Intuitionistic fuzzy set-based strong stackelberg game strategy generation method
By using a strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets, the problem of decision-maker fuzziness and uncertainty in complex network games is solved, enabling strategy optimization and outcome analysis in complex network games and broadening the scope of application.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NAT UNIV OF DEFENSE TECH
- Filing Date
- 2023-06-14
- Publication Date
- 2026-05-29
AI Technical Summary
Existing research cannot effectively integrate the ambiguity and uncertainty of decision-makers in complex network games, cannot reflect subjective judgments in actual game problems, and existing methods cannot express the ambiguity and uncertainty of game problems.
A strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets is adopted. The intuitionistic fuzzy two-person zero-sum payoff matrix is generated by using hyperbolic membership/non-membership functions, and the game problem is transformed into a nonlinear programming problem to obtain the mixed strategy Nash equilibrium solution.
By combining fuzzy mathematics theory, we provide reasonable strategy selection under fuzzy conditions, which can better cope with the uncertainty in complex network games and broaden the practical application of complex network games.
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Figure CN116822169B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of complex network game technology in systems engineering, and in particular to a strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets. Background Technology
[0002] In the current field of game theory research, there is a special type of network game where the network is not a real-world computer system, but rather a network topology that abstracts critical infrastructure, such as train stations and airports, into nodes, and the connections between different sites into edges, creating a complex infrastructure network. In the security field, critical nodes in infrastructure are vulnerable to attacks, which can impact public order and normal social life. Security departments need to protect these nodes. Complex network game theory can be used to study the attack and protection of critical nodes in infrastructure networks, helping to formulate optimal protection strategies and explore the importance of these nodes.
[0003] Current research on this type of problem exists, but existing studies only provide objective evaluation methods based on network topology. For example, in a fully informed static or dynamic game framework, these methods use the network connectivity performance index—maximum connected component size—to calculate the payoff matrices for attackers and defenders, and then calculate the corresponding Nash equilibrium strategies. However, in real-world game problems, the understanding of the problem by both sides is uncertain, information is insufficient, and the decision-making environment is unpredictable. Existing methods cannot adequately incorporate the subjective judgments of decision-makers and cannot express the ambiguity and uncertainty of real-world game problems.
[0004] In today's networked society, Stackelberg games on complex networks are an important research direction in game theory. Research mainly involves the construction of game models, strategy analysis and solution, and the expansion of application areas. Current research focuses on designing efficient algorithms and tools for analysis and solution, promoting their application in different fields, and further expanding theoretical research to better address the challenges of complex systems in reality. In recent years, this field has made many important advances, providing new ideas and methods for game theory research and application. In the field of fuzzy mathematics, Professor Zadeh proposed fuzzy set theory, providing a reasonable approach to solving such problems. Addressing the limitations of fuzzy set theory and the practical need to express hesitation, Atanassov proposed intuitionistic fuzzy set theory, which uses two scales (membership and non-membership) to represent support, opposition, and hesitation in fuzzy phenomena. This theory provides inspiration for solving more complex game problems. Currently, there is relatively little research on introducing intuitionistic fuzzy set theory into Stackelberg games, making this research of significant importance. Summary of the Invention
[0005] This invention aims to at least solve one of the technical problems existing in the prior art. To this end, this invention discloses a method for generating strategies in strong Stackelberg games based on intuitionistic fuzzy sets. The method, based on strong Stackelberg games, utilizes hyperbolic membership / non-membership functions to generate an intuitionistic fuzzy two-person zero-sum payoff matrix, transforming the solution of complex network strong Stackelberg games based on intuitionistic fuzzy sets into a nonlinear programming problem, thereby obtaining a hybrid strategy Nash equilibrium solution.
[0006] The objective of this invention is achieved through the following technical solution: a strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets, the method comprising:
[0007] Step 1: Obtain the topology of the infrastructure network, designate the defender as the first party, determine the strategy sets of the attacker and defender, and construct a strong Stackelberg game model for complex networks.
[0008] Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the complex network strong Stackelberg game model, and thus obtain the payoff matrix.
[0009] Step 3: Construct hyperbolic membership functions and non-membership functions using the intuitionistic fuzzy set determination method;
[0010] Step 4: Using hyperbolic membership functions and non-membership functions, the revenue matrix is converted into an intuitionistic fuzzy set revenue matrix;
[0011] Step 5: The solution of the complex network strong Stackelberg game model is transformed into a nonlinear programming problem to obtain the mixed strategy Nash equilibrium solution, that is, to obtain the strategy optimization results of the attacker and defender.
[0012] The infrastructure network is represented as a simple undirected graph G(V,E), where Represents the set of all nodes in the network, where, Indicates the number of nodes in the network. It is the set of all edges in the network.
[0013] Furthermore, the attacker's attack strategy set is as follows: For an attack strategy vector , Indicate whether the i-th infrastructure has been attacked, denoted by . Let i be the set of attacking nodes, and let i be the i-th node. Being attacked, i.e. , ,otherwise The set of defense strategies of the defending side is as follows: A defense strategy vector of the defending side , Indicates whether the i-th infrastructure is defended, denoted by . Let i be the set of defense nodes, and let i be the i-th node. Being defended, i.e. , ,otherwise For the i-th node ,remember and These represent the attack cost for the attacker and the defense cost for the defender, respectively, for each node. The attack cost and defense cost are expressed as follows: , ;
[0014] in, Represents a node The degree, This represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's degree on the attacker's cost. This represents the cost sensitivity coefficient of the defender, that is, the impact of the node's degree on the defender's cost;
[0015] Let the attacker's attack cost constraint be... The defender's defense cost constraint is For the attacked node If both exist and That is, if a node is attacked but not protected, then... This will be removed; the attacker's attack strategy set includes three typical strategies, namely, the maximum attack strategy. Minimum attack strategy Random attack strategy The maximum-degree attack strategy attacks nodes in descending order of degree, the minimum-degree attack strategy attacks nodes in ascending order of degree, and the random attack strategy randomly selects nodes that meet the cost constraints for attack. The defender's set of defense strategies includes three typical strategies: maximum-degree defense strategy... Minimal defense strategy Random defense strategy The maximum-degree defense strategy defends nodes in descending order of degree, the minimum-degree defense strategy defends nodes in ascending order of degree, and the random defense strategy randomly selects nodes that meet the cost constraints. For both attackers and defenders, these three typical strategies should select as many nodes as possible without exceeding their cost constraints. and .
[0016] Specifically, the profit matrix is divided into the attacker's profit matrix and the defender's profit matrix. Let be the attacker's payoff function, then This represents the attacker's gain when choosing attack strategy X and the defender choosing defense strategy Y. This represents the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y:
[0017] ;
[0018] ;
[0019] in, Let G represent the maximum connected component size of the initial infrastructure network G, and let the set of all removed nodes be denoted as . The network formed after the nodes are removed is , It is the set of all edges in the network formed after nodes are removed. Let represent the maximum connected component size of the network after one round of gameplay, and satisfy . .
[0020] Specifically, the hyperbolic membership function and the aforementioned non-membership function The form is:
[0021] ;
[0022] ;
[0023] in, Indicates the highest acceptable level. Indicates the minimum acceptable level;
[0024] The attacker selects a set of attack strategies. One of the attack strategies The defender selects a set of defense strategies. One of the defensive strategies The original benefit value of the defending side is expressed as , This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in the text refers to the maximum connected slice size of the initial network. This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in this context is to determine the maximum connected component size of the network after game theory, and to transform the defender's payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The attacker's losses were... Therefore, the intuitionistic fuzzy set payoff matrix of the defender under different pure strategy situations is expressed as:
[0025] ;
[0026] Under the hybrid strategy, the defender's intuitionistic fuzzy set equilibrium payoff is:
[0027] ;
[0028] in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Membership degree of the time-return value This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Non-membership degree of time-based payoffs, and other payoff symbol values. Indicates membership degree, where 'v' in the symbol represents non-membership degree, and the subscript... Indicates the attacker's maximum-degree attack strategy, minimum-degree attack strategy, and random attack strategy, with subscripts. This represents the defender's maximum-degree defense strategy, minimum-degree defense strategy, and random defense strategy. This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy. This represents the membership degree of the payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected The non-membership degree of the payoff matrix is represented by the i-th defensive strategy. If selected, the attack strategy is chosen as the j-th one. Selected.
[0029] Specifically, the nonlinear programming problem described in step 5 is:
[0030] ;
[0031] This can then be transformed into:
[0032] ;
[0033] Where i represents the i-th strategy of the defender, and j represents the j-th strategy of the attacker. Represents the relative weights of membership / non-membership function constraints. Once determined, the Nash equilibrium solution is:
[0034] , This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy.
[0035] This represents the defensive side's benefit value. The values representing the attacker's gains are all in the form of intuitionistic fuzzy sets;
[0036] This represents the membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected
[0037] This represents the non-membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected;
[0038] This represents the membership degree of the defender's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected;
[0039] This represents the non-membership degree of the defender's payoff matrix, where the defense strategy is selected at the ith level. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected, in the subscript The time represents the attacker's optimal response strategy to the defender, which is the pure strategy that maximizes the attacker's own benefit, derived from the above formula.
[0040] Compared with existing methods, the advantages of this invention are as follows: In recent years, Stackelberg dynamic games based on complex networks have attracted widespread attention from scholars both domestically and internationally. However, existing research cannot reflect the fuzziness in decision-makers' understanding of the game problem. This invention, combining fuzzy mathematics theory, proposes a method for generating optimal strategies for strong Stackelberg games based on intuitionistic fuzzy sets. This method yields reasonable strategy choices for both players under fuzzy conditions and analyzes the results. Explaining the uncertainty of strong Stackelberg games based on complex networks using intuitionistic fuzzy theory can greatly broaden the practical application of complex network game research. Attached Figure Description
[0041] Figure 1 A flowchart illustrating an embodiment of the present invention is shown;
[0042] Figure 2 A schematic diagram of the infrastructure network in an embodiment of the present invention is shown. Detailed Implementation
[0043] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this invention, and not all embodiments. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.
[0044] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
[0045] This embodiment considers only one attacker and one defender, both of whom have complete knowledge of the existing network topology. A critical infrastructure, such as a railway network, can be abstracted as a simple undirected graph G(V,E), where... This represents the set of all nodes in the network, i.e., the stations in a railway network. Indicates the number of nodes in the network. It is the set of all edges in a network, i.e., the railway lines in a railway network.
[0046] Consider only one attacker and one defender, both with complete knowledge of the existing network topology. All attacks and defenses target nodes within the network. A node is considered successfully compromised when it is attacked by the attacker and not protected by the defender. This is a two-player zero-sum game; the more important the node, the higher the cost of attacking or defending.
[0047] like Figure 1 As shown, a strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets is described, the method comprising:
[0048] Step 1: Obtain the topology of the infrastructure network, designate the defender as the first party, determine the strategy sets of the attacker and defender, and construct a strong Stackelberg game model for complex networks.
[0049] Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the complex network strong Stackelberg game model, and thus obtain the payoff matrix.
[0050] Step 3: Construct hyperbolic membership functions and non-membership functions using the intuitionistic fuzzy set determination method;
[0051] Step 4: Using hyperbolic membership functions and non-membership functions, the revenue matrix is converted into an intuitionistic fuzzy set revenue matrix;
[0052] Step 5: The solution of the complex network strong Stackelberg game model is transformed into a nonlinear programming problem to obtain the mixed strategy Nash equilibrium solution, that is, to obtain the strategy optimization results of the attacker and defender.
[0053] The attacker's attack strategy set is as follows For an attack strategy vector , Indicate whether the i-th infrastructure has been attacked, denoted by . Let i be the set of attacking nodes, and let i be the i-th node. Being attacked, i.e. , ,otherwise The set of defense strategies of the defending side is as follows: A defense strategy vector of the defending side , Indicate whether the i-th infrastructure is defended, denoted as . Let i be the set of defense nodes, and let i be the i-th node. Being defended, i.e. , ,otherwise For the i-th node ,remember and These represent the attack cost for the attacker and the defense cost for the defender, respectively, for each node. The attack cost and defense cost are expressed as follows: , ;
[0054] in, Represents a node The degree, This represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's degree on the attacker's cost. This represents the cost sensitivity coefficient of the defender, that is, the impact of the node's degree on the defender's cost;
[0055] Let the attacker's attack cost constraint be... The defender's defense cost constraint is For the attacked node If both exist and That is, if a node is attacked but not protected, then... This will be removed; the attacker's attack strategy set includes three typical strategies, namely, the maximum attack strategy. Minimum attack strategy Random attack strategy The maximum-degree attack strategy attacks nodes in descending order of degree, the minimum-degree attack strategy attacks nodes in ascending order of degree, and the random attack strategy randomly selects nodes that meet the cost constraints for attack. The defender's set of defense strategies includes three typical strategies: maximum-degree defense strategy... Minimal defense strategy Random defense strategy The maximum-degree defense strategy defends nodes in descending order of degree, the minimum-degree defense strategy defends nodes in ascending order of degree, and the random defense strategy randomly selects nodes that meet the cost constraints. For both attackers and defenders, these three typical strategies should select as many nodes as possible without exceeding their cost constraints. and Taking the attacker as an example, the maximum degree strategy means attacking nodes in descending order of degree (this allows attacking more important nodes), the minimum degree strategy means attacking nodes in ascending order of degree (this allows attacking more nodes), and the random strategy means randomly selecting nodes that meet the cost constraints for attack (this makes the attack more unpredictable). For both the attacker and defender, all three typical strategies should select as many nodes as possible without exceeding the cost constraints of either side. and .
[0056] The payoff matrix is divided into the payoff matrix for the attacker and the payoff matrix for the defender. Let be the attacker's payoff function, then This represents the attacker's gain when choosing attack strategy X and the defender choosing defense strategy Y. This represents the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y:
[0057] ;
[0058] ;
[0059] in, Let G represent the maximum connected component size of the initial infrastructure network G, and let the set of all removed nodes be denoted as . The network formed after the nodes are removed is , It is the set of all edges in the network formed after nodes are removed. Let represent the maximum connected component size of the network after one round of gameplay, and satisfy . Since the interests of the two parties are fundamentally opposed, the aforementioned gains are opposites.
[0060] In generating intuitionistic fuzzy sets, consideration should be given to how to more accurately reflect the cognitive preferences of decision-makers. Membership / non-membership functions are the cornerstone of intuitionistic fuzzy set theory. From the definition of intuitionistic fuzzy sets, determining the intuitionistic fuzzy set on the universe of discourse X, the most important aspect is determining the two functions X→[0,1] (membership degree) and X→[0,1] (non-membership degree). Hyperbolic membership function. and the aforementioned non-membership function The form is:
[0061] ;
[0062] ;
[0063] in, Indicates the highest acceptable level. Indicates the minimum acceptable level;
[0064] The attacker selects a set of attack strategies. One of the attack strategies The defender selects a set of defense strategies. One of the defensive strategies The original gains of the defending side are expressed as , This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in the text refers to the maximum connected slice size of the initial network. This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in this context is to determine the maximum connected component size of the network after game theory, and to transform the defender's payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The attacker's losses were... Therefore, the intuitionistic fuzzy set payoff matrix of the defender under different pure strategy situations is expressed as:
[0065] ;
[0066] Under the hybrid strategy, the defender's intuitionistic fuzzy set equilibrium payoff is:
[0067] ;
[0068] in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Membership degree of the time-return value This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Non-membership degree of time-based payoffs, and other payoff symbol values. Indicates membership degree, where 'v' in the symbol represents non-membership degree, and the subscript... Indicates the attacker's maximum-degree attack strategy, minimum-degree attack strategy, and random attack strategy, with subscripts. This represents the defender's maximum-degree defense strategy, minimum-degree defense strategy, and random defense strategy. This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy. This represents the membership degree of the payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected The non-membership degree of the payoff matrix is represented by the i-th defensive strategy. If selected, the attack strategy is chosen as the j-th one. Selected.
[0069] The solution model for the above intuitionistic fuzzy set two-person zero-sum strong Stackelberg game can ultimately be transformed into a nonlinear programming problem:
[0070] ;
[0071] This can then be transformed into:
[0072] ;
[0073] Where i represents the i-th strategy of the defender, and j represents the j-th strategy of the attacker. Represents the relative weights of membership / non-membership function constraints. Once determined, the Nash equilibrium solution is:
[0074] , This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy.
[0075] This represents the defensive side's benefit value. The values representing the attacker's gains are all in the form of intuitionistic fuzzy sets;
[0076] This represents the membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected
[0077] This represents the non-membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected;
[0078] This represents the membership degree of the defender's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected;
[0079] This represents the non-membership degree of the defender's payoff matrix, where the defense strategy is selected at the ith level. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected, in the subscript The time represents the attacker's optimal response strategy to the defender, which is the pure strategy that maximizes the attacker's own benefit, derived from the above formula.
[0080] In real life, infrastructure network structures vary widely. This experiment uses a scale-free network structure with 300 nodes as an example. Figure 2 As shown, and assuming that both sides have limited resources.
[0081] Intuitive fuzzy theory is applied to complex network game theory, and an initial payoff matrix is obtained based on the payoff function. Since the judgment of membership and non-membership functions is highly subjective, the membership / non-membership function is derived by simulating the decision-maker's subjective preferences, based on the network topology in this example, as shown in the following equation: =6, =1. ;
[0082] .
[0083] After obtaining the intuitionistic fuzzy set payoff matrix, the model solution process can yield the Nash equilibrium mixed strategy solution.
[0084] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can 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.) containing computer-usable program code.
Claims
1. A strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets, characterized in that, The method includes: Step 1: Obtain the topology of the infrastructure network, designate the defender as the first party, determine the strategy sets of the attacker and defender, and construct a strong Stackelberg game model for complex networks. Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the complex network strong Stackelberg game model, and thus obtain the payoff matrix. Step 3: Construct hyperbolic membership functions and non-membership functions using the intuitionistic fuzzy set determination method; Step 4: Using hyperbolic membership functions and non-membership functions, the revenue matrix is converted into an intuitionistic fuzzy set revenue matrix; Step 5: The solution of the complex network strong Stackelberg game model is transformed into a nonlinear programming problem to obtain the mixed strategy Nash equilibrium solution, that is, to obtain the strategy optimization results of the attacker and defender. The infrastructure network is represented as a simple undirected graph G(V,E), where Represents the set of all nodes in the network, where, Indicates the number of nodes in the network. It is the set of all edges in the network; The hyperbolic membership function and the aforementioned non-membership function The form is: in, Indicates the highest acceptable level. Indicates the minimum acceptable level; The attacker selects a set of attack strategies. One of the attack strategies The defender selects a set of defense strategies. One of the defensive strategies The original gains of the defending side are expressed as , This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in the text refers to the maximum connected slice size of the initial network. This indicates that the defender selects a set of defense strategies. The attacker selects a set of attack strategies. The strategy in this context is to determine the maximum connected component size of the network after game theory, and to transform the defender's payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The attacker's losses were... Therefore, the intuitionistic fuzzy set payoff matrix of the defender under different pure strategy situations is expressed as: Under the hybrid strategy, the defender's intuitionistic fuzzy set equilibrium payoff is: in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Membership degree of the time-return value This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Non-membership degree of time-based payoffs, and other payoff symbol values. Indicates membership degree, in the symbol Indicates non-membership degree, subscript Indicates the attacker's maximum-degree attack strategy, minimum-degree attack strategy, and random attack strategy, with subscripts. This represents the defender's maximum-degree defense strategy, minimum-degree defense strategy, and random defense strategy. This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy. This represents the membership degree of the payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected The non-membership degree of the payoff matrix is represented by the i-th defensive strategy. If selected, the attack strategy is chosen as the j-th one. Selected.
2. The method for generating strategies in strong Stackelberg games based on intuitionistic fuzzy sets according to claim 1, characterized in that, The attack strategy set of the attacker is as follows: For an attack strategy vector , Indicates the first Whether the infrastructure has been attacked, record Let be the set of attacking nodes, if the first... Nodes Being attacked, i.e. , ,otherwise ; The set of defense strategies of the defending side is as follows: A defense strategy vector of the defending side , Indicates the first Whether the infrastructure is defended, record Let be the set of defense nodes, if the first... Nodes Being defended, i.e. , ,otherwise For the first Nodes ,remember and These represent the attack cost for the attacker and the defense cost for the defender, respectively, for each node. The attack cost and defense cost are expressed as follows: ， in, Represents a node The degree, This represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's degree on the attacker's cost. This represents the cost sensitivity coefficient of the defender, that is, the impact of the node's degree on the defender's cost; Let the attacker's attack cost constraint be... The defender's defense cost constraint is For the attacked node If both exist and That is, if a node is attacked but not protected, then... This will be removed; the attacker's attack strategy set includes three typical strategies, namely, the maximum attack strategy. Minimum attack strategy Random attack strategy The maximum-degree attack strategy attacks nodes in descending order of degree, the minimum-degree attack strategy attacks nodes in ascending order of degree, and the random attack strategy randomly selects nodes that meet the cost constraints for attack. The defender's set of defense strategies includes three typical strategies: maximum-degree defense strategy... Minimal defense strategy Random defense strategy The maximum degree defense strategy defends nodes in descending order of degree, the minimum degree defense strategy defends nodes in ascending order of degree, and the random defense strategy randomly selects nodes that meet the cost constraints for defense.
3. The method for generating strategies in strong Stackelberg games based on intuitionistic fuzzy sets according to claim 1 or 2, characterized in that, The aforementioned profit matrix is divided into the attacker's profit matrix and the defender's profit matrix. Let be the attacker's payoff function, then This represents the attacker's gain when choosing attack strategy X and the defender choosing defense strategy Y. This represents the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y: in, Let G represent the maximum connected component size of the initial infrastructure network G, and let be the set of all removed nodes. The network formed after the nodes are removed is , It is the set of all edges in the network formed after nodes are removed. Let represent the maximum connected component size of the network after one round of gameplay, and satisfy . .
4. The strategy generation method for strong Stackelberg games based on intuitionistic fuzzy sets according to claim 1, characterized in that, The nonlinear programming problem described in step 5 is: This can then be transformed into: in, The first one represents the defending side One strategy, Indicates the attacker's... One strategy, Represents the relative weights of membership / non-membership function constraints. Once determined, the Nash equilibrium solution is: , This represents the probability vector of the defender's hybrid strategy. This represents the probability vector of the attacker's mixed strategy. This represents the defensive side's benefit value. The values representing the attacker's gains are all in the form of intuitionistic fuzzy sets; This represents the membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected This represents the non-membership degree of the attacker's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected; This represents the membership degree of the defender's payoff matrix, where the defense strategy is selected at the i-th position. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected This represents the non-membership degree of the defender's payoff matrix, where the defense strategy is selected at the ith level. If selected, the j-th attack strategy is chosen, and the attack strategy is in... Selected, in the subscript The time represents the attacker's optimal response strategy to the defender, which is the pure strategy that maximizes the attacker's own benefit, derived from the above formula.