 . 基于作战网络可靠度的体系贡献率评估 [J ] . 系统工程与电子技术 , 2021 , 43 ( 7 ): 1875 - 1883 . DOI: 10.12305/j.issn.1001-506X.2021.07.19 http://doi.org/10.12305/j.issn.1001-506X.2021.07.19 针对武器装备体系中日益复杂的装备种类、交互关系和大量兼具多种功能的综合型武器装备, 建立了包含综合型装备的武器装备体系作战环网络描述模型。首先，结合网络可靠度思想对现有的作战环网络能力求解方法进行改进, 分别给出了基于不交最小路的精确求解方法和基于谱半径的蒙特卡罗近似求解方法。之后, 给出了一种改善的综合型装备体系贡献率评估方法。最后, 利用作战实例对上述建模求解流程进行检验, 并利用程序生成随机网络对两种算法进行了比较。结果表明，上述评估流程具有良好的可行性和准确性，两种算法对不同规模的网络在运算时间和精度方面则具有不同的适用性。

ZHOU C , SONG B F , SHANG B L , et al . System of systems contribution rate evaluation based on operational network reliability [J ] . Systems Engineering and Electronics , 2021 , 43 ( 7 ): 1875 - 1883 . (in Chinese) DOI: 10.12305/j.issn.1001-506X.2021.07.19 http://doi.org/10.12305/j.issn.1001-506X.2021.07.19 In view of the increasingly complex equipment types, interaction relations and a large number of multifunctional equipment in the armament system of systems, the operation loop network description model of the armament system of systems containing the multifunctional equipment is established. Firstly, based on the network reliability theory, the method of calculating the armament system of systems operation capabilities is improved, and two methods of the min-path method and the Monte Carlo method based on spectral radius are presented. Then, an improved evaluation method of contribution rate is presented. Finally, an operation system example is used to verify the above modeling and solving process. The two algorithms are compared by using the random network generated by the program.The result shows that the evaluation process is feasible and accurate, and the two algorithms have different applicability to networks of different sizes in terms of calculation time and accuracy.

LI J C , ZHAO D L , JIANG J , et al . Capability oriented equipment contribution analysis in temporal combat networks [J ] . IEEE Transactions on Systems, Man, and Cybernetics:Systems , 2018 , 51 ( 2 ): 696 - 704 . DOI: 10.1109/TSMC.6221021 http://doi.org/10.1109/TSMC.6221021 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6221021 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6221021

聂俊峰 , 陈行军 , 史红权 . 面向任务驱动的海上编队云作战体系动态超网络模型 [J ] . 兵工学报 , 2021 , 42 ( 11 ): 2513 - 2521 . DOI: 10.3969/j.issn.1000-1093.2021.11.024 http://doi.org/10.3969/j.issn.1000-1093.2021.11.024 针对云作战新型作战模式，提出了基于作战任务域、逻辑功能域和资源实体域，以及情报侦察网、主动辅助决策网和火力打击网的海上编队云作战体系分析模型。抽象表征了海上编队云作战体系的功能节点、信息关系以及关联映射规则，建立了面向任务驱动的海上编队云作战体系动态超网络模型。面向超网络要素特征、超网络关联特征和超网络弹性特征构建了海上编队云作战体系动态超网络特征参数空间。通过海上编队防空典型作战任务对模型进行了仿真实现，并基于特征参数对仿真结果进行了对比分析。仿真结果表明，海上编队云作战体系动态超网络具有较好的鲁棒性和网络弹性，云作战体系相比于传统作战体系具有更好的抗毁性和打击效率，该方法能够科学有效地对海上编队云作战体系进行动态建模。

NIE J F , CHEN X J , SHI H Q . Dynamic super-network model of task-oriented naval fleet “Cloud Combat” system [J ] . Acta Armamentarii , 2021 , 42 ( 11 ): 2513 - 2521 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2021.11.024 http://doi.org/10.3969/j.issn.1000-1093.2021.11.024 For the new operation mode of “cloud combat”，a analysis model of naval fleet “cloud combat” system is proposed，which includes operational mission domain，logical function domain, resource entity domain， intelligence reconnaissance network，active assistant decision network and fire strike network. The functional nodes，information relations and association mapping rules of naval fleet “cloud combat” system are abstractly represented，and a dynamic super-network model of task-oriented naval fleet “cloud combat” system is constructed. The feature parameter space of dynamic super-network of naval fleet “cloud combat” system is constructed based on the features of super-network elements，super-network association and super-network elasticity. The model is simulated for the typical combat task of air defense. The simulated results demonstrate that the dynamic super-network of naval fleet “cloud combat” system has good robustness and network elasticity，and the “cloud combat” system has better invulnerability and attack efficiency compared to the traditional combat system. This method can solve the modeling problem of naval fleet “cloud combat” system scientifically and effectively.

LI J C , ZHAO D L , GE B F , et al . Disintegration of operational capability of heterogeneous combat networks under incomplete information [J ] . IEEE Transactions on Systems Man Cybernetics-System , 2020 , 50 ( 12 ): 5172 - 5179 . DOI: 10.1109/TSMC.6221021 http://doi.org/10.1109/TSMC.6221021 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6221021 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6221021

王耀祖 , 尚柏林 , 宋笔锋 , 等 . 基于杀伤链的作战体系网络关键节点识别方法 [J ] . 系统工程与电子技术 , 2023 ( 3 ): 736 - 744 . DOI: 10.12305/j.issn.1001-506X.2023.03.14 http://doi.org/10.12305/j.issn.1001-506X.2023.03.14 针对作战体系(operational system-of-system, SOS)因装备功能各异、交互关系复杂、高对抗性导致的关键装备难以识别的问题, 引入杀伤链的分析思想对其进行求解。首先, 提出了作战体系网络模型构建方法, 给出了杀伤链的定义与基本类型。其次, 提出了基于蒙特卡罗抽样的网络状态转化方法, 实现了网络边权值与边存在性的转化; 根据子图同构匹配理论, 给出了基于改进Ullmann算法的杀伤链搜索方法; 以对杀伤链形成的综合影响为切入点, 构建了节点重要度指标R。最后, 利用所提方法对某空中拦截任务体系网络的关键节点进行识别, 并与现有方法的结果进行对比, 验证了所提方法的合理性和有效性。

WANG Y Z , SHANG B L , SONG B F , et al . Identification method of key node in operational system-of-systems network based on kill chain [J ] . Systems Engineering and Electronics , 2023 ( 3 ): 736 - 744 . (in Chinese) DOI: 10.12305/j.issn.1001-506X.2023.03.14 http://doi.org/10.12305/j.issn.1001-506X.2023.03.14 Aiming at the problem that the key equipment is difficult to identify for operational system-of-systems (SoS) due to different equipment functions, complex interaction relationship and high confrontation, the analysis idea of kill chain is introduced to solve the above problem. Firstly, the method of constructing network model of operational SoS is proposed, and the definition and basic types of kill chains are introduced. Secondly, a network state transformation method based on Monte Carlo sampling is proposed to realize the transformation of the edge w eight and edge existence. According to subgraph isomorphism matching theory, a kill chain search method is presented based on the improved Ullmann algorithm. Taking the comprehensive impact on the formation of kill chains as the entry point, the node importance index R is constructed. Finally, the proposed method is used to identify the key node of an air interception mission SoS network, and the results of the existing methods are compared. The comparision results verify the rationality and effectiveness of the proposed method.

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DAVILA-FRIAS A , YADAV O P . All-terminal network reliability estimation using convolutional neural networks [J ] . Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability , 2022 , 236 ( 4 ): 584 - 597 . DOI: 10.1177/1748006X20969465 http://doi.org/10.1177/1748006X20969465 http://journals.sagepub.com/doi/10.1177/1748006X20969465 http://journals.sagepub.com/doi/10.1177/1748006X20969465 Estimating the all-terminal network reliability by using artificial neural networks (ANNs) has emerged as a promissory alternative to classical exact NP-hard algorithms. Approaches based on traditional ANNs have usually considered the network reliability upper bound as part of the inputs, which implies additional time-consuming calculations during both training and testing phases. This paper proposes the use of Convolutional Neural Networks (CNNs), without the reliability upper-bound as an input, to address the all-terminal network reliability estimation problem. The present study introduces a multidimensional matrix format to embed the topological and link reliability information of networks. The unique contribution of this article is the method to capture the topology of a network in terms of its adjacency matrix, link reliability, and topological attributes providing a novel use of CNN beyond image classification. Since CNNs have been successful for image classification, appropriate modifications are needed and introduced to use them in the estimation of network reliability. A regression output layer is proposed, preceded by a sigmoid layer to achieve predictions within the range of reliability characteristic, a feature that some previous ANN-based works lack. Several training parameters together with a filter multiplier (CNN architecture parameter) were investigated. The actual values and the ones predicted with the best trained CNN were compared in the light of RMSE (0.04406) and p-value (0.3) showing non-significant difference. This study provides evidence supporting the hypothesis that the network reliability can be estimated by CNNs from its topology and link reliability information, embedded as an image-like multidimensional matrix. Another important result of the proposed approach is the significant reduction in computational time. An average of 1.18 ms/network was achieved by the CNN, whereas backtracking exact algorithm took around 500 s/network.

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BROWN J , DEGAGNÉ C . Roots of two-terminal reliability polynomials [J ] Networks , 2021 , 78 ( 2 ): 153 - 163 . DOI: 10.1002/net.v78.2 http://doi.org/10.1002/net.v78.2 https://onlinelibrary.wiley.com/toc/10970037/78/2 https://onlinelibrary.wiley.com/toc/10970037/78/2

DOHMEN K . Inclusion-exclusion and network reliability [J ] . The Electronic Journal of Combinatorics , 1998 , 5 ( 1 ): R36 . DOI: 10.37236/1374 http://doi.org/10.37236/1374 https://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r36 https://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r36 Based on a recent improvement of the inclusion-exclusion principle, we present a new approach to network reliability problems. In particular, we give a new proof of a result of Shier, which expresses the reliability of a network as an alternating sum over chains in a semilattice, and a new proof of a result of Provan and Ball, which provides an algorithm for computing network reliability in pseudopolynomial time. Moreover, some results on $k$-out-of-$n$ systems are established.

BALL M O . Complexity of network reliability computations [J ] . Networks , 1980 , 10 ( 2 ): 153 - 165 . DOI: 10.1002/net.v10:2 http://doi.org/10.1002/net.v10:2 https://onlinelibrary.wiley.com/toc/10970037/10/2 https://onlinelibrary.wiley.com/toc/10970037/10/2

杨圩生 , 王钰 , 杨洋 , 等 . 基于作战环的不同节点攻击策略下的作战网络效能评估 [J ] . 系统工程与电子技术 , 2021 , 43 ( 11 ): 3220 - 3228 . DOI: 10.12305/j.issn.1001-506X.2021.11.22 http://doi.org/10.12305/j.issn.1001-506X.2021.11.22 基于作战环思想, 以作战网络抽象模型为基础, 围绕作战环的分类、定义、形式化描述、数学模型等基本问题开展了理论研究, 进一步丰富了作战环理论体系。同时, 以能否快速有效打击对方目标实体为根本依据, 建立了基于目标节点打击率和基于目标节点打击效率的作战网络效能评估指标。通过仿真实验, 将作战环相关理论运用于不同节点攻击策略下的作战网络效能评估, 为实际作战体系对抗中双方的攻击与防护提供应用指导, 也为进一步开展基于作战环的作战网络应用研究提供参考借鉴。

YANG W S , WANG Y , YANG Y , et al . Combat network effectiveness evaluation under different node attackstrategies based on operation loop [J ] . Systems Engineering and Electronics , 2021 , 43 ( 11 ): 3220 - 3228 . (in Chinese)