技术领域technical field
本发明涉及系统工程中的复杂网络博弈技术领域，尤其涉及基于双曲隶属函数的复杂网络博弈防御方策略优化方法。The invention relates to the technical field of complex network games in system engineering, in particular to a complex network game defense strategy optimization method based on hyperbolic membership functions.
背景技术Background technique
在当前博弈论的研究领域，有一类特殊的网络博弈，其中的网络并不是计算机系统等实际存在的网络，而是将关键基础设施，比如火车站点、飞机场等抽象成网络拓扑结构中的节点，将不同站点之间的联系抽象成边，建立基础设施复杂网络。在安防领域，基础设施的关键节点容易受到攻击，对治安管理和正常社会生活会造成影响，安防部门需要对这些节点进行防护。可以采用复杂网络博弈来研究基础设施网络中关键节点的攻击和防护问题，有助于制定最佳防护策略，探索节点的重要性。In the current research field of game theory, there is a special kind of network game, in which the network is not the actual network such as computer system, but the key infrastructure, such as railway station, airport, etc., is abstracted into nodes in the network topology , abstract the connection between different sites into edges, and build a complex network of infrastructure. In the field of security, key nodes of infrastructure are vulnerable to attacks, which will affect public security management and normal social life. Security departments need to protect these nodes. Complex network games can be used to study the attack and protection of key nodes in the infrastructure network, which helps to formulate the best protection strategy and explore the importance of nodes.
目前已有关于此类问题的研究，但是现有研究只给出了基于网络拓扑结构的客观评价方法，比如在完全信息静态或动态博弈框架下，利用网络连通性能指标——最大连通片规模计算攻击方和防御方的收益矩阵，并计算相应的纳什均衡策略。但是在此类实际的博弈问题中，博弈参与双方对问题的理解并不确定，获得信息不足，决策环境变幻莫测，现有方法并不能很好地融合决策者的主观判断，不能表达实际博弈问题的模糊性和不确定性。There have been researches on this kind of problem, but the existing research only gives an objective evaluation method based on the network topology structure, such as using the network connectivity performance index - the calculation of the largest connected slice size under the framework of complete information static or dynamic game The profit matrix of the attacker and the defender, and calculate the corresponding Nash equilibrium strategy. However, in such actual game problems, the understanding of the problem by both parties involved in the game is uncertain, the information obtained is insufficient, and the decision-making environment is unpredictable. The existing methods cannot well integrate the subjective judgment of the decision maker and cannot express the actual game. Ambiguity and uncertainty of the problem.
Zadeh教授提出了模糊集理论，为解决这类问题提供了一条合理的途径。针对模糊集合论的局限性和表达犹豫的实际需要，Atanassov提出了直觉模糊集理论，它用两个尺度(隶属度和非隶属度)来表示模糊现象的支持、反对和犹豫。这一理论的提出为解决更复杂的博弈问题提供了启示。目前将直觉模糊集理论引入基础设施复杂网络博弈中的相关研究较少，这方面的研究具有重要意义。Professor Zadeh proposed fuzzy set theory, which provides a reasonable way to solve this kind of problems. Aiming at the limitations of fuzzy set theory and the actual needs of expressing hesitation, Atanassov proposed the intuitionistic fuzzy set theory, which uses two scales (membership degree and non-membership degree) to express the support, opposition and hesitation of fuzzy phenomena. The proposal of this theory provides enlightenment for solving more complex game problems. At present, there are few related studies on the introduction of intuitionistic fuzzy set theory into infrastructure complex network games, and this research is of great significance.
发明内容Contents of the invention
本发明旨在至少解决现有技术中存在的技术问题之一。为此，本发明公开了基于双曲隶属函数的复杂网络博弈防御方策略优化方法。所述方法利用直觉模糊集的二人零和矩阵对策，提出了一种利用双曲隶属度/非隶属度函数生成直觉模糊收益矩阵的方法，在直觉模糊条件下，采用一种有效的算法求解纳什均衡解，最终获得攻击方策略的优化方法和结果。The present invention aims to solve at least one of the technical problems existing in the prior art. For this reason, the invention discloses a complex network game defense strategy optimization method based on a hyperbolic membership function. The method uses the two-person zero-sum matrix strategy of intuitionistic fuzzy sets, and proposes a method for generating an intuitionistic fuzzy benefit matrix using a hyperbolic membership/non-membership function. Under intuitionistic fuzzy conditions, an effective algorithm is used to solve Nash equilibrium solution, and finally obtain the optimization method and result of the attacker's strategy.
本发明的目的是通过如下技术方案实现的，基于双曲隶属函数的复杂网络博弈防御方策略优化方法，所述方法包括：The object of the present invention is to be realized by following technical scheme, based on hyperbolic membership function complex network game defender strategy optimization method, said method comprises:
步骤1，获取基础设施网络的拓扑结构，确定攻击方和防御方的策略集合，构建复杂网络博弈的基本模型；Step 1. Obtain the topology of the infrastructure network, determine the strategy set of the attacker and the defender, and build a basic model of the complex network game;
步骤2，确定表示网络连通性的指标，计算复杂网络博弈的基本模型中攻击方和防御方在各种策略剖面下的收益，由此得到收益矩阵；Step 2, determine the indicators representing network connectivity, and calculate the income of the attacker and the defender under various strategy profiles in the basic model of complex network games, and thus obtain the income matrix;
步骤3，利用直觉模糊数确定方法，构建双曲线隶属度函数和非隶属度函数；Step 3, using the intuitionistic fuzzy number determination method to construct a hyperbolic membership function and a non-membership function;
步骤4，利用双曲线隶属度函数和非隶属度函数，将所述的收益矩阵转换为直觉模糊集收益矩阵；Step 4, using the hyperbolic membership function and non-membership function, converting the income matrix into an intuitionistic fuzzy set income matrix;
步骤5，将复杂网络博弈的基本模型的求解转换为线性规划问题求解，得到混合纳什均衡解，即获得防御方策略优化结果。In step 5, the solution of the basic model of the complex network game is converted into a solution of a linear programming problem, and a mixed Nash equilibrium solution is obtained, that is, the strategy optimization result of the defender is obtained.
所述的基础设施网络表示为简单无向图G(V,E)，其中V＝{v1,v2,...,vN}代表网络中所有节点的集合，其中N＝|V|表示网络中的节点数量，是网络中所有边的集合。The infrastructure network is expressed as a simple undirected graph G(V,E), where V={v1 ,v2 ,...,vN } represents the set of all nodes in the network, where N=|V| Indicates the number of nodes in the network, is the set of all edges in the network.
具体地，所述的攻击方的策略集合为SA，对于一个攻击策略向量sA＝[x1,x2,...,xN]∈SA，xi表示第i个基础设施是否被攻击，记为攻击节点的集合，如果第i个节点vi被攻击，即vi∈VA，xi＝1，否则xi＝0；防御方的策略向量sD＝[y1,y2,...,yN]∈SD，yi表示第i个基础设施是否被防守，/>为防守节点的集合，如果第i个节点vi被防守，即vi∈VD，yi＝1，否则yi＝0，对于被攻击的节点vi，若同时存在xi＝1且yi＝0，即虽被攻击但是未被保护，则节点vi将被移除，由此，攻击向量的总成本为：Specifically, the attacker's strategy set is SA , for an attack strategy vector sA =[x1 ,x2 ,...,xN ]∈SA ,xi represents whether the i-th infrastructure was attacked, remember is the set of attack nodes, if the i-th node vi is attacked, that is, vi ∈ VA , xi =1, otherwise xi =0; the strategy vector sD of the defender =[y1 ,y2 ,. ..,yN ]∈SD , yi represents whether the i-th infrastructure is defended, /> is the set of defending nodes, if the i-th node vi is defended, that is, vi ∈ VD , yi = 1, otherwise yi = 0, for the attacked node vi , if xi = 1 and yi = 0, that is, although it is attacked but not protected, the node vi will be removed. Therefore, the total cost of the attack vector is:
其中，表示对于攻击方来说第i个节点的成本，ri表示第i个节点的度，qA表示攻击方对于节点度的成本敏感系数；in, Represents the cost of the i-th node for the attacker, ri represents the degree of the i-th node, and qA represents the cost sensitivity coefficient of the attacker to the node degree;
攻击方的总成本是有限的，因此应满足约束：The total cost of the attacker is bounded, so the constraints should be satisfied:
其中，CA表示攻击方的总成本约束，θA表示攻击方的成本约束系数，取值范围为[0,1]；Among them, CA represents the total cost constraint of the attacker, θA represents the cost constraint coefficient of the attacker, and the value range is [0,1];
ωA与ωD分别表示攻击方与防御方对可用成本资源的最低利用率，对于攻击方，应满足约束：ωA and ωD represent the minimum utilization rate of available cost resources by the attacker and the defender, respectively. For the attacker, the constraint should be satisfied:
防御方应满足约束：The defender should satisfy the constraints:
其中，CD表示防御方的总成本约束，θD表示防御方的成本约束系数，取值范围为[0,1]，ri表示第i个节点的度，qD表示防御方对于节点度的成本敏感系数。Among them, CD represents the total cost constraint of the defender, θD represents the cost constraint coefficient of the defender, and the value range is [0,1], ri represents the degree of the i-th node, and qD represents the degree of the defender for the node cost sensitivity coefficient.
具体地，所述的收益矩阵分为攻击方的收益矩阵和防御方的收益矩阵，UA:|SA|×|SD|为攻击方的收益函数，则UA(X,Y)表示攻击方在选择策略X和防御方选择策略Y时攻击方的收益，UD(X,Y)表示攻击方在选择策略X和防御方选择策略Y时防御方的收益：Specifically, the income matrix is divided into the income matrix of the attacker and the income matrix of the defender, UA :|SA |×|SD | is the income function of the attacker, then UA (X, Y) represents The attacker’s income when the attacker chooses strategy X and the defender chooses strategy Y, UD (X, Y) represents the defender’s income when the attacker chooses strategy X and the defender chooses strategy Y:
其中，Γ(G)表示初始的基础设施网络G的最大连通片规模，记所有被移除的节点组成的集合为节点被移除后组成的网络为/>是节点被移除后组成的网络中所有边的集合，/>表示进行一轮博弈后网络的最大连通片规模，且满足/>Among them, Γ(G) represents the maximum connected slice size of the initial infrastructure network G, and the set composed of all removed nodes is The network formed after the node is removed is /> is the set of all edges in the network formed after the node is removed, /> Indicates the maximum connected slice size of the network after a round of game, and satisfies />
具体地，所述的双曲线隶属度函数u(x)和所述的非隶属度函数v(x)的形式为：Specifically, the forms of the hyperbolic membership function u(x) and the non-membership function v(x) are:
其中，α表示最高可接受水平，β表示最低可接受水平；Among them, α indicates the highest acceptable level, and β indicates the lowest acceptable level;
攻击方选取第i个纯策略sAi∈SA(i＝1,2,...,m)，防御方选择第j个纯策略sDj∈SD(j＝1,2,...,n)，攻击方的原有收益值可表示为Γ(G)ij表示攻击方选择策略i，防御方选择策略j时的原有最大联通片规模,/>表示攻击方选择策略i，防御方选择策略j时进行一轮博弈后的最大联通片规模，通过双曲隶属函数将攻击方收益值转化为直觉模糊集<μij,νij>，而防御方的损失也为<μij,νij>，μij表示攻击方选择策略i，防御方选择策略j时对于攻击方的隶属度，νij表示攻击方选择策略i，防御方选择策略j时对于攻击方的非隶属度，由此，攻击方在不同纯策略局势下的直觉模糊集收益矩阵表示为The attacker chooses the i-th pure strategy sAi ∈ SA (i=1,2,...,m), and the defender chooses the j-th pure strategy sDj ∈ SD (j=1,2,... ,n), the original revenue value of the attacker can be expressed as Γ(G)ij represents the original maximum size of China Unicom when the attacker chooses strategy i and the defender chooses strategy j,/> Indicates the maximum size of connected slices after a round of games when the attacker chooses strategy i and the defender chooses strategy j, and transforms the attacker’s revenue value into an intuitionistic fuzzy set <μij ,νij > through the hyperbolic membership function, while the defender The loss is also <μij ,νij >, μij represents the membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j, νij represents the attacker chooses strategy i, and when the defender chooses strategy j for The non-membership degree of the attacking party, thus, the intuitionistic fuzzy set benefit matrix of the attacking party in different pure strategy situations is expressed as
则混合策略下，攻击方的直觉模糊集均衡收益为：Then under the mixed strategy, the attacker’s intuitionistic fuzzy set equilibrium income is:
其中p＝(p1,p2,...,pm)T表示攻击方混合策略的概率向量，q＝(q1,q2,...,qn)T表示防御方混合策略的概率向量。Where p=(p1 ,p2 ,...,pm )T represents the probability vector of the attacker's mixed strategy, q=(q1 ,q2 ,...,qn )T represents the defense's mixed strategy probability vector.
具体地，步骤5中所述的线性规划问题为：Specifically, the linear programming problem described in step 5 is:
和and
其中，i表示攻击方的第i个策略，共m个策略；j表示防御方的第j个策略，共n个策略,λ表示隶属/非隶属函数约束的相对权重，λ确定后，得到的纳什均衡解为：(p,q,<μ,ν>,<σ,ρ>)，<μ,ν>表示攻击方的收益值，<σ,ρ>表示防御方的收益值，形式均为直觉模糊集，μ表示攻击方收益的隶属度，ν表示攻击方收益的非隶属度，σ表示防御方收益的隶属度，ρ表示防御方收益的非隶属度。Among them, i represents the i-th strategy of the attacker, a total of m strategies; j represents the j-th strategy of the defender, a total of n strategies, λ represents the relative weight of the membership/non-membership function constraints, after λ is determined, the obtained The Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>), <μ,ν> represents the income value of the attacker, and <σ,ρ> represents the income value of the defender, in the form of For intuitionistic fuzzy sets, μ represents the membership degree of the attacker’s revenue, ν represents the non-membership degree of the attacker’s revenue, σ represents the membership degree of the defender’s revenue, and ρ represents the non-membership degree of the defender’s revenue.
与现有方法相比，本发明方法的优点在于：复杂网络博弈是近年来的研究热点，然而现有的研究无法反映决策者对博弈问题认识的模糊性，本发明方法提出了基于直觉模糊集的复杂网络博弈模型，给出了直觉模糊集收益矩阵的生成方法和求解思路，最后得防御方在模糊情况下应该做出的合理策略选择，并对结果进行分析。用直觉模糊理论解释复杂网络博弈的不确定性可以极大地拓宽网络攻防博弈研究在实际中的应用。Compared with the existing method, the advantage of the method of the present invention is that: complex network game is a research hotspot in recent years, but the existing research cannot reflect the ambiguity of the decision maker's understanding of the game problem, and the method of the present invention proposes a method based on intuition The complex network game model of the paper gives the generation method and solution idea of the intuitionistic fuzzy set profit matrix, and finally obtains the reasonable strategy choice that the defender should make in the fuzzy situation, and analyzes the results. Using intuitionistic fuzzy theory to explain the uncertainty of complex network games can greatly broaden the application of network attack and defense game research in practice.
附图说明Description of drawings
图1示出了本发明实施例的流程示意图；Fig. 1 shows a schematic flow chart of an embodiment of the present invention;
图2示出了本发明实施例中基础设施网络的示意图；Fig. 2 shows the schematic diagram of infrastructure network in the embodiment of the present invention;
图3示出了本发明实施例的攻方的概率分配示意图；Fig. 3 shows a schematic diagram of the probability distribution of the attacking party in the embodiment of the present invention;
图4示出了本发明实施例的防方的概率分配示意图。Fig. 4 shows a schematic diagram of the probability distribution of the defense party according to the embodiment of the present invention.
具体实施方式Detailed ways
为了使本发明的目的、技术方案和优点更加清楚，下面将结合附图对本发明作进一步地详细描述，显然，所描述的实施例仅仅是本发明一部份实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其它实施例，都属于本发明保护的范围。In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments . Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.
应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.
本实施例中只考虑一个攻击方和一个防御方，且攻防双方均完全了解现有网络的拓扑结构信息。一个关键基础设施，如铁路网络，可抽象为简单无向图G(V,E)，其中V＝{v1,v2,...,vN}代表网络中所有节点的集合，即铁路网络中的车站，其中N＝|V|表示网络中的节点数量。是网络中所有边的集合，即铁路网络中的铁路线。In this embodiment, only one attacker and one defender are considered, and both the attacker and the defender fully understand the topology information of the existing network. A key infrastructure, such as a railway network, can be abstracted as a simple undirected graph G(V,E), where V={v1 ,v2 ,...,vN } represents the set of all nodes in the network, that is, the railway Stations in the network, where N = |V| represents the number of nodes in the network. is the set of all edges in the network, i.e. the railway lines in the railway network.
只考虑一个攻击方和一个防御方，且攻防双方均完全了解现有网络的拓扑结构信息。所有攻击和防守的对象均为网络中的节点。节点被成功攻下的判断准则为：节点被攻击方攻击而没有被防御方保护。二人零和博弈，节点越重要攻击或防御所需要的成本越高。Only one attacker and one defender are considered, and both attackers and defenders fully understand the topology information of the existing network. The objects of all attacks and defenses are nodes in the network. The criterion for judging that a node is successfully captured is: the node is attacked by the attacker but not protected by the defender. Two-person zero-sum game, the more important the node, the higher the cost of attack or defense.
如图1所示，基于双曲隶属函数的复杂网络博弈防御方策略优化方法，所述方法包括：As shown in Figure 1, the complex network game defender strategy optimization method based on hyperbolic membership function, the method includes:
步骤1，获取基础设施网络的拓扑结构，确定攻击方和防御方的策略集合，构建复杂网络博弈的基本模型；Step 1. Obtain the topology of the infrastructure network, determine the strategy set of the attacker and the defender, and build a basic model of the complex network game;
步骤2，确定表示网络连通性的指标，计算复杂网络博弈的基本模型中攻击方和防御方在各种策略剖面下的收益，由此得到收益矩阵；Step 2, determine the indicators representing network connectivity, and calculate the income of the attacker and the defender under various strategy profiles in the basic model of complex network games, and thus obtain the income matrix;
步骤3，利用直觉模糊数确定方法，构建双曲线隶属度函数和非隶属度函数；Step 3, using the intuitionistic fuzzy number determination method to construct a hyperbolic membership function and a non-membership function;
步骤4，利用双曲线隶属度函数和非隶属度函数，将所述的收益矩阵转换为直觉模糊集收益矩阵；Step 4, using the hyperbolic membership function and non-membership function, converting the income matrix into an intuitionistic fuzzy set income matrix;
步骤5，将复杂网络博弈的基本模型的求解转换为线性规划问题求解，得到混合纳什均衡解，即获得防御方策略优化结果。In step 5, the solution of the basic model of the complex network game is converted into a solution of a linear programming problem, and a mixed Nash equilibrium solution is obtained, that is, the strategy optimization result of the defender is obtained.
所述的攻击方的策略集合为SA，对于一个攻击策略向量sA＝[x1,x2,...,xN]∈SA，xi表示第i个基础设施是否被攻击，记为攻击节点的集合，如果第i个节点vi被攻击，即vi∈VA，xi＝1，否则xi＝0；防御方的策略向量sD＝[y1,y2,...,yN]∈SD，yi表示第i个基础设施是否被防守，记/>为防守节点的集合，如果第i个节点vi被防守，即vi∈VD，yi＝1，否则yi＝0，对于被攻击的节点vi，若同时存在xi＝1且yi＝0，即虽被攻击但是未被保护，则节点vi将被移除，由此，攻击向量的总成本为：The strategy set of the attacker is SA , for an attack strategy vector sA =[x1 ,x2 ,...,xN ]∈SA ,x irepresents whether the i-th infrastructure is attacked, remember is the set of attack nodes, if the i-th node vi is attacked, that is, vi ∈ VA , xi =1, otherwise xi =0; the strategy vector sD of the defender =[y1 ,y2 ,. ..,yN ]∈SD , yi indicates whether the i-th infrastructure is defended, remember /> is the set of defending nodes, if the i-th node vi is defended, that is, vi ∈ VD , yi = 1, otherwise yi = 0, for the attacked node vi , if xi = 1 and yi = 0, that is, although it is attacked but not protected, the node vi will be removed. Therefore, the total cost of the attack vector is:
其中，表示对于攻击方来说第i个节点的成本，ri表示第i个节点的度，qA表示攻击方对于节点度的成本敏感系数；in, Represents the cost of the i-th node for the attacker, ri represents the degree of the i-th node, and qA represents the cost sensitivity coefficient of the attacker to the node degree;
而攻击方的总成本是有限的，因此应满足约束：The total cost of the attacker is limited, so the constraint should be satisfied:
其中，CA表示攻击方的总成本约束，θA表示攻击方的成本约束系数，取值范围为[0,1]；Among them, CA represents the total cost constraint of the attacker, θA represents the cost constraint coefficient of the attacker, and the value range is [0,1];
如果仅按照上述的约束条件选取合适的策略，那么尽可能少地选择攻击节点即可在较大程度上满足约束，但是在实际情况中，攻击方对现有的资源应尽可能利用从而达到对网络的预期破坏效果。为解决这一矛盾，本实施例提出“最低资源利用率”的概念，ωA与ωD分别表示攻击方与防御方对可用成本资源的最低利用率，对于攻击方，整体应满足约束：If the appropriate strategy is only selected according to the above constraints, the constraints can be satisfied to a large extent by selecting as few attack nodes as possible, but in actual situations, the attacker should use the existing resources as much as possible to achieve The expected disruptive effect of the network. In order to solve this contradiction, this embodiment proposes the concept of "minimum resource utilization rate". ωA and ωD respectively represent the minimum utilization rate of available cost resources by the attacker and the defender. For the attacker, the overall constraints should be satisfied:
防御方应满足约束：The defender should satisfy the constraints:
其中，CD表示防御方的总成本约束，θD表示防御方的成本约束系数，取值范围为[0,1]，ri表示第i个节点的度，qD表示防御方对于节点度的成本敏感系数。Among them, CD represents the total cost constraint of the defender, θD represents the cost constraint coefficient of the defender, and the value range is [0,1], ri represents the degree of the i-th node, and qD represents the degree of the defender for the node cost sensitivity coefficient.
收益函数刻画了博弈模型中各参与者在各种策略剖面下的收益。令UA:|SA|×|SD|为攻击方的收益函数，则UA(X,Y)表示攻击方在选择策略X和防御方选择策略Y时攻击方的收益，同理，防御方的收益表示为UD(X,Y)。UD(X,Y)表示攻击方在选择策略X和防御方选择策略Y时防御方的收益：The income function describes the income of each participant in the game model under various strategy profiles. Let UA :|SA |×|SD | be the income function of the attacker, then UA (X,Y) represents the income of the attacker when the attacker chooses strategy X and the defender chooses strategy Y. Similarly, The defender's payoff is denoted UD (X,Y). UD (X, Y) represents the defender's income when the attacker chooses strategy X and the defender chooses strategy Y:
其中，Γ(G)表示初始的基础设施网络G的最大连通片规模，记所有被移除的节点组成的集合为节点被移除后组成的网络为/>是节点被移除后组成的网络中所有边的集合，/>表示进行一轮博弈后网络的最大连通片规模，且满足/>Among them, Γ(G) represents the maximum connected slice size of the initial infrastructure network G, and the set composed of all removed nodes is The network formed after the node is removed is /> is the set of all edges in the network formed after the node is removed, /> Indicates the maximum connected slice size of the network after a round of game, and satisfies />
由于攻防双方的利益关系是根本对立的，因此上述收益互为相反数。Since the interests of the offensive and defensive parties are fundamentally opposed, the above benefits are opposite to each other.
在生成直觉模糊集的过程中，应考虑如何更加准确地反映决策者的认知偏好。隶属/非隶属函数是直觉模糊集理论的奠基石，由直觉模糊集的定义(加参考文献)，确定论域X上的直觉模糊集最重要的是确定其中的X→[0,1](隶属度)和X→[0,1](非隶属度)这两个函数。隶属度函数的选择有很多，如线性、指数、双曲等形式，具体选择哪种模型取决于应用场景以及决策者的自身偏好。而在实际应用中，隶属度函数对目标的边际满意度或不满意度不是恒定的，用简单的线性隶属函数难以表示。双曲隶属函数作为一种非线性的隶属度函数，其形状包括部分凹和其余部分凸。凸形描述了决策者边际满意度不断增高，而凹面部分则反映了边际满意度不断降低，因此，以攻击方为例，在双曲隶属度的情况下，决策者在对攻击收益的满意度较差时，他往往对攻击收益有较高的边际满意度；而决策者在对攻击收益的满意度较高时，往往攻击收益的边际满意度较小。结合实际的网络攻防场景，站在攻击方的视角考虑问题，在攻击收益较小时，难以达到决策者对网络的预期破坏效果，网络在很大程度上还是可以发挥相应的功能，因此提高收益的愿望就更加强烈；而在攻击收益较大时，对网络的预期破坏效果已经达到，此时网络往往已经无法发挥相应的功能，提高收益虽然有利，但是攻击方的愿望已经不强烈了。关于非隶属度的解释同理。In the process of generating intuitionistic fuzzy sets, how to reflect the cognitive preferences of decision makers more accurately should be considered. The membership/non-membership function is the cornerstone of the intuitionistic fuzzy set theory, and the intuitionistic fuzzy set on the domain X is determined by the definition of the intuitionistic fuzzy set (plus references). The most important thing is to determine the two functions of X→[0,1] (membership degree) and X→[0,1] (non-membership degree). There are many choices of membership function, such as linear, exponential, hyperbolic, etc. The specific model to choose depends on the application scenario and the decision maker's own preference. However, in practical applications, the marginal satisfaction or dissatisfaction of the target with the membership function is not constant, and it is difficult to express it with a simple linear membership function. As a nonlinear membership function, the hyperbolic membership function has a shape that is partially concave and the rest is convex. The convex shape describes the increasing marginal satisfaction of the decision maker, while the concave part reflects the decreasing marginal satisfaction. Therefore, taking the attacker as an example, in the case of hyperbolic membership, the decision maker’s satisfaction with the attack benefits When it is poor, he tends to have higher marginal satisfaction with attack benefits; while decision makers tend to have less marginal satisfaction with attack benefits when they are more satisfied with attack benefits. Combined with the actual network attack and defense scenarios, consider the problem from the perspective of the attacker. When the attack benefits are small, it is difficult to achieve the expected damage effect of the decision-maker on the network. The desire is stronger; and when the attack benefits are large, the expected damage effect on the network has been achieved. At this time, the network is often unable to perform the corresponding functions. Although it is beneficial to increase the profit, the attacker's desire is not strong anymore. The explanation for non-membership degree is the same.
所述的双曲线隶属度函数u(x)和所述的非隶属度函数v(x)的形式为：The forms of the hyperbolic membership degree function u(x) and the non-membership degree function v(x) are:
其中，α表示最高可接受水平，β表示最低可接受水平；Among them, α indicates the highest acceptable level, and β indicates the lowest acceptable level;
其中，α表示最高可接受水平(the desired or the most acceptable leveldenoted by m)，β表示最低可接受水平(the worst acceptable level of achievement)，对于隶属度与非隶属度函数，其判断具有较强的主观性，应综合实际情况，历史数据以及决策者主观偏好得出。Among them, α represents the desired or the most acceptable level denoted by m, and β represents the worst acceptable level of achievement. For the membership and non-membership functions, its judgment has a strong Subjectivity should be based on the actual situation, historical data and subjective preferences of decision makers.
假设攻击方选取第i个纯策略sAi∈SA(i＝1,2,...,m)，防御方选择第j个纯策略sDj∈SD(j＝1,2,...,n)，攻击方的原有收益值可表示为Γ(G)ij表示攻击方选择策略i，防御方选择策略j时的原有最大联通片规模,/>表示攻击方选择策略i，防御方选择策略j时进行一轮博弈后的最大联通片规模，通过双曲隶属函数将攻击方收益值转化为直觉模糊集<μij,νij>，而防御方的损失也为<μij,νij>，μij表示攻击方选择策略i，防御方选择策略j时对于攻击方的隶属度，νij表示攻击方选择策略i，防御方选择策略j时对于攻击方的非隶属度，由此，攻击方在不同纯策略局势下的直觉模糊集收益矩阵表示为Suppose the attacker chooses the i-th pure strategy sAi ∈ SA (i=1,2,...,m), and the defender chooses the j-th pure strategy sDj ∈ SD (j=1,2,...,m). .,n), the original revenue value of the attacker can be expressed as Γ(G)ij represents the original maximum size of China Unicom when the attacker chooses strategy i and the defender chooses strategy j,/> Indicates the maximum size of connected slices after a round of games when the attacker chooses strategy i and the defender chooses strategy j, and transforms the attacker’s revenue value into an intuitionistic fuzzy set <μij ,νij > through the hyperbolic membership function, while the defender The loss is also <μij ,νij >, μij represents the membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j, νij represents the attacker chooses strategy i, and when the defender chooses strategy j for The non-membership degree of the attacking party, thus, the intuitionistic fuzzy set benefit matrix of the attacking party in different pure strategy situations is expressed as
则混合策略下，攻击方的直觉模糊集均衡收益为：Then under the mixed strategy, the attacker’s intuitionistic fuzzy set equilibrium income is:
其中p＝(p1,p2,...,pm)T表示攻击方混合策略的概率向量，q＝(q1,q2,...,qn)T表示防御方混合策略的概率向量。Where p=(p1 ,p2 ,...,pm )T represents the probability vector of the attacker's mixed strategy, q=(q1 ,q2 ,...,qn )T represents the defense's mixed strategy probability vector.
对于上述直觉模糊集二人零和博弈的问题，其求解模型最终可转化为：For the above-mentioned intuitionistic fuzzy set two-person zero-sum game problem, its solution model can finally be transformed into:
和and
其中，i表示攻击方的第i个策略，共m个策略；j表示防御方的第j个策略，共n个策略,λ表示隶属/非隶属函数约束的相对权重，λ确定后，得到的纳什均衡解为：(p,q,<μ,ν>,<σ,ρ>)，<μ,ν>表示攻击方的收益值，<σ,ρ>表示防御方的收益值，形式均为直觉模糊集，μ表示攻击方收益的隶属度，ν表示攻击方收益的非隶属度，σ表示防御方收益的隶属度，ρ表示防御方收益的非隶属度。Among them, i represents the i-th strategy of the attacker, a total of m strategies; j represents the j-th strategy of the defender, a total of n strategies, λ represents the relative weight of the membership/non-membership function constraints, after λ is determined, the obtained The Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>), <μ,ν> represents the income value of the attacker, and <σ,ρ> represents the income value of the defender, in the form of For intuitionistic fuzzy sets, μ represents the membership degree of the attacker’s revenue, ν represents the non-membership degree of the attacker’s revenue, σ represents the membership degree of the defender’s revenue, and ρ represents the non-membership degree of the defender’s revenue.
现实生活中基础设施网络结构多种多样，实验以一个10节点的网络结构为例，如图2所示，并假设攻防双方资源有限，可选择的节点数量不超过3。There are various infrastructure network structures in real life. The experiment takes a 10-node network structure as an example, as shown in Figure 2, and assumes that the offensive and defensive parties have limited resources, and the number of selectable nodes does not exceed 3.
将直觉模糊理论应用于网络攻防博弈，由上一章的模型可得到攻防双方的策略集合SA，SD(|SA|＝120，|SD|＝120)，并根据收益函数得到初始收益矩阵。由于对隶属度与非隶属度函数的判断具有较强的主观性，因此结合本例中的网络拓扑结构，模拟决策者主观偏好得出隶属/非隶属函数如下式，其中m＝6,n＝1Applying the intuitionistic fuzzy theory to the network attack-defense game, the strategy sets SA and SD (|SA |=120, |SD |=120) of the attack and defense sides can be obtained from the model in the previous chapter, and the initial Benefit matrix. Since the judgment of the membership and non-membership functions is highly subjective, combined with the network topology in this example, the subjective preference of the decision maker is simulated to obtain the membership/non-membership function as follows, where m=6,n= 1
在得到直觉模糊集收益矩阵后，模型求解过程可得到纳什均衡混合策略解，由于每个策略对应一个选择概率，因此可将不同纯策略对应的概率映射到不同节点上，映射方式如下：After obtaining the intuitionistic fuzzy set income matrix, the model solution process can obtain the Nash equilibrium mixed strategy solution. Since each strategy corresponds to a selection probability, the probabilities corresponding to different pure strategies can be mapped to different nodes. The mapping method is as follows:
其中和/>是两个参与者单个节点的概率分布，/>和/>是所有攻击和防御策略的概率分布。in and /> is the probability distribution of a single node of two participants, /> and /> is the probability distribution of all attack and defense strategies.
根据上述内容，可作出攻方的概率分配图如图3所示，防方的概率分配图如图4所示。According to the above content, the probability distribution diagram of the attacking side can be drawn as shown in Figure 3, and the probability distribution diagram of the defending side is shown in Figure 4.
根据图3，对不同节点的攻击概率分配进行分析：According to Figure 3, the attack probability distribution of different nodes is analyzed:
可以看出随着λ的变化不同节点的概率分配并没有明显改变，分析出现这样情况的原因，是因为λ反映的是隶属度函数约束和非隶属度函数约束的相对权重，而此结果对应的隶属度与非隶属度函数是对称的，因此权重的变化并不会引起对某些策略选择概率的改变，因此这些策略所包含的节点被选择的概率便不会改变。It can be seen that the probability distribution of different nodes does not change significantly with the change of λ. The reason for this situation is that λ reflects the relative weight of the membership function constraint and the non-membership function constraint, and this result corresponds to The functions of membership degree and non-membership degree are symmetrical, so the change of weight will not cause the change of the selection probability of certain strategies, so the probability of the nodes included in these strategies will not be changed.
可以看出添加隶属度函数与不添加隶属度函数时不同节点的攻击概率有所变化，但是整体相差不大，分析可能是由于在隶属度函数的作用下收益值发生改变，因此不同策略选择的概率也发生轻微改变，从而影响不同节点的概率分布。It can be seen that the attack probability of different nodes changes when adding the membership function and not adding the membership function, but the overall difference is not large. The analysis may be due to the change of the income value under the effect of the membership function, so the choice of different strategies The probabilities are also changed slightly, affecting the probability distributions at different nodes.
根据图4，对不同节点的防守概率分配进行分析：According to Figure 4, the defense probability distribution of different nodes is analyzed:
可以看出随着λ的变化不同节点的概率分配并没有明显改变，原因与攻击节点相同。It can be seen that the probability distribution of different nodes does not change significantly with the change of λ, and the reason is the same as that of the attacking node.
可以看出添加隶属度函数与不添加隶属度函数时不同节点的防守概率有所变化，但是整体相差不大，分析可能是由于在隶属度函数的作用下收益值发生改变，因此不同策略选择的概率也发生轻微改变，从而影响不同节点的概率分布。It can be seen that the defense probability of different nodes changes when the membership function is added or not, but the overall difference is not large. The analysis may be due to the change of the income value under the effect of the membership function, so the choice of different strategies The probabilities are also changed slightly, affecting the probability distributions at different nodes.
表1λ等于0.5时各节点的攻防概率选择与节点指标的对比Table 1 Comparison of attack and defense probability selection of each node and node index when λ is equal to 0.5
表1是通过计算网络的各种现有指标综合展示不同节点的重要程度，可见对于节点1这样相对重要的节点，攻击方选择的概率反而降低；但对于防御方来讲，在网络拓扑结构中相对重要的节点其选择防守的概率也较高，这是由于攻击方“预料”到重要性大的节点防御方防守力量也相对较强，而对于防御方来说，一旦小概率事件发生，其损失将会使其难以承受。这一结论与不加直觉模糊得到的结论相似，符合博弈理论的正常逻辑。Table 1 comprehensively shows the importance of different nodes by calculating various existing indicators of the network. It can be seen that for a relatively important node such as node 1, the probability of the attacker’s choice is reduced; but for the defender, in the network topology structure Relatively important nodes have a higher probability of choosing defense. This is because the attacker "anticipates" that the defensive power of the more important nodes is also relatively strong. For the defender, once a small probability event occurs, other Losses would make it unbearable. This conclusion is similar to the conclusion obtained without intuitionistic fuzziness, and conforms to the normal logic of game theory.
本领域技术人员应明白，本申请的实施例可提供为方法、系统或计算机程序产品。因此，本申请可采用完全硬件实施例、完全软件实施例或结合软件和硬件方面的实施例的形式。而且，本申请可采用在一个或多个其中包含有计算机可用程序代码的计算机可用存储介质(包括但不限于磁盘存储器、CD-ROM、光学存储器等)上实施的计算机程序产品的形式。Those skilled in the art should understand that the embodiments of the present application may be provided as methods, systems or computer program products. Accordingly, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202310382005.4ACN116455626A (en) | 2023-04-11 | 2023-04-11 | Strategy Optimization Method of Defender in Complex Network Game Based on Hyperbolic Membership Function |
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202310382005.4ACN116455626A (en) | 2023-04-11 | 2023-04-11 | Strategy Optimization Method of Defender in Complex Network Game Based on Hyperbolic Membership Function |
| Publication Number | Publication Date |
|---|---|
| CN116455626Atrue CN116455626A (en) | 2023-07-18 |
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202310382005.4APendingCN116455626A (en) | 2023-04-11 | 2023-04-11 | Strategy Optimization Method of Defender in Complex Network Game Based on Hyperbolic Membership Function |
| Country | Link |
|---|---|
| CN (1) | CN116455626A (en) |
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN118797943A (en)* | 2024-07-12 | 2024-10-18 | 中国人民解放军国防科技大学 | Strategy generation method for interval-valued intuitionistic fuzzy Stackelberg game based on weighted relationship |
| CN118917057A (en)* | 2024-07-12 | 2024-11-08 | 中国人民解放军国防科技大学 | Fuzzy Stark game strategy generation method based on word order |
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN118797943A (en)* | 2024-07-12 | 2024-10-18 | 中国人民解放军国防科技大学 | Strategy generation method for interval-valued intuitionistic fuzzy Stackelberg game based on weighted relationship |
| CN118917057A (en)* | 2024-07-12 | 2024-11-08 | 中国人民解放军国防科技大学 | Fuzzy Stark game strategy generation method based on word order |
| Publication | Publication Date | Title |
|---|---|---|
| CN110138764B (en) | Attack path analysis method based on hierarchical attack graph | |
| CN107395430B (en) | Cloud platform dynamic risk access control method | |
| CN111931242A (en) | Data sharing method, computer equipment applying same and readable storage medium | |
| CN108833401A (en) | Method and device for network active defense strategy selection based on Bayesian evolutionary game | |
| CN111770073A (en) | A fog network offloading decision-making and resource allocation method based on blockchain technology | |
| CN116455626A (en) | Strategy Optimization Method of Defender in Complex Network Game Based on Hyperbolic Membership Function | |
| WO2022151654A1 (en) | Random greedy algorithm-based horizontal federated gradient boosted tree optimization method | |
| CN113392919B (en) | Deep belief network DBN detection method of attention mechanism | |
| CN108040062B (en) | Network security situation assessment method based on evidence reasoning rule | |
| CN108696534B (en) | Real-time network security threat early warning analysis method and device | |
| Kuang et al. | A privacy protection model of data publication based on game theory | |
| CN118246545A (en) | A method for generating network game strategies based on node average path constraints | |
| CN116822169A (en) | Strong Takeberg game strategy generation method based on intuitionistic fuzzy set | |
| CN118916878A (en) | Combined audit security defense method for federal learning multi-element poisoning attack | |
| CN116647364A (en) | Complex network game attacker strategy optimization method based on intuitionistic ambiguity | |
| CN119544523A (en) | A method for solving network games based on greedy algorithm | |
| CN118036748A (en) | A network game method based on the strategic difficulty level of nodes comprehensive evaluation index | |
| CN116956551A (en) | Infrastructure network game strategy optimization method based on linear constraints | |
| CN110430526A (en) | Method for secret protection based on credit assessment | |
| CN112070315B (en) | A method for terrorist attack network analysis and event prediction based on centrality measurement | |
| Saini et al. | Identifying collusion attacks in P2P trust and reputation systems | |
| Dong et al. | Attack and Defense Game with Intuitionistic Fuzzy Payoffs in Infrastructure Networks | |
| Zhang et al. | GT‐Bidding: Group Trust Model of P2P Network Based on Bidding | |
| CN111767567A (en) | Social information security management method | |
| CN119743281B (en) | Node-based infrastructure network game strategy generation method |
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination |