Infrastructure network attack and defense game strategy generation method based on multi-index fusion

Abstract

The application discloses a method for generating infrastructure network attack and defense game strategy based on multi-index fusion, and has the characteristics that the method comprises the following steps: acquiring the topology structure of an infrastructure network, determining the strategy set of player 1 and player 2, and constructing an infrastructure network game model; calculating the payoff matrix of player 1 and player 2 in the infrastructure network game under various strategy profiles respectively by taking three network performance indexes of the maximum connected piece scale, network efficiency and clustering coefficient; constructing three pairs of membership functions and non-membership functions according to the preference of a decision maker; converting the payoff matrix into an intuitionistic fuzzy set payoff matrix by using the three pairs of membership functions and non-membership functions, and obtaining a final payoff matrix by using an intuitionistic fuzzy weighted aggregation operator for weighted fusion; and combining the intuitionistic fuzzy set theory to convert the solution of the infrastructure network game model into the solution of a nonlinear programming problem, so as to obtain the mixed strategy Nash equilibrium solution of player 1 and player 2.

Description

Technical Field

[0001] This invention relates to the field of complex network game technology in systems engineering, and in particular to a method for generating attack and defense game strategies for infrastructure networks based on multi-index fusion. 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] Existing research on network attack and defense game theory only provides an objective evaluation method based on network topology, failing to incorporate the subjective judgment of decision-makers and express the ambiguity and complexity of real-world problems. For example, in a complete information static or dynamic game framework, the payoff matrices of the attacker and defender are calculated using the network connectivity performance index—the size of the largest connected component—and the corresponding Nash equilibrium strategies. However, in such real-world game problems, the understanding of the problem by both parties is uncertain, they have insufficient information, the decision-making environment is unpredictable, and there are many factors to consider. Existing methods cannot effectively integrate the subjective judgment of decision-makers and cannot express the ambiguity and complexity 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. Furthermore, to comprehensively consider the subjective judgments of decision-makers, merging multiple indicators according to weights to form a new payoff matrix can better account for the relationships and weights between factors, reduce the impact of uncertainty on the game outcome, and make the results closer to reality. 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 generating attack and defense game strategies for infrastructure networks based on multi-index fusion. The method is based on a two-player zero-sum game, using hyperbolic membership / non-membership functions to generate an intuitionistic fuzzy set payoff matrix. Based on this, multiple indicators are fused, and new payoff matrices are generated with different weights. An efficient algorithm is then employed to solve for the Nash equilibrium, ultimately obtaining the optimization method and results for both sides' strategies.

[0006] The objective of this invention is achieved through the following technical solution: a method for generating attack and defense game strategies for infrastructure networks based on multi-index fusion, the method comprising:

[0007] Step 1: Obtain the topology of the infrastructure network, with the attacker and defender being Player 1 and Player 2, respectively. Determine the strategy sets of Player 1 and Player 2, and construct the infrastructure network game model.

[0008] Step 2: Using three network performance metrics—maximum connected component size, network efficiency, and clustering coefficient—calculate the payoff matrices for Player 1 and Player 2 in the infrastructure network game under various strategy profiles.

[0009] Step 3: Based on the decision-maker's preferences, construct three pairs of membership functions and non-membership functions corresponding to the three network performance indicators: maximum connected component size, network efficiency, and clustering coefficient.

[0010] Step 4: Using three pairs of membership functions and non-membership functions, the payoff matrix described in Step 2 is converted into an intuitionistic fuzzy set payoff matrix, and then the intuitionistic fuzzy weighted set is used for sub-weighted fusion to obtain the final payoff matrix.

[0011] Step 5: Combining intuitionistic fuzzy set theory, the solution of the infrastructure network game model is transformed into the solution of a nonlinear programming problem, and the mixed strategy Nash equilibrium solution of player 1 and player 2 is obtained.

[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 infrastructure network, where N = |V| represents the number of nodes in the infrastructure network. It is the set of all edges in the infrastructure network.

[0013] Specifically, the attacker's attack 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 set of defense strategies of the defending side is S. D A defense strategy vector s of the defending side D =[y1,y2,...,y N ]∈S D y i Indicate whether the i-th infrastructure is defended, denoted as . Let i be the set of defense 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;

[0014] For the i-th node v i ,remember and These represent the attack cost for the attacker and the defense cost for the defender, respectively, node v. i The attack cost and defense cost are expressed as follows:

[0015]

[0016] Where, r i Represents node v i Structural attribute index, q A q represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's structural attribute index on the attacker's attack cost. D This represents the cost sensitivity coefficient of the defender, i.e., the impact of the structural attribute index of the node on the defender's defense cost;

[0017] Define the available resources for both the attacker and defender as follows:

[0018]

[0019]

[0020] Among them, C A C represents the attacker's available resources. D Indicates the available resources of the defending side; θ A∈[0,1],θ D ∈[0,1] are the attack budget constraint coefficient and the defense budget constraint coefficient, respectively;

[0021] Let the attacker's total attack cost be C. X ,but:

[0022]

[0023] The attacker's total attack cost is finite, therefore the following constraints must be met:

[0024]

[0025] The total defense cost for the defender is C. Y ,but:

[0026]

[0027] The strategy set includes the attacker's attack strategy set and the defender's defense strategy set. The attacker's attack strategy set includes three typical strategies, namely, the maximum attack strategy s. Amax ∈S A Minimal attack strategy Amin ∈S A Random attack strategy Arand ∈S A 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 constraint for attack. The defender's set of defense strategies includes three typical strategies: the maximum-degree defense strategy s. Dmax ∈S D Minimal defense strategy Dmax ∈S D Random defense strategy Drand ∈S D 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.

[0028] Furthermore, the structural attribute indicators are degree, betweenness, or eigenvector centrality.

[0029] Specifically, the profit matrix includes the attacker's profit matrix and the defender's profit matrix. Let i be the attacker's payoff function under network performance metric i, then This represents the attacker's gain when choosing attack strategy X and the defender's gain when choosing defense strategy Y under network performance metric i. Let represent the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y under network performance index i. Let i = 1 represent the maximum connected component size, i = 2 represent network efficiency, and i = 3 represent the clustering coefficient. Then:

[0030]

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] Where Γ(G), E(G), and H(G) represent the maximum connected component size, network efficiency, and clustering coefficient of the initial infrastructure network, respectively, and the set of all removed nodes is denoted as . The network formed after the nodes are removed is It is the set of all edges in the infrastructure network formed after nodes are removed. and Represents the maximum connected component size, network efficiency, and clustering coefficient of the infrastructure network after one round of game playing, and satisfies...

[0037] Furthermore, the membership function and non-membership function corresponding to network performance index i adopt the hyperbolic membership function μ. i (x) and non-membership function ν i The form of (x):

[0038]

[0039]

[0040] Where α represents the highest acceptable level and β represents the lowest acceptable level;

[0041] The attacker selects a set of attack strategies S A One of the attack strategies Amax ,s Amin ,s Arand ∈S A The defender selects a set of defense strategies S. D One of the defense strategies Dmax ,s Dmin ,sDrand ∈S D When the network performance metric is the maximum connected component size, i = 1, and the attacker's original gain is expressed as:

[0042]

[0043] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The largest connected segment size before the game begins. This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The maximum connected component size after the game is played is determined by transforming the attacker's original payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different strategy situations corresponding to the performance index of the largest connected component network is expressed as:

[0044]

[0045] Where, μ 1AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax Membership degree of revenue value of the largest connected slice size, ν 1AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax The revenue value of the maximum connected piece size is not a membership degree. In other symbols, μ represents membership degree and ν represents non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 1 indicates that the maximum connected piece size is used as the network performance indicator.

[0046] When the network performance metric is network efficiency, i = 2, and the attacker's original gain is expressed as:

[0047]

[0048] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D Network efficiency before the game begins. This indicates that the attacker's attack strategy is s. AThe defender's defense strategy is s D The network efficiency after the game is discussed, and the attacker's original payoff value is transformed into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the network efficiency and network performance indicators, is expressed as:

[0049]

[0050] Where, μ 2AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax At that time, the membership degree of the network efficiency gain value, ν 2AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax When the network efficiency gain value is used, the non-membership degree is used. In other symbols, μ represents membership degree and ν represents non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 2 indicates that network efficiency is used as a network performance indicator.

[0051] When the network performance metric is the clustering coefficient, i = 3, the attacker's original gain is expressed as:

[0052]

[0053] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The aggregation coefficient before the game begins. This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The aggregation coefficients after the game are used to transform the attacker's original payoff into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the clustering coefficient network performance index, is expressed as:

[0054]

[0055] Where, μ 3AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. AmaxThe defense strategy is the maximum defense strategy. Dmax When, the membership degree of the clustering coefficient return value, ν 3AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax When the clustering coefficient is used, it represents the degree of non-membership. In other symbols, μ represents membership, ν represents non-membership, and subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. Subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. Subscript 3 indicates that the clustering coefficient is used as a network performance indicator.

[0056] Let a j =<μ j ,ν j >(j=1,2,…,n) is a set of intuitionistic fuzzy sets, where n represents the number of indicators. The intuitionistic fuzzy weighted set sub-calculation is defined as:

[0057]

[0058] In the formula, ω=(ω1,ω2,…,ω n ) is a j The weight vector (j = 1, 2, ..., n) satisfies the condition 0 ≤ ω j ≤1, Using Intuitive Fuzzy Weighted Set Calculation Subsidiary IFWA ω Weighted sets of M1, M2, and M3 are combined into a comprehensive intuitionistic fuzzy set payoff matrix, i.e.

[0059]

[0060] Where, μ AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax Membership degree of the comprehensive return value, ν AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax The non-membership degree of the comprehensive return value is given by the symbol μ, which represents the membership degree, and ν, which represents the non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy, respectively. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy, respectively.

[0061] Specifically, for the comprehensive intuitionistic fuzzy set payoff matrix, the nonlinear programming problem described in step 5 is:

[0062]

[0063] and

[0064]

[0065] Where i represents the attacking side's i-th attack strategy, with a total of m attack strategies; j represents the defending side's j-th defense strategy, with a total of n defense strategies; μ ij This represents the membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy. ij Let represent the non-membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy; λ represents the relative weights of the membership / non-membership function constraints. Once λ is determined, the resulting Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>); p=(p1,p2,...,p m ) T Let q = (q1, q2, ..., q...) represent the probability vector of the attacker's mixed strategy. m ) T Let <μ,ν> represent the probability vector of the defender's mixed strategy; let <σ,ρ> represent the attacker's payoff in the game problem; and let <σ,ρ> represent the defender's payoff in the game problem. Both are intuitionistic fuzzy sets.

[0066] Compared with existing methods, the advantages of this invention are as follows: Network attack-defense game theory has been a research hotspot in recent years. However, existing research cannot reflect the fuzziness and comprehensiveness of decision-makers' understanding of game problems. This invention introduces an intuitionistic fuzzy set payoff matrix, which better reflects the players' subjective feelings and experiences during the game process, making the payoff matrix more consistent with reality and improving the accuracy and credibility of the game results. Furthermore, fusing multiple indicators according to weights to form a new payoff matrix can better consider the relationships and weights between factors, reducing the impact of uncertainty on the game results. Finally, it yields the reasonable strategy choices that both attackers and defenders should make under fuzzy conditions, and analyzes the results. Using intuitionistic fuzzy theory to explain the uncertainty of network attack-defense games can greatly broaden the practical application of network attack-defense game research. Attached Figure Description

[0067] Figure 1 A flowchart illustrating an embodiment of the present invention is shown;

[0068] Figure 2 A schematic diagram of the infrastructure network in an embodiment of the present invention is shown;

[0069] Figure 3The diagram shows the Nash equilibrium results of the attacker's mixed strategies;

[0070] Figure 4 The diagram shows the Nash equilibrium result of the defender's hybrid strategy. Detailed Implementation

[0071] 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.

[0072] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0073] 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 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.

[0074] 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.

[0075] like Figure 1 As shown, a method for generating attack and defense game strategies for infrastructure networks based on multi-indicator fusion is described, the method comprising:

[0076] Step 1: Obtain the topology of the infrastructure network, with the attacker and defender being Player 1 and Player 2, respectively. Determine the strategy sets of Player 1 and Player 2, and construct the infrastructure network game model.

[0077] Step 2: Using three network performance metrics—maximum connected component size, network efficiency, and clustering coefficient—calculate the payoff matrices for Player 1 and Player 2 in the infrastructure network game under various strategy profiles.

[0078] Step 3: Based on the decision-maker's preferences, construct three pairs of membership functions and non-membership functions corresponding to the three network performance indicators: maximum connected component size, network efficiency, and clustering coefficient.

[0079] Step 4: Using three pairs of membership functions and non-membership functions, the payoff matrix described in Step 2 is converted into an intuitionistic fuzzy set payoff matrix, and then the intuitionistic fuzzy weighted set is used for sub-weighted fusion to obtain the final payoff matrix.

[0080] Step 5: Combining intuitionistic fuzzy set theory, the solution of the infrastructure network game model is transformed into the solution of a nonlinear programming problem, and the mixed strategy Nash equilibrium solution of player 1 and player 2 is obtained.

[0081] 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 infrastructure network, where N = |V| represents the number of nodes in the infrastructure network. It is the set of all edges in the infrastructure network.

[0082] Specifically, the attacker's attack 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 set of defense strategies of the defending side is S. D A defense strategy vector s of the defending side D =[y1,y2,...,y N ]∈S D y i Indicates whether the i-th infrastructure is defended, denoted by . Let i be the set of defense 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. iIt will be removed;

[0083] For the i-th node v i ,remember and These represent the attack cost for the attacker and the defense cost for the defender, respectively, node v. i The attack cost and defense cost are expressed as follows:

[0084]

[0085] Where, r i Represents node v i Structural attribute index, q A q represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's structural attribute index on the attacker's attack cost. D This represents the cost sensitivity coefficient of the defender, that is, the impact of the structural attribute index of the node on the defender's defense cost;

[0086] Define the available resources for both the attacker and defender as follows:

[0087]

[0088]

[0089] Among them, C A C represents the attacker's available resources. D Indicates the available resources of the defending side; θ A ∈[0,1], θ D ∈[0,1] are the attack budget constraint coefficient and the defense budget constraint coefficient, respectively;

[0090] Let the attacker's total attack cost be C. X ,but:

[0091]

[0092] The attacker's total attack cost is finite, therefore the following constraints must be met:

[0093]

[0094] The total defense cost for the defender is C. Y ,but:

[0095]

[0096] The strategy set includes the attacker's attack strategy set and the defender's defense strategy set. The attacker's attack strategy set includes three typical strategies, namely, the maximum attack strategy s. Amax ∈S A Minimal attack strategyAmin ∈S A Random attack strategy Arand ∈S A 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 constraint for attack. The defender's set of defense strategies includes three typical strategies: the maximum-degree defense strategy s. Dmax ∈S D Minimal defense strategy Dmax ∈S D Random defense strategy Drand ∈S D 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.

[0097] Furthermore, the structural attribute indicators are degree, betweenness, or eigenvector centrality.

[0098] Specifically, the profit matrix includes the attacker's profit matrix and the defender's profit matrix. Let i be the attacker's payoff function under network performance metric i, then This represents the attacker's gain when choosing attack strategy X and the defender's gain when choosing defense strategy Y under network performance metric i. Let represent the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y under network performance index i. Let i = 1 represent the maximum connected component size, i = 2 represent network efficiency, and i = 3 represent the clustering coefficient. Then:

[0099]

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] Where Γ(G), E(G), and H(G) represent the maximum connected component size, network efficiency, and clustering coefficient of the initial infrastructure network, respectively, and the set of all removed nodes is denoted as . The network formed after the nodes are removed is It is the set of all edges in the infrastructure network formed after nodes are removed. and Represents the maximum connected component size, network efficiency, and clustering coefficient of the infrastructure network after one round of game playing, and satisfies...

[0106] Furthermore, the membership function and non-membership function corresponding to network performance index i adopt the hyperbolic membership function μ. i (x) and non-membership function ν i The form of (x):

[0107]

[0108]

[0109] Where α represents the highest acceptable level and β represents the lowest acceptable level;

[0110] The attacker selects a set of attack strategies S A One of the attack strategies Amax ,s Amin ,s Arand ∈S A The defender selects a set of defense strategies S. D One of the defense strategies Dmax ,s Dmin ,s Drand ∈S D When the network performance metric is the maximum connected component size, i = 1, and the attacker's original gain is expressed as:

[0111]

[0112] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The largest connected segment size before the game begins. This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The maximum connected component size after the game is played is determined by transforming the attacker's original payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different strategy situations corresponding to the performance index of the largest connected component network is expressed as:

[0113]

[0114] Where, μ 1AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax Membership degree of revenue value of the largest connected slice size, ν 1AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax The revenue value of the maximum connected piece size is not a membership degree. In other symbols, μ represents membership degree and ν represents non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 1 indicates that the maximum connected piece size is used as the network performance indicator.

[0115] When the network performance metric is network efficiency, i = 2, and the attacker's original gain is expressed as:

[0116]

[0117] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D Network efficiency before the game begins. This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The network efficiency after the game is discussed, and the attacker's original payoff value is transformed into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the network efficiency and network performance indicators, is expressed as:

[0118]

[0119] Where, μ 2AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax At that time, the membership degree of the network efficiency gain value, ν 2AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. DmaxWhen the network efficiency gain value is used, the non-membership degree is used. In other symbols, μ represents membership degree and ν represents non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 2 indicates that network efficiency is used as a network performance indicator.

[0120] When the network performance metric is the clustering coefficient, i = 3, the attacker's original gain is expressed as:

[0121]

[0122] in, This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The aggregation coefficient before the game begins. This indicates that the attacker's attack strategy is s. A The defender's defense strategy is s D The aggregation coefficients after the game are used to transform the attacker's original payoff into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the clustering coefficient network performance index, is expressed as:

[0123]

[0124] Where, μ 3AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax When, the membership degree of the clustering coefficient return value, ν 3AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax When the clustering coefficient is used, it represents the degree of non-membership. In other symbols, μ represents membership, ν represents non-membership, and subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy. Subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. Subscript 3 indicates that the clustering coefficient is used as a network performance indicator.

[0125] Let a j =<μ j ,ν j>(j=1,2,…,n) is a set of intuitionistic fuzzy sets, where n represents the number of indicators. The intuitionistic fuzzy weighted set sub-calculation is defined as:

[0126]

[0127] In the formula, ω=(ω1,ω2,…,ω n ) is a j The weight vector (j = 1, 2, ..., n) satisfies the condition 0 ≤ ω j ≤1, Using Intuitive Fuzzy Weighted Set Calculation Subsidiary IFWA ω Weighted sets of M1, M2, and M3 are combined into a comprehensive intuitionistic fuzzy set payoff matrix, i.e.

[0128]

[0129] Where, μ AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax Membership degree of the comprehensive return value, ν AmaxDmax This indicates that the attack strategy is a maximum attack strategy s. Amax The defense strategy is the maximum defense strategy. Dmax The non-membership degree of the comprehensive return value is given by the symbol μ, which represents the membership degree, and ν, which represents the non-membership degree. The subscripts Amax, Amin, and Arand represent the attacker's maximum degree attack strategy, minimum degree attack strategy, and random attack strategy, respectively. The subscripts Dmax, Dmin, and Drand represent the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy, respectively.

[0130] Specifically, for the comprehensive intuitionistic fuzzy set payoff matrix, the nonlinear programming problem described in step 5 is:

[0131]

[0132] and

[0133]

[0134] Where i represents the attacking side's i-th attack strategy, with a total of m attack strategies; j represents the defending side's j-th defense strategy, with a total of n defense strategies; μ ij This represents the membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy. ijLet represent the non-membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy; λ represents the relative weights of the membership / non-membership function constraints. Once λ is determined, the resulting Nash equilibrium solution is: (p,q,<μ,ν>,<σ,ρ>); p=(p1,p2,...,p m ) T Let q = (q1, q2, ..., q...) represent the probability vector of the attacker's mixed strategy. m ) T Let <μ,ν> represent the probability vector of the defender's mixed strategy; let <σ,ρ> represent the attacker's payoff in the game problem; and let <σ,ρ> represent the defender's payoff in the game problem. Both are intuitionistic fuzzy sets.

[0135] In real life, infrastructure network structures vary widely. This embodiment uses a scale-free network structure with 300 nodes as an example. Figure 2 As shown, and assuming that both sides have limited resources;

[0136] In this embodiment, three network metrics (n=3) are considered: maximum connected component size, network efficiency, and clustering coefficient; the weights ω = (ω1, ω2, ..., ω n Depending on the decision-maker's own preferences, in this embodiment, ω1 = 0.2, ω2 = 0.2, ω3 = 0.6;

[0137] Let the attacker's attack cost be C. A The defender's defense cost is C. D For the attacker, the following constraints must be met:

[0138]

[0139] The defending side must satisfy the following constraints:

[0140]

[0141] In this embodiment, θ A With θ D For real numbers in [0.1, 0.9], iterate through them with a step size of 0.1;

[0142] Furthermore, each index is converted into an intuitionistic fuzzy set. The membership function of the largest connected component size index is as follows, where α = 0.35 and β = 0.2:

[0143]

[0144] The non-membership function is as follows, where α = 0.8 and β = 0.2:

[0145]

[0146] The membership function for the network efficiency index is as follows, where m = 0.5 and n = 0.3:

[0147]

[0148] The non-membership function is as follows, where m = 0.5 and n = 0.05:

[0149]

[0150] The membership and non-membership functions of the clustering coefficient index are as follows, where T1 = 0.3 and T2 = 0.6.

[0151]

[0152] The non-membership functions are as follows, where T1 = -0.05 and T2 = 0.5.

[0153]

[0154] Next, by integrating the above three indicators, the intuitionistic fuzzy weighted set subroutine can be defined as:

[0155]

[0156] In the formula, ω=(ω1,ω2,…,ω n ) is a j The weight vector (j = 1, 2, ..., n) satisfies the condition 0 ≤ ω j ≤1, In this embodiment, ω1 = 0.2, ω2 = 0.2, and ω3 = 0.6, thereby obtaining the payoff matrix of the comprehensive index. According to the method of this patent, the optimal strategies of the two game players are finally obtained.

[0157] Taking the maximum degree strategy as an example, the Nash equilibrium result of the attacker's hybrid strategy is as follows: Figure 3 As shown, it can be seen that as θ A and θ D As θ changes, the probability of the attacker choosing the maximum degree strategy fluctuates, where, when θ A θ is 0.9 D When the value is 0.1, the probability of choosing the maximum degree strategy is 0.96;

[0158] The Nash equilibrium result of the defender's hybrid strategy is as follows: Figure 4 As shown, it can be seen that as θ A Increase, θ D As θ decreases, the probability of choosing the maximum-degree strategy decreases, where, when θ... A θ is 0.9 DWhen the value is 0.1, the probability of choosing the maximum degree strategy is 0.02.

[0159] 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 generating attack and defense game strategies for infrastructure networks based on multi-index fusion, characterized in that, The method includes: Step 1: Obtain the topology of the infrastructure network, with the attacker and defender being Player 1 and Player 2, respectively. Determine the strategy sets of Player 1 and Player 2, and construct the infrastructure network game model. Step 2: Using three network performance metrics—maximum connected component size, network efficiency, and clustering coefficient—calculate the payoff matrices for Player 1 and Player 2 in the infrastructure network game under various strategy profiles. Step 3: Based on the decision-maker's preferences, construct three pairs of membership functions and non-membership functions corresponding to the three network performance indicators: maximum connected component size, network efficiency, and clustering coefficient. Step 4: Using three pairs of membership functions and non-membership functions, the payoff matrix described in Step 2 is converted into an intuitionistic fuzzy set payoff matrix, and then the intuitionistic fuzzy weighted set is used for sub-weighted fusion to obtain the final payoff matrix. Step 5: Combining intuitionistic fuzzy set theory, the solution of the infrastructure network game model is transformed into the solution of a nonlinear programming problem, and the mixed strategy Nash equilibrium solution of player 1 and player 2 is obtained. The infrastructure network is represented as a simple undirected graph G(V,E), where Represents the set of all nodes in the infrastructure network, where, This indicates the number of nodes in the infrastructure network. It is the set of all edges in the infrastructure network; The membership function and non-membership function corresponding to network performance index i are adopted using the hyperbolic membership function. Non-membership function Format: 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 When the network performance metric is the maximum connected component size, i=1, and the attacker's original gain is expressed as: in, This indicates that the attacker's attack strategy is... The defender's defense strategy is The largest connected segment size before the game begins. This indicates that the attacker's attack strategy is... The defender's defense strategy is The maximum connected component size after the game is played is determined by transforming the attacker's original payoff into an intuitionistic fuzzy set using a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different strategic situations corresponding to the performance index of the largest connected component network is expressed as: in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Membership degree of revenue value based on the largest connected area size. This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. The revenue value of the largest connected slice size is not a membership degree, and other symbols are not included. 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 indicates the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 1 indicates that the network performance index is based on the maximum connected slice size. When the network performance metric is network efficiency, i=2, and the attacker's original gain is expressed as: in, This indicates that the attacker's attack strategy is... The defender's defense strategy is Network efficiency before the game begins. This indicates that the attacker's attack strategy is... The defender's defense strategy is The network efficiency after the game is discussed, and the attacker's original payoff value is transformed into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the network efficiency and network performance indicators, is expressed as: in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. At that time, the membership degree of the network efficiency gain value. This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. At that time, the non-membership degree of the network efficiency gain value, and other symbols. 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 indicates the defender's maximum-degree defense strategy, minimum-degree defense strategy, and random defense strategy. The subscript 2 indicates that network efficiency is used as a network performance indicator. When the network performance metric is the clustering coefficient, i=3, the attacker's original gain is expressed as: in, This indicates that the attacker's attack strategy is... The defender's defense strategy is The aggregation coefficient before the game begins. This indicates that the attacker's attack strategy is... The defender's defense strategy is The aggregation coefficients after the game are used to transform the attacker's original payoff into an intuitionistic fuzzy set through a hyperbolic membership function. The defenders also suffered losses. Therefore, the intuitionistic fuzzy set payoff matrix of the attacker under different pure strategy situations, corresponding to the clustering coefficient network performance index, is expressed as: in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. At that time, the membership degree of the clustering coefficient return value, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. At that time, the non-membership degree of the clustering coefficient return value, and other symbols. 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 indicates the defender's maximum degree defense strategy, minimum degree defense strategy, and random defense strategy. The subscript 3 indicates that the clustering coefficient is used as a network performance indicator. set up For a set of intuitive fuzzy sets, The intuitionistic fuzzy weighted set subroutine, representing the number of indicators, is defined as follows: In the formula, for The weight vector satisfies the condition , ; Use intuition-based fuzzy weighted set to calculate the numerator Will , , The weighted set is aggregated into a comprehensive intuitionistic fuzzy set payoff matrix, i.e. in, This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Membership degree of the overall return value at that time This indicates that the attack strategy is a maximum attack strategy. The defense strategy is a maximum defense strategy. Non-membership degree of the comprehensive return value, other symbols 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 indicates the defender's maximum defense strategy, minimum defense strategy, and random defense strategy.

2. The method for generating attack and defense game strategies for infrastructure networks based on multi-index fusion 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 attacked node If both exist and That is, if a node is attacked but not protected, then... It will be removed; For the 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 Structural attribute indicators, This represents the attacker's cost sensitivity coefficient, i.e., the impact of the node's structural attribute indicators on the attacker's attack costs. This represents the cost sensitivity coefficient of the defender, i.e., the impact of the structural attribute index of the node on the defender's defense cost; Define the available resources for both the attacker and defender as follows: in, This indicates the attacker's available resources. Indicates the available resources of the defending side; , These are the attack budget constraint coefficient and the defense budget constraint coefficient, respectively. Let the attacker's total attack cost be... ,but: The attacker's total attack cost is finite, therefore the following constraints must be met: The total defense cost for the defender is ,but: The strategy set includes the attacker's attack strategy set and the defender's defense strategy set. The attacker's attack strategy set includes three typical strategies: maximum attack strategy, maximum attack strategy, and 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 attack and defense game strategies for infrastructure networks based on multi-index fusion according to claim 2, characterized in that, The structural attribute indicators mentioned are degree, betweenness, or eigenvector centrality.

4. The method for generating attack and defense game strategies for infrastructure networks based on multi-index fusion according to claim 2, characterized in that, The aforementioned profit matrix includes the attacker's profit matrix and the defender's profit matrix. Let i be the attacker's payoff function under network performance metric i, then This represents the attacker's gain when choosing attack strategy X and the defender's gain when choosing defense strategy Y under network performance metric i. Let represent the defender's gain when the attacker chooses attack strategy X and the defender chooses defense strategy Y under network performance index i. Let i=1 represent the maximum connected component size, i=2 represent network efficiency, and i=3 represent clustering coefficient. Then: in, , and Let represent the maximum connected component size, network efficiency, and clustering coefficient of the initial infrastructure network, respectively. 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 infrastructure network formed after nodes are removed. , and Represents the maximum connected component size, network efficiency, and clustering coefficient of the infrastructure network after one round of game playing, and satisfies... , , .

5. The method for generating attack and defense game strategies for infrastructure networks based on multi-index fusion according to claim 1, characterized in that, For the comprehensive intuitionistic fuzzy set payoff matrix, the nonlinear programming problem described in step 5 is: and in, Indicates the attacker's... There are 10 attack strategies, totaling 1 One attack strategy; The first one represents the defending side One defense strategy, total One defense strategy; This represents the membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy. Let represent the non-membership function value in the attacker's intuitionistic fuzzy set payoff matrix when the attacker chooses the i-th attack strategy and the defender chooses the j-th defense strategy; λ represents the relative weights of the membership / non-membership function constraints. Once λ is determined, the resulting Nash equilibrium solution is: ; This represents the probability vector of the attacker's mixed strategy. This represents the probability vector of the defender's hybrid strategy; This represents the payoff value for the attacker in a game theory problem. The values ​​represent the payoffs of the defender in a game problem, and are all in the form of intuitionistic fuzzy sets.

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