Intuitionistic fuzzy-based strategy optimization method for complex network game attacker

Abstract

The application discloses a complex network game attack strategy optimization method based on intuitionistic fuzzy, and the method comprises the following steps: acquiring the topological structure of an infrastructure network, determining the strategy set of an attacker and a defender, and constructing a basic model of a complex network game; determining an index representing network connectivity, calculating the income of the attacker and the defender in the basic model of the complex network game under various strategy profiles, and thus obtaining an income matrix; using an intuitionistic fuzzy number determination method to construct a hyperbolic membership function and a non-membership function; converting the income matrix into an intuitionistic fuzzy set income matrix by using the hyperbolic membership function and the non-membership function; converting the solution of the basic model of the complex network game into a linear programming problem solution, obtaining a mixed Nash equilibrium solution, and thus obtaining the attack strategy optimization result.

Description

Technical Field

[0001] This invention relates to the field of complex network game technology in systems engineering, and in particular to an attack strategy optimization method based on intuitionistic fuzzy logic in complex network games. 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] 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 offers insights into solving more complex game theory problems. Currently, there is limited research on applying intuitionistic fuzzy set theory to complex network games in infrastructure, making this area of ​​research significant. Summary of the Invention

[0005] This invention aims to address at least one of the technical problems existing in the prior art. To this end, this invention discloses a method for optimizing the attacker's strategy in complex network games based on intuitionistic fuzzy logic. The method utilizes the two-person zero-sum matrix game of intuitionistic fuzzy sets, proposes a method for generating the intuitionistic fuzzy payoff matrix using hyperbolic membership / non-membership functions, and employs an efficient algorithm to solve for the Nash equilibrium under intuitionistic fuzzy conditions, ultimately obtaining the optimization method and results for the attacker's strategy.

[0006] The objective of this invention is achieved through the following technical solution: a method for optimizing the attacker's strategy in complex network games based on intuitionistic fuzzy logic, the method comprising:

[0007] Step 1: Obtain the topology of the infrastructure network, determine the strategy sets of the attacker and defender, and construct a basic model of complex network game.

[0008] Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the basic model of complex network games, and thus obtain the payoff matrix.

[0009] Step 3: Using the intuitionistic fuzzy number determination method, construct hyperbolic membership functions and non-membership functions;

[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 basic model of complex network game is transformed into a linear programming problem to obtain the mixed Nash equilibrium solution, that is, the optimization result of the attacker's strategy.

[0012] The infrastructure network is represented as a simple undirected graph G(V,E), where V={v1,v2,...,v...} N} represents the set of all nodes in the network, where N = |V| represents the number of nodes in the network. It is the set of all edges in the network.

[0013] Specifically, the attacker's strategy set is S. A For an attack strategy vector s A =[x1,x2,...,x N ]∈S A x i Indicate whether the i-th infrastructure has been attacked, denoted by . Let i be the set of attacking nodes, and let v be the i-th node. i Being attacked, i.e., v i ∈V A x i =1, otherwise x i =0; the defender's strategy vector s D =[y1,y2,...,y N ]∈S D y i Indicate whether the i-th infrastructure is defended, denoted by . Let i be the set of defensive nodes, and let v be the i-th node. i Being defended, i.e., v i ∈V D yi =1, otherwise y i =0, for the attacked node v i If x exists simultaneously i =1 and y i =0, meaning that although attacked, node v is not protected. i It will be removed; therefore, the total cost of the attack vector is:

[0014]

[0015] in, Let r represent the cost of the i-th node from the attacker's perspective. i Let q represent the degree of the i-th node. A This represents the attacker's cost sensitivity coefficient to node degree;

[0016] The attacker's total cost is finite, therefore the following constraints must be met:

[0017]

[0018] Among them, C A θ represents the attacker's total cost constraint. A This represents the cost constraint coefficient of the attacker, with a value range of [0,1].

[0019] ω A With ω D These represent the minimum utilization rates of available cost resources by the attacker and defender, respectively. For the attacker, the following constraints must be satisfied:

[0020]

[0021] The defending side must satisfy the following constraints:

[0022]

[0023] Among them, C D θ represents the total cost constraint for the defender. D This represents the cost constraint coefficient for the defender, with a value range of [0,1]. i Let q represent the degree of the i-th node. D This represents the defender's cost sensitivity coefficient to node degree.

[0024] Specifically, the profit matrix is ​​divided into the attacker's profit matrix and the defender's profit matrix, U A :|S A |×|S D | is the attacker's payoff function, then U A (X,Y) represents the attacker's payoff when choosing strategy X and the defender chooses strategy Y.D (X,Y) represents the defender's gain when the attacker chooses strategy X and the defender chooses strategy Y:

[0025]

[0026]

[0027] Where Γ(G) represents the maximum connected component size of the initial infrastructure network G, and denoted as 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 .

[0028] Specifically, the hyperbolic membership function u(x) and the non-membership function v(x) are in the following forms:

[0029]

[0030]

[0031] Where α represents the highest acceptable level and β represents the lowest acceptable level;

[0032] The attacker selects the i-th pure strategy s Ai ∈S A (i = 1, 2, ..., m), the defender chooses the j-th pure strategy s. Dj ∈S D (j=1,2,...,n), the attacker's original gain can be expressed as Γ(G) ij This represents the original maximum connected component size when the attacker chooses strategy i and the defender chooses strategy j. This represents the maximum connected component size after one round of gameplay when the attacker chooses strategy i and the defender chooses strategy j. The attacker's payoff is transformed into an intuitionistic fuzzy set <μ using a hyperbolic membership function. ij ,ν ij >, while the defender's losses were also <μ ij ,ν ij >,μ ij ν represents the degree of membership of the attacker to the defender when the attacker chooses strategy i and the defender chooses strategy j. ij This represents the non-membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations is represented as follows:

[0033]

[0034] Under the hybrid strategy, the attacker's intuitionistic fuzzy set equilibrium payoff is:

[0035]

[0036] Where p = (p1, p2, ..., p m ) T Let q = (q1, q2, ..., q...) represent the probability vector of the attacker's mixed strategy. n ) T This represents the probability vector of the defender's hybrid strategy.

[0037] Specifically, the linear programming problem described in step 5 is:

[0038] min{(1-μ) λ ν 1-λ}

[0039]

[0040] and

[0041] min{(1-σ) λ ρ 1-λ}

[0042]

[0043] Where i represents the i-th strategy of the attacker, and there are m strategies in total; j represents the j-th strategy of the defender, and there are n strategies in total; λ represents the relative weight of the membership / non-membership function constraint. After λ is determined, the obtained Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>), where <μ,ν> represents the payoff value of the attacker, and <σ,ρ> represents the payoff value of the defender. Both are in the form of intuitionistic fuzzy sets. μ represents the membership degree of the attacker's payoff, ν represents the non-membership degree of the attacker's payoff, σ represents the membership degree of the defender's payoff, and ρ represents the non-membership degree of the defender's payoff.

[0044] Compared with existing methods, the advantages of this invention are as follows: Complex network game theory has been a research hotspot in recent years; however, existing research cannot reflect the ambiguity of decision-makers' understanding of the game problem. This invention proposes a complex network game model based on intuitionistic fuzzy sets, provides a method for generating and solving the intuitionistic fuzzy set payoff matrix, and finally obtains the reasonable strategy choice that the attacker should make under fuzzy conditions, and analyzes the results. Using intuitionistic fuzzy theory to explain the uncertainty of complex network games can greatly broaden the practical application of complex network game research. Attached Figure Description

[0045] Figure 1 A flowchart illustrating an embodiment of the present invention is shown;

[0046] Figure 2 A schematic diagram of the infrastructure network in an embodiment of the present invention is shown;

[0047] Figure 3 A schematic diagram illustrating the probability allocation for the attacker in an embodiment of the present invention is shown;

[0048] Figure 4 A schematic diagram illustrating the probability allocation of the defender in an embodiment of the present invention is shown. Detailed Implementation

[0049] 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.

[0050] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0051] This embodiment considers only one attacker and one defender, and both parties 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 V={v1,v2,...,v...} N} represents the set of all nodes in the network, i.e., the stations in the railway network, where N = |V| represents 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.

[0052] Consider only one attacker and one defender, both of whom have 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.

[0053] like Figure 1 As shown, an attack strategy optimization method based on intuitionistic fuzzy logic in complex network games is proposed, the method comprising:

[0054] Step 1: Obtain the topology of the infrastructure network, determine the strategy sets of the attacker and defender, and construct a basic model of complex network game.

[0055] Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the basic model of complex network games, and thus obtain the payoff matrix.

[0056] Step 3: Using the intuitionistic fuzzy number determination method, construct hyperbolic membership functions and non-membership functions;

[0057] Step 4: Using hyperbolic membership functions and non-membership functions, the revenue matrix is ​​converted into an intuitionistic fuzzy set revenue matrix;

[0058] Step 5: The solution of the basic model of complex network game is transformed into a linear programming problem to obtain the mixed Nash equilibrium solution, that is, the optimization result of the attacker's strategy.

[0059] The attacker's strategy set is S. A For an attack strategy vector s A =[x1,x2,...,x N ]∈S A x i Indicate whether the i-th infrastructure has been attacked, denoted by . Let i be the set of attacking nodes, and let v be the i-th node. i Being attacked, i.e., v i ∈V A x i =1, otherwise x i =0; the defender's strategy vector s D =[y1,y2,...,y N ]∈S D y i Indicate whether the i-th infrastructure is defended, denoted by . Let i be the set of defensive nodes, and let v be the i-th node. i Being defended, i.e., v i ∈V D y i =1, otherwise y i =0, for the attacked node v i If x exists simultaneously i =1 and y i =0, meaning that although attacked, node v is not protected. i It will be removed; therefore, the total cost of the attack vector is:

[0060]

[0061] in, Let r represent the cost of the i-th node from the attacker's perspective. i Let q represent the degree of the i-th node. AThis represents the attacker's cost sensitivity coefficient to node degree;

[0062] The attacker's total cost is finite, therefore the following constraints must be met:

[0063]

[0064] Among them, C A θ represents the attacker's total cost constraint. A This represents the cost constraint coefficient of the attacker, with a value range of [0,1].

[0065] If a suitable strategy is selected based solely on the constraints described above, minimizing the number of attack nodes can largely satisfy the constraints. However, in reality, attackers should utilize existing resources to the fullest extent to achieve the desired disruptive effect on the network. To resolve this contradiction, this embodiment proposes the concept of "minimum resource utilization," ω A With ω D These represent the minimum utilization rates of available cost resources by the attacker and defender, respectively. For the attacker, the overall constraint should be satisfied as follows:

[0066]

[0067] The defending side must satisfy the following constraints:

[0068]

[0069] Among them, C D θ represents the total cost constraint for the defender. D This represents the cost constraint coefficient for the defender, with a value range of [0,1]. i Let q represent the degree of the i-th node. D This represents the defender's cost sensitivity coefficient to node degree.

[0070] The payoff function characterizes the payoffs of each participant in a game theory model under various strategy profiles. Let U A :|S A |×|S D | is the attacker's payoff function, then U A (X,Y) represents the attacker's payoff when choosing strategy X and the defender's payoff when choosing strategy Y. Similarly, the defender's payoff is represented by U. D (X,Y). U D (X,Y) represents the defender's gain when the attacker chooses strategy X and the defender chooses strategy Y:

[0071]

[0072]

[0073] Where Γ(G) represents the maximum connected component size of the initial infrastructure network G, and denoted as 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 .

[0074] Since the interests of the offensive and defensive sides are fundamentally opposed, the aforementioned gains are opposites.

[0075] 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. The definition of an intuitionistic fuzzy set (see references) determines 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). There are many choices for membership functions, such as linear, exponential, and hyperbolic forms. The specific model chosen depends on the application scenario and the decision-maker's preferences. In practical applications, the marginal satisfaction or dissatisfaction of the membership function with respect to the target is not constant, making it difficult to represent with a simple linear membership function. The hyperbolic membership function, as a non-linear membership function, has a shape that is partially concave and partially convex. The convex shape describes the decision-maker's continuously increasing marginal satisfaction, while the concave part reflects the continuously decreasing marginal satisfaction. Therefore, taking the attacker as an example, in the case of hyperbolic membership, when the decision-maker's satisfaction with the attack's gains is poor, they often have a higher marginal satisfaction with the attack's gains; conversely, when the decision-maker's satisfaction with the attack's gains is high, the marginal satisfaction with the attack's gains is often low. Considering complex network game scenarios in practice, and from the attacker's perspective, when the attack gain is small, it's difficult to achieve the decision-maker's expected disruptive effect on the network. The network can still largely function, thus the desire to increase the gain is stronger. However, when the attack gain is large, the expected disruptive effect on the network has already been achieved, and the network often cannot function properly. While increasing the gain is still beneficial, the attacker's desire is less intense. The explanation for non-membership degrees is similar.

[0076] The hyperbolic membership function u(x) and the non-membership function v(x) are in the following forms:

[0077]

[0078]

[0079] Where α represents the highest acceptable level and β represents the lowest acceptable level;

[0080] Here, α represents the desired or most acceptable level (denoted by m), and β represents the worst acceptable level of achievement. The judgment of membership and non-membership functions is highly subjective and should be derived by comprehensively considering the actual situation, historical data, and the decision-maker's subjective preferences.

[0081] Assume the attacker chooses the i-th pure policy s Ai ∈S A (i = 1, 2, ..., m), the defender chooses the j-th pure strategy s. Dj ∈S D (j=1,2,...,n), the attacker's original gain can be expressed as Γ(G) ij This represents the original maximum connected component size when the attacker chooses strategy i and the defender chooses strategy j. This represents the maximum connected component size after one round of gameplay when the attacker chooses strategy i and the defender chooses strategy j. The attacker's payoff is transformed into an intuitionistic fuzzy set <μ using a hyperbolic membership function. ij ,ν ij >, while the defender's losses were also <μ ij ,ν ij >,μ ij ν represents the degree of membership of the attacker to the defender when the attacker chooses strategy i and the defender chooses strategy j. ij This represents the non-membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations is represented as follows:

[0082]

[0083] Under the hybrid strategy, the attacker's intuitionistic fuzzy set equilibrium payoff is:

[0084]

[0085] Where p = (p1, p2, ..., p m ) T Let q = (q1, q2, ..., q...) represent the probability vector of the attacker's mixed strategy. n ) T This represents the probability vector of the defender's hybrid strategy.

[0086] For the aforementioned intuitionistic fuzzy set two-player zero-sum game problem, the solution model can ultimately be transformed into: min{(1-μ)} λ ν 1-λ}

[0087]

[0088] and

[0089] min{(1-σ) λ ρ 1-λ}

[0090]

[0091] Where i represents the i-th strategy of the attacker, and there are m strategies in total; j represents the j-th strategy of the defender, and there are n strategies in total; λ represents the relative weight of the membership / non-membership function constraint. After λ is determined, the obtained Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>), where <μ,ν> represents the payoff value of the attacker, and <σ,ρ> represents the payoff value of the defender. Both are in the form of intuitionistic fuzzy sets. μ represents the membership degree of the attacker's payoff, ν represents the non-membership degree of the attacker's payoff, σ represents the membership degree of the defender's payoff, and ρ represents the non-membership degree of the defender's payoff.

[0092] In real life, infrastructure network structures vary widely. This experiment uses a 10-node network structure as an example. Figure 2 As shown, it is assumed that both the attacker and defender have limited resources and the number of nodes that can be selected does not exceed 3.

[0093] Applying intuitionistic fuzzy theory to complex network games, the strategy set S for both attackers and defenders can be obtained from the model in the previous chapter. A S D (|S A |=120,|S D |=120), and obtain the initial payoff matrix based on the payoff function. Since the judgment of membership and non-membership functions is highly subjective, the membership / non-membership functions are derived by simulating the decision-maker's subjective preferences, as shown in Equations 1 and 2, based on the network topology in this example, where m=6, n=1.

[0094]

[0095]

[0096] After obtaining the intuitionistic fuzzy set payoff matrix, the model solution process yields the Nash equilibrium mixed policy solution. Since each policy corresponds to a choice probability, the probabilities of different pure policies can be mapped to different nodes, as follows:

[0097]

[0098]

[0099] in and It represents the probability distribution of individual nodes for two participants. and It is the probability distribution of all attack and defense strategies.

[0100] Based on the above, a probability distribution diagram for the attacking side can be drawn as follows: Figure 3 As shown, the probability distribution diagram for the defending side is as follows: Figure 4 As shown.

[0101] according to Figure 3 An analysis was conducted on the attack probability distribution of different nodes:

[0102] It can be seen that the probability distribution of different nodes does not change significantly with the change of λ. The reason for this is that λ reflects the relative weights of membership function constraints and non-membership function constraints. The membership and non-membership functions corresponding to this result are symmetrical. Therefore, the change of weights will not cause a change in the selection probability of some strategies. Thus, the probability of the nodes included in these strategies being selected will not change.

[0103] It can be seen that the attack probability of different nodes changes when a membership function is added versus not added, but the overall difference is not significant. The analysis suggests that the payoff value changes under the influence of the membership function, thus slightly altering the probability of different strategy choices and affecting the probability distribution of different nodes.

[0104] according to Figure 4 An analysis of the defense probability distribution at different nodes was conducted.

[0105] It can be seen that the probability distribution of different nodes does not change significantly with the change of λ, for the same reason as the attack node.

[0106] It can be seen that the defense probability of different nodes changes when a membership function is added or not, but the overall difference is not significant. The analysis suggests that the payoff value changes under the influence of the membership function, thus slightly altering the probability of different strategy choices and affecting the probability distribution of different nodes.

[0107] Table 1. Comparison of attack and defense probability selection and node indicators for each node when λ equals 0.5.

[0108]

[0109] Table 1 comprehensively demonstrates the importance of different nodes by calculating various existing network indicators. It shows that for relatively important nodes like node 1, the probability of an attacker choosing to defend them is actually lower. However, for the defender, the probability of defending relatively important nodes in the network topology is also higher. This is because the attacker "anticipates" that the defender's defensive strength is relatively strong for nodes with high importance. For the defender, the losses from a low-probability event would be unbearable. This conclusion is similar to that obtained without intuitionistic fuzzy logic, consistent with the normal logic of game theory. Therefore, it proves the effectiveness of two-person zero-sum game theory based on intuitionistic fuzzy logic in complex network game applications.

[0110] 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 method for optimizing attacker strategies in complex network games based on intuitionistic fuzzy logic, characterized in that, The method includes: Step 1: Obtain the topology of the infrastructure network, determine the strategy sets of the attacker and defender, and construct a basic model of complex network game. Step 2: Determine the index representing network connectivity, calculate the payoffs of the attacker and defender under various strategy profiles in the basic model of complex network games, and thus obtain the payoff matrix. Step 3: Using the intuitionistic fuzzy number determination method, construct hyperbolic membership functions and non-membership functions; 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 basic model of complex network game is transformed into a linear programming problem to obtain the mixed Nash equilibrium solution, that is, the optimization result of the attacker's strategy. 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 attacker's strategy set 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 ; Defender's strategy vector , Indicates the first Whether the infrastructure is defended, remember. For the set of defensive nodes, if the first... Nodes Being defended, that is , ,otherwise For the attacked node If both exist and That is, if a node is attacked but not protected, then... It will be removed; therefore, the total cost of the attack vector is: in, This represents the cost of the i-th node from the attacker's perspective. This represents the degree of the i-th node. This represents the attacker's cost sensitivity coefficient to node degree; The attacker's total cost is finite, therefore the following constraints must be met: in, This represents the attacker's total cost constraint. This represents the cost constraint coefficient of the attacker, with a value range of [0,1]. and These represent the minimum utilization rates of available cost resources by the attacker and defender, respectively. For the attacker, the following constraints must be satisfied: The defending side must satisfy the following constraints: in, This represents the total cost constraint for the defending side. This represents the cost constraint coefficient for the defender, with a value range of [0,1]. This represents the degree of the i-th node. This represents the defender's cost sensitivity coefficient to node degree.

2. The method for optimizing attacker strategies in complex network games based on intuitionistic fuzzy logic as described in claim 1, 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 strategy X and the defender choosing strategy Y. This represents the defender's gain when the attacker chooses strategy X and the defender chooses strategy Y: 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 . .

3. The method for optimizing attacker strategies in complex network games based on intuitionistic fuzzy logic as described in claim 2, characterized in that, 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 the first A pure strategy The defending side chooses the first A pure strategy The attacker's original gain can be expressed as , This represents the original maximum connected component size when the attacker chooses strategy i and the defender chooses strategy j. This represents the maximum connected component size after one round of gameplay when the attacker chooses strategy i and the defender chooses strategy j. The attacker's payoff is transformed into an intuitionistic fuzzy set using a hyperbolic membership function. The defenders also suffered losses. , This represents the degree of membership of the attacker to the defender when the attacker chooses strategy i and the defender chooses strategy j. This represents the non-membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations is represented as follows: Under the hybrid strategy, the attacker's intuitionistic fuzzy set equilibrium payoff is: in This represents the probability vector of the attacker's mixed strategy. This represents the probability vector of the defender's hybrid strategy.

4. The method for optimizing attacker strategies in complex network games based on intuitionistic fuzzy logic as described in claim 3, characterized in that, The linear programming problem described in step 5 is: in, Indicates the attacker's... One strategy, total One strategy; The first one represents the defending side One strategy, total One strategy, Represents the relative weights of membership / non-membership function constraints. Once determined, the attacker's Nash equilibrium solution is: , The attacker's gain is represented by an intuitionistic fuzzy set. This indicates the degree of membership that represents the attacker's gain. This represents the non-membership degree of the attacker's gain.

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