Hyperbolic membership function-based complex network game defense strategy optimization method

Abstract

The application discloses a complex network game defense strategy optimization method based on a hyperbolic membership function, and the method comprises the following steps: acquiring a topological structure of an infrastructure network, determining a strategy set of an attack side and a defense side, and constructing a basic model of a complex network game; determining an index representing network connectivity, calculating the income of the attack side and the defense side in the basic model of the complex network game under various strategy profiles, and thus obtaining an income matrix; constructing a hyperbolic membership function and a non-membership function by using an intuitionistic fuzzy number determination method; 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 a defense side strategy optimization result.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of complex network game in system engineering, and particularly relates to a complex network game defense strategy optimization method based on hyperbolic membership function. BACKGROUND

[0002] In the current research field of game theory, there is a special network game, in which the network is not an actual existing network such as a computer system, but a key infrastructure such as a train station, an airport, etc. is abstracted into a node in the network topology structure, and the connection between different stations is abstracted into an edge to establish an infrastructure complex network. In the security field, the key nodes of the infrastructure are vulnerable to attack, which will affect the public security management and normal social life, and the security department needs to protect these nodes. The complex network game can be used to study the attack and protection of key nodes in the infrastructure network, which helps to develop the best protection strategy and explore the importance of nodes.

[0003] At present, there are some researches on this kind of problem, but the existing researches only give an objective evaluation method based on the network topology structure, such as using the network connectivity performance index, the maximum connected component size, to calculate the payoff matrix of the attacker and the defender in the complete information static or dynamic game framework, and to calculate the corresponding Nash equilibrium strategy. However, in this kind of actual game problem, the understanding of the problem by the two parties of the game is not certain, the information obtained is insufficient, and the decision-making environment is unpredictable. The existing method cannot well integrate the subjective judgment of the decision maker and cannot express the fuzziness and uncertainty of the actual game problem.

[0004] Professor Zadeh proposed the fuzzy set theory, which provides a reasonable way to solve this kind of problem. In view of the limitations of fuzzy set theory and the actual need to express hesitation, Atanassov proposed the intuitionistic fuzzy set theory, which uses two scales (membership and non-membership) to represent the support, opposition and hesitation of fuzzy phenomenon. The proposal of this theory provides an inspiration for solving more complex game problems. At present, there are few related researches on introducing the intuitionistic fuzzy set theory into the infrastructure complex network game, and this kind of research has important significance. SUMMARY

[0005] The present application aims at at least solving one of the technical problems existing in the prior art. To this end, the present application discloses a complex network game defense strategy optimization method based on hyperbolic membership function. The method uses the two-person zero-sum matrix game of intuitionistic fuzzy set, proposes a method of generating intuitionistic fuzzy payoff matrix using hyperbolic membership / non-membership function, solves the Nash equilibrium solution under the intuitionistic fuzzy condition by using an effective algorithm, and finally obtains the optimization method and result of the attacker strategy.

[0006] The objective of this invention is achieved through the following technical solution: a method for optimizing the defensive strategy in complex network games based on hyperbolic membership functions, 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 defender'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 This indicates whether the i-th infrastructure element is defended. 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 i = 1 and y i = 0, i.e., although attacked but not protected, then the node v i will be removed, thus the total cost of the attack vector is:

[0014]

[0015] where, denotes the cost of the ith node for the attacker, r i denotes the degree of the ith node, q A denotes the cost sensitivity coefficient of the attacker to the node degree;

[0016] The total cost of the attacker is limited, thus the constraint should be satisfied:

[0017]

[0018] where, C A denotes the total cost constraint of the attacker, θ A denotes the cost constraint coefficient of the attacker, with the value range of [0, 1];

[0019] ω A and ω D respectively denote the minimum utilization rate of the available cost resources by the attacker and the defender, for the attacker, the constraint should be satisfied:

[0020]

[0021] The defender should satisfy the constraint:

[0022]

[0023] where, C D denotes the total cost constraint of the defender, θ D denotes the cost constraint coefficient of the defender, with the value range of [0, 1], r i denotes the degree of the ith node, q D denotes the cost sensitivity coefficient of the defender to the node degree.

[0024] Specifically, the payoff matrix is divided into the payoff matrix of the attacker and the payoff matrix of the defender, U A : S A | x | S D | is the payoff function of the attacker, then U A (X, Y) denotes the payoff of the attacker when the attacker selects the strategy X and the defender selects the strategy Y, UD (X, Y) represents the payoff of the defender 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 let the set of all removed nodes be The network after the nodes are removed is is the set of all edges in the network after the nodes are removed, Γ(G) represents the maximum connected component size of the network after one round of game, and satisfies

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

[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), and the defender selects the j-th pure strategy s Dj ∈ S D (j = 1, 2,..., n), and the original payoff value of the attacker can be represented as Γ(G) ij Γ(G) represents the original maximum connected component size when the attacker chooses strategy i and the defender chooses strategy j, Γ(G) represents the maximum connected component size after one round of game when the attacker chooses strategy i and the defender chooses strategy j, and the payoff value of the attacker is converted into the intuitionistic fuzzy set <μ ij ,ν ij > by the hyperbolic membership function, and the loss of the defender is also <μ ij ,ν ij >, μ ij represents the membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j, and v ij represents the non-membership degree of the attacker when the attacker chooses strategy i and the defender chooses strategy j, and thus the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations is represented as

[0033]

[0034] Then under the mixed strategy, the intuitionistic fuzzy set equilibrium payoff of the attacker is:

[0035]

[0036] where p=(p1, p2,..., p m ) T is the probability vector of the mixed strategy of the attacker, q=(q1, q2,..., q n ) T is the probability vector of the mixed strategy of the defender.

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

[0038]

[0039] and

[0040]

[0041] where i represents the ith strategy of the attacker, a total of m strategies; j represents the jth strategy of the defender, a total of n strategies, and λ represents the relative weight of the membership / non-membership function constraint. After λ is determined, the Nash equilibrium solution obtained is (p, q, <μ,ν>, <σ,ρ>), <μ,ν> represents the payoff value of the attacker, <σ,ρ> represents the payoff value of the defender, both 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.

[0042] Compared with the prior art, the method has the advantages that complex network games are a research hotspot in recent years, however, the existing research cannot reflect the fuzziness of the understanding of the game problem by the decision maker, the method proposes a complex network game model based on intuitionistic fuzzy sets, gives a generation method and solution idea of the payoff matrix of intuitionistic fuzzy sets, finally obtains the reasonable strategy selection of the defender under the fuzzy condition, and analyzes the result. The uncertainty of the complex network game is explained by using the intuitionistic fuzzy theory, which can greatly broaden the application of network attack and defense game research in practice. BRIEF DESCRIPTION OF DRAWINGS

[0043] Figure 1 A flowchart of an embodiment of the application is shown;

[0044] Figure 2 A schematic diagram of an infrastructure network in an embodiment of the application is shown;

[0045] Figure 3A schematic diagram illustrating the probability allocation for the attacker in an embodiment of the present invention is shown;

[0046] Figure 4 A schematic diagram illustrating the probability allocation of the defender in an embodiment of the present invention is shown. Detailed Implementation

[0047] 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.

[0048] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0049] 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.

[0050] 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.

[0051] like Figure 1 As shown, a method for optimizing the defender's strategy in complex network games based on hyperbolic membership functions is presented. The method includes:

[0052] 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.

[0053] 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.

[0054] Step 3: Using the intuitionistic fuzzy number determination method, construct hyperbolic membership functions and non-membership functions;

[0055] Step 4, the revenue matrix is converted into an intuitionistic fuzzy set revenue matrix by using hyperbolic membership function and non-membership function;

[0056] Step 5, the solution of the basic model of complex network game is converted into the solution of linear programming problem, and a mixed Nash equilibrium solution is obtained, that is, the optimization result of the defense strategy is obtained.

[0057] The strategy set of the attack party is S A For an attack strategy vector s A =[x1,x2,...,x N ]∈S A , x i represents whether the ith infrastructure is attacked, and V i represents the set of attack nodes, if the ith node v i is attacked, that is, v A ∈V i , x i =1, otherwise x D =0; the strategy vector of the defense party s N =[y1,y2,...,y D ]∈S i , y i represents whether the ith infrastructure is defended, and V i represents the set of defense nodes, if the ith node v D is defended, that is, v i ∈V i , y i =1, otherwise y i =0, for the attacked node v i , if there exists x i =1 and y i =0 at the same time, that is, although attacked but not protected, then the node v A will be removed, thus the total cost of the attack vector is:

[0058]

[0059] wherein, represents the cost of the ith node for the attack party, r A represents the degree of the ith node, and q A represents the cost sensitivity coefficient of the attack party to the node degree;

[0060] And the total cost of the attack party is limited, so the constraint should be met:

[0061]

[0062] wherein, C Aθ represents the attacker's total cost constraint. A This represents the cost constraint coefficient of the attacker, with a value range of [0,1].

[0063] 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:

[0064]

[0065] The defending side must satisfy the following constraints:

[0066]

[0067] 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.

[0068] 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:

[0069]

[0070]

[0071] 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 G (k) denote the size of the largest connected component of the network after k rounds of the game, and satisfy

[0072] Since the interests of the attacker and the defender are fundamentally opposed, the above benefits are opposite to each other.

[0073] In the process of generating intuitionistic fuzzy sets, we should consider how to more accurately reflect the cognitive preferences of decision makers. The membership / non-membership function is the cornerstone of intuitionistic fuzzy set theory. According to the definition of intuitionistic fuzzy set (reference), the intuitionistic fuzzy set on the deterministic domain X is determined as The most important thing is to determine the two functions X→[0, 1] (membership degree) and X→[0, 1] (non-membership degree) in it. There are many choices for the membership degree function, such as linear, exponential, hyperbolic, etc. The specific choice depends on the application scenario and the decision maker's own preferences. In practical applications, the marginal satisfaction or dissatisfaction of the membership degree function to the target is not constant, and it is difficult to represent with a simple linear membership function. The hyperbolic membership function is a kind of nonlinear membership function, and its shape includes part of the concave and the rest of the convex. The convex shape describes the decision maker's marginal satisfaction increasing, while the concave part reflects the marginal satisfaction decreasing. Therefore, taking the attacker as an example, in the case of hyperbolic membership, when the decision maker is not satisfied with the attack benefit, he often has a higher marginal satisfaction for the attack benefit; when the decision maker is satisfied with the attack benefit, the marginal satisfaction of the attack benefit is smaller. Combined with the actual network attack and defense scene, considering the problem from the perspective of the attacker, when the attack benefit is small, it is difficult to achieve the decision maker's expected damage effect on the network, and the network can still function to a certain extent, so the desire to improve the benefit is stronger; when the attack benefit is large, the expected damage effect on the network has been achieved, and at this time the network often cannot function accordingly, so although improving the benefit is beneficial, the attacker's desire is not strong. The explanation of non-membership degree is the same.

[0074] The form of the hyperbolic membership function u(x) and the non-membership function v(x) is:

[0075]

[0076]

[0077] Where α represents the highest acceptable level, and β represents the lowest acceptable level.

[0078] Where, α represents the highest acceptable level (the desired or the most acceptable level noted by m), β represents the lowest acceptable level (the worst acceptable level of achievement), for membership and non-membership function, its judgment has strong subjectivity, should be comprehensive actual situation, historical data and decision maker subjective preference.

[0079] Assume that the attacker selects the i-th pure strategy s Ai ∈ S A (j = 1, 2,..., n), the original income value of the attacker can be expressed as Dj ∈ S D (j = 1, 2,..., n), the original income value of the attacker can be expressed as Γ(G) ij represents the original maximum connected piece size when the attacker selects strategy i and the defender selects strategy j, represents the maximum connected piece size after a round of game when the attacker selects strategy i and the defender selects strategy j, the income value of the attacker is converted into the intuitionistic fuzzy set <μ ij ,ν ij > by the hyperbolic membership function, and the loss of the defender is also <μ ij ,ν ij >, μ ij represents the membership degree of the attacker when the attacker selects strategy i and the defender selects strategy j, and ν ij represents the non-membership degree of the attacker when the attacker selects strategy i and the defender selects strategy j, thus, the intuitionistic fuzzy set income matrix of the attacker under different pure strategy situations is represented as

[0080]

[0081] Then, the intuitionistic fuzzy set equilibrium income of the attacker under mixed strategy is:

[0082]

[0083] Where p = (p1, p2,..., p m ) T represents the probability vector of the attacker's mixed strategy, q = (q1, q2,..., q n ) T represents the probability vector of the defender's mixed strategy.

[0084] For the above intuitionistic fuzzy set two-person zero-sum game problem, its solution model can be finally transformed as:

[0085]

[0086] and

[0087]

[0088] where i represents the ith strategy of the attacker, a total of m strategies; j represents the jth strategy of the defender, a total of n strategies, λ represents the relative weight of the membership / non-membership function constraint, after λ is determined, the Nash equilibrium solution obtained is: (p, q, <μ, v>, <σ, p>), <μ, v> represents the income value of the attacker, <σ, p> represents the income value of the defender, both in the form of intuitionistic fuzzy sets, μ represents the membership degree of the attacker's income, v represents the non-membership degree of the attacker's income, σ represents the membership degree of the defender's income, and p represents the non-membership degree of the defender's income.

[0089] In real life, infrastructure network structures are various, and an experiment takes a 10-node network structure as an example, as shown in Figure 2 , and it is assumed that the resources of the attacker and the defender are limited, and the number of nodes that can be selected does not exceed 3.

[0090] Intuitionistic fuzzy theory is applied to network attack and defense game, and the strategy set S A of the attacker and the defender can be obtained from the model in the last chapter D (|S A |=120, |S D |=120), and the initial income matrix is obtained according to the income function. Since the judgment of the membership degree and the non-membership degree function has strong subjectivity, the membership / non-membership function is obtained according to the network topology structure in this example and the simulation of the subjective preference of the decision maker, where m=6, n=1

[0091]

[0092]

[0093] After obtaining the intuitionistic fuzzy set income matrix, the Nash equilibrium mixed strategy solution can be obtained in the solving process of the model. Since each strategy corresponds to a selection probability, the probabilities corresponding to different pure strategies can be mapped to different nodes in the following manner:

[0094]

[0095]

[0096] where and are the probability distributions of a single node of the two participants, and are the probability distributions of all attack and defense strategies.

[0097] According to the above, the probability distribution of the attack side is shown in Figure 3 and the probability distribution of the defense side is shown in Figure 4 .

[0098] According to Figure 3 , the probability distribution of the attack side is analyzed as follows:

[0099] It can be seen that the probability distribution of different nodes does not change significantly with the change of λ. The reason for this analysis is that λ reflects the relative weight of the membership function constraint and the non-membership function constraint, and the membership and non-membership functions corresponding to this result are symmetrical, so the change of weight will not cause the change of the probability of selecting some strategies, and therefore the probability of selecting the nodes contained in these strategies will not change.

[0100] It can be seen that the attack probability of different nodes changes slightly when the membership function is added and when the membership function is not added, but the overall difference is not large. The analysis may be due to the change of the payoff value under the action of the membership function, so the probability of selecting different strategies also changes slightly, thereby affecting the probability distribution of different nodes.

[0101] According to Figure 4 , the probability distribution of the defense side is analyzed as follows:

[0102] 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 attack node.

[0103] It can be seen that the defense probability of different nodes changes slightly when the membership function is added and when the membership function is not added, but the overall difference is not large. The analysis may be due to the change of the payoff value under the action of the membership function, so the probability of selecting different strategies also changes slightly, thereby affecting the probability distribution of different nodes.

[0104] Table 1 Comparison of attack and defense probability selection of each node and node index when λ is equal to 0.5

[0105]

[0106] Table 1 is a comprehensive display of the importance of different nodes through the calculation of various existing indicators of the network. It can be seen that for a relatively important node such as node 1, the probability of selection by the attack side is actually reduced; but for the defense side, the probability of selecting defense for the relatively important node in the network topology is also relatively high, because the attack side "anticipates" that the node with high importance has relatively strong defense strength, and for the defense side, once a small probability event occurs, its loss will be difficult to bear. This conclusion is similar to the conclusion obtained without intuitionistic fuzzy, and conforms to the normal logic of game theory.

[0107] Those skilled in the art will appreciate that embodiments of the present application can be readily used as a method, a system or a computer program product. 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 can take the form of a computer program product on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROMs, optical storage devices, etc.) embodying computer-readable program code thereon for use by a computer.

Claims

1. A method for optimizing the defensive strategy in complex network games based on hyperbolic membership functions, 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 defender'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; 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 expressed as: ， 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; 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 Nash equilibrium solution for the defender is: , The value representing the defender's gain is in the form of an intuitionistic fuzzy set. This indicates the degree of membership that represents the defender's benefit. This represents the degree of non-membership of the defender's gains.

2. The method for optimizing the defensive strategy in complex network games based on hyperbolic membership functions according to 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 . .

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