A Method for Generating Optimization Strategies in Network Game Theory Based on a Hybrid Node and Edge Strategy

Abstract

This invention discloses a method for generating network game optimization strategies based on a hybrid node and edge strategy. The method includes: acquiring the network topology, determining the strategy sets of the attacker and defender, and constructing a basic model of the network game; calculating the attack and defense costs of nodes and edges in the network based on the network topology and network metrics, and determining the cost constraints of the attacker and defender; considering the hybrid attack and defense strategies of nodes and edges, obtaining the strategy sets of the attacker and defender; using the maximum connected component size index to represent network performance, calculating the payoffs of the attacker and defender under each strategy profile, and obtaining the payoff matrix of the network game model; and converting the basic model of the network game into a linear programming problem for solution, obtaining the hybrid strategy Nash equilibrium solution of the attacker and defender.

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 network game optimization strategies based on a hybrid strategy of nodes and edges. Background Technology

[0002] Critical infrastructure networks are the backbone of modern society, functioning like a blood transfusion network to maintain the normal operation of all parts of society. Once these critical infrastructures fail, are attacked, or malfunction, the impact extends beyond specific sectors, potentially affecting the entire society and posing a serious threat to security, the economy, and social stability. Therefore, effectively preventing deliberate attacks on critical infrastructure networks with limited resources has always been a key concern for security agencies worldwide.

[0003] To address the threats of random failures and deliberate attacks on critical infrastructure networks, researchers have attempted to analyze these threats using various theoretical frameworks, with game theory, as a mathematical theory and method for studying competitive problems, gaining favor among researchers.

[0004] Existing research has yielded significant results in attack-defense game theory studies targeting critical infrastructure networks, using nodes or edges as attack and defense objectives. However, in reality, nodes and edges in critical infrastructure networks typically face the same failure risk. Existing research rarely addresses hybrid attack strategies involving nodes and edges, nor does it integrate non-uniform costs with these hybrid attack-defense strategies. Therefore, this paper analyzes the impact of hybrid attack-defense strategies involving nodes and edges on the equilibrium outcome under non-uniform cost conditions, and derives the optimal strategy choices for both attackers and defenders under different cost constraint coefficients and resource allocation ratios. Summary of the Invention

[0005] This invention aims to at least solve one of the technical problems existing in the prior art. To this end, this invention discloses a method for generating network game optimization strategies based on a hybrid strategy of nodes and edges. The method is based on a two-player zero-sum game, considers the hybrid strategy of nodes and edges, and combines resource constraints for modeling. The model is then transformed into a linear programming problem for solution, yielding a hybrid strategy Nash equilibrium solution for both the attacker and defender.

[0006] The objective of this invention is achieved through the following technical solution: a method for generating network game optimization strategies based on a hybrid node and edge strategy, the method comprising:

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

[0008] Step 2: Based on the network topology and network metrics, calculate the attack and defense costs of nodes and edges in the network, and determine the cost constraints for the attacker and defender.

[0009] Step 3: Consider the hybrid attack and defense strategies of nodes and edges to obtain the attacking side's strategy set and the defending side's strategy set.

[0010] Step 4: Using the maximum connected component size as the index to represent network performance, calculate the payoffs of the attacker and defender under each policy profile to obtain the payoff matrix of the network game model.

[0011] Step 5: Replace the basic model of the network game with a linear programming problem to obtain the mixed strategy Nash equilibrium solution for the attacker and defender.

[0012] The network is represented as a simple undirected graph G(V,E), where V represents the set of all nodes in the network, N = |V| represents the number of nodes in the network, and E represents the set of connections.

[0013] Specifically, the attack and defense cost of a node is measured by its degree value, which determines the node's importance and, consequently, its attack and defense cost. The attack and defense costs of a node are as follows:

[0014]

[0015] Among them, Y i ND and Y i NA They represent nodes v respectively i The cost of defense versus the cost of attack, k i Representative node v i The degree of θ∈(-∞,∞) is the node cost sensitivity coefficient. When θ<0, the attack and defense cost of a node is inversely proportional to the importance of the node; when θ=0, the attack and defense cost of a node is unrelated to the importance of the node, that is, the attack and defense cost of all nodes is the same and is 1. A considerable number of experiments on complex network attack and defense games are conducted under this condition [3,6]; when θ>0, the attack and defense cost of a node is directly proportional to the importance of the node, which also conforms to most real-world situations.

[0016] The attack and defense costs of the aforementioned edge are:

[0017]

[0018] in, and Representing edge e respectively ij The cost of defense versus the cost of attack, v i and v j For edge e ij The two endpoints, ki and k j Ω represents the degree at both endpoints. i Indicates the relationship with node v i There exists a set of nodes connected by edges, Ω j Indicates the relationship with node v j There exists a set of nodes connected by edges, where 'a' represents the node connected to node 'v'. j Connected nodes, b represents the node v i Connected nodes.

[0019] Specifically, the cost constraints for both attackers and defenders refer to the fact that both sides have limited resources and can only protect or attack a subset of nodes and edges in the critical infrastructure network. Based on the definition of the attack and defense costs for nodes and edges, we can derive the following equation:

[0020]

[0021] Given a total available resource quantity of C, the attacker and defender can achieve a complete attack or defense on a node or edge in one go. Therefore, the attack resources available to the attacker are:

[0022] C A =α A C

[0023] Where, α A ∈[0,1] represents the attack cost constraint coefficient for the attacker; similarly, the available defense resources for the defender are:

[0024] C D =α D C

[0025] Where, α D ∈[0,1] represents the defense cost constraint coefficient of the defender.

[0026] Furthermore, the hybrid attack and defense strategy of nodes and edges refers to the fact that both the attacker and the defender can simultaneously choose nodes and edges as game strategies. The attacker's attack strategy is defined as follows:

[0027]

[0028] And it satisfies:

[0029]

[0030] Where vector X represents an attack strategy of the attacker, and S A Let N be the set of all attack strategies, and C be the total number of nodes in the critical infrastructure network. A For the attack resources available to the attacker, if the attacker chooses to attack node v i,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise,

[0031] The defender's defense strategy is defined as follows:

[0032]

[0033] And it satisfies:

[0034]

[0035] Where vector Y represents a defense strategy of the defender, and S D C is the set of all defense strategies. D For the defender's available defense resources, if the defender selects defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise,

[0036] Furthermore, the aforementioned profit matrix is ​​divided into the attacker's profit matrix and the defender's profit matrix, U A (X,Y) represents the attacker's gain, U D (X,Y) represents the defender's gains:

[0037]

[0038] Define U A (X,Y) is the attacker's payoff function, where X is the attacker's chosen attack strategy and Y is the defender's chosen defense strategy. The calculation formula is as follows:

[0039]

[0040] Similarly, define U D (X,Y) is the payoff function for the defender, calculated as follows:

[0041]

[0042] Where Γ is the network performance evaluation function, which monotonically does not increase as the number of network nodes decreases, and G is the initial target network. For the target network after a round of attack and defense, in critical infrastructure networks, the connectivity between nodes is a key factor in whether the critical infrastructure network can function properly. Therefore, in this paper, we choose the maximum connected component size as the network performance evaluation function to reflect the loss of network performance. At this point, the attacker's payoff function is:

[0043]

[0044] The defender's payoff function is:

[0045]

[0046] Among them, L max The maximum connected component size of the initial network. This represents the largest interconnected segment size of the network after a round of competition. C represents the network connectivity scale after one round of game play. i denoted by , where n represents the size of the network connectivity slice before the game, and n represents the total number of network nodes.

[0047] Specifically, the linear programming problem in step 5 is:

[0048] For the attacker, the goal is to destroy critical infrastructure network nodes and edges with a limited number of available resources, thereby minimizing the connectivity of the critical infrastructure network. Therefore, the optimization problem is defined as follows:

[0049]

[0050] Where max represents maximization, U A (X,Y) represents the attacker's gain, ∑ represents the summation symbol, and C A For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise,

[0051] For the defender, the goal is to effectively protect critical infrastructure network nodes and edges with a limited number of available resources, and maintain the normal functioning of the critical infrastructure network. Therefore, its optimization problem is defined as follows:

[0052]

[0053] Where max represents maximization, U D (X,Y) represents the attacker's gain, ∑ represents the summation symbol, and C D For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise, By solving the two optimization problems mentioned above, we can obtain the optimal strategy choices for both the attacker and defender in the game.

[0054] Compared with existing methods, the advantages of this invention are as follows: This invention establishes a critical infrastructure network attack and defense game model with a hybrid attack and defense strategy of nodes and edges under non-uniform cost conditions. The game model of this invention considers more realistic conditions, can better provide reference for relevant departments, and expands the research ideas for critical infrastructure network attack and defense game models. Attached Figure Description

[0055] Figure 1 A flowchart illustrating an embodiment of the present invention is shown;

[0056] Figure 2 A schematic diagram of the network topology in an embodiment of the present invention is shown;

[0057] Figure 3 A schematic diagram of a Nash equilibrium solution under specified resource constraints according to an embodiment of the present invention is shown.

[0058] Figure 4 A schematic diagram illustrating resource allocation according to an embodiment of the present invention is shown. Detailed Implementation

[0059] 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.

[0060] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0061] In this embodiment, the attacker is the attacker and the defender is the defender. Taking the railway network as an example, the critical infrastructure 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, N = |V| represents the number of nodes in the network, and a node represents a station in the railway network; Denotes the set concatenation, A(G) = (a ij ) N×N Let the adjacency matrix of graph G be such that if node V i and V j If there is a railway line between them, then a ij =a ji =1, if node V i and V j If there is no railway line between them, then a ij =a ji =0.

[0062] This game involves 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 game consists of only one round.

[0063] like Figure 1 As shown, a network game strategy generation method based on node average path constraints is described, the method comprising:

[0064] Step 1: Obtain the network topology, determine the strategy sets of the attacker and defender, and construct the basic model of network game.

[0065] Step 2: Based on the network topology and network metrics, calculate the attack and defense costs of nodes and edges in the network, and determine the cost constraints for the attacker and defender.

[0066] Step 3: Consider the hybrid attack and defense strategies of nodes and edges to obtain the attacking side's strategy set and the defending side's strategy set.

[0067] Step 4: Using the maximum connected component size as the index to represent network performance, calculate the payoffs of the attacker and defender under each policy profile to obtain the payoff matrix of the network game model.

[0068] Step 5: Replace the basic model of the network game with a linear programming problem to obtain the mixed strategy Nash equilibrium solution for the attacker and defender.

[0069] The network is represented as a simple undirected graph G(V,E), where V represents the set of all nodes in the network, N = |V| represents the number of nodes in the network, and E represents the set of connections.

[0070] Specifically, the attack and defense cost of a node is measured by its degree value, which determines the node's importance and, consequently, its attack and defense cost. The attack and defense costs of a node are as follows:

[0071]

[0072] Among them, Y i ND and Y i NA They represent nodes v respectively i The cost of defense versus the cost of attack, k i Representative node v i The degree of θ∈(-∞,∞) is the node cost sensitivity coefficient. When θ<0, the attack and defense cost of a node is inversely proportional to the importance of the node; when θ=0, the attack and defense cost of a node is unrelated to the importance of the node, that is, the attack and defense cost of all nodes is the same and is 1. A considerable number of experiments on complex network attack and defense games are conducted under this condition [3,6]; when θ>0, the attack and defense cost of a node is directly proportional to the importance of the node, which also conforms to most real-world situations.

[0073] The attack and defense costs of the aforementioned edge are:

[0074]

[0075] in, and Representing edge e respectively ij The cost of defense versus the cost of attack, v i and v j For edge e ij The two endpoints, k i and k j Ω represents the degree at both endpoints. i Indicates the relationship with node v i There exists a set of nodes connected by edges, Ω. j Indicates the relationship with node v j There exists a set of nodes connected by edges;

[0076] Specifically, the cost constraints for both attackers and defenders refer to the fact that both sides have limited resources and can only protect or attack a subset of nodes and edges in the critical infrastructure network. Based on the definition of the attack and defense costs for nodes and edges, we can derive the following equation:

[0077]

[0078] Given a total available resource quantity of C, the attacker and defender can achieve a complete attack or defense on a node or edge in one go. Therefore, the attack resources available to the attacker are:

[0079] C A =α A C

[0080] Where, α A ∈[0,1] represents the attack cost constraint coefficient for the attacker; similarly, the available defense resources for the defender are:

[0081] C D =α D C

[0082] Where, α D ∈[0,1] represents the defense cost constraint coefficient of the defender.

[0083] Furthermore, the hybrid attack and defense strategy of nodes and edges refers to the fact that both the attacker and the defender can simultaneously choose nodes and edges as game strategies. The attacker's attack strategy is defined as follows:

[0084]

[0085] And it satisfies:

[0086]

[0087] Where vector X represents an attack strategy of the attacker, and S A Let N be the set of all attack strategies, and C be the total number of nodes in the critical infrastructure network. A For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise,

[0088] The defender's defense strategy is defined as follows:

[0089]

[0090] And it satisfies:

[0091]

[0092] Where vector Y represents a defense strategy of the defender, and S D C is the set of all defense strategies. D For the defender's available defense resources, if the defender selects defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise,

[0093] Furthermore, the aforementioned profit matrix is ​​divided into the attacker's profit matrix and the defender's profit matrix, U A (X,Y) represents the attacker's gain, U D (X,Y) represents the defender's gains:

[0094]

[0095] Define U A (X,Y) is the attacker's payoff function, where X is the attacker's chosen attack strategy and Y is the defender's chosen defense strategy. The calculation formula is as follows:

[0096]

[0097] Similarly, define U D (X,Y) is the payoff function for the defender, calculated as follows:

[0098]

[0099] Where Γ is the network performance evaluation function, which monotonically does not increase as the number of network nodes decreases, and G is the initial target network. For the target network after a round of attack and defense, in critical infrastructure networks, the connectivity between nodes is a key factor in whether the critical infrastructure network can function properly. Therefore, in this paper, we choose the maximum connected component size as the network performance evaluation function to reflect the loss of network performance. At this point, the attacker's payoff function is:

[0100]

[0101] The defender's payoff function is:

[0102]

[0103] Among them, L maxThe maximum connected component size of the initial network. This represents the largest connected segment size of the network after one round of competition.

[0104] Specifically, the linear programming problem in step 5 is:

[0105] For the attacker, the goal is to destroy critical infrastructure network nodes and edges with a limited number of available resources, thereby minimizing the connectivity of the critical infrastructure network. Therefore, the optimization problem is defined as follows:

[0106]

[0107] Where max represents maximization, U A (X,Y) represents the attacker's gain, ∑ represents the summation symbol, and C A For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise,

[0108] For the defender, the goal is to effectively protect critical infrastructure network nodes and edges with a limited number of available resources, and maintain the normal functioning of the critical infrastructure network. Therefore, its optimization problem is defined as follows:

[0109]

[0110] Where max represents maximization, U D (X,Y) represents the attacker's gain, ∑ represents the summation symbol, and C D For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge eij ,So otherwise, By solving the two optimization problems mentioned above, we can obtain the optimal strategy choices for both the attacker and defender in the game.

[0111] In real life, infrastructure network structures vary widely. This experiment uses a 14-node network structure as an example. Figure 2 As shown, this target network has 14 nodes and 16 edges, with a total available resource C = 32. The average attack / defense cost of a node is approximately 2.29, and the average attack / defense cost of an edge is approximately 2. In the experiment, α is set... A =α D =0.3, and satisfies the following conditions:

[0112]

[0113] The results obtained after solving the problem are shown in the table below.

[0114] Attack strategy probability Defense strategy probability {v4,v7,e5-6} 0.039 {v4,v7,e5-6} 0.211 {v7,e4-7,e13-14} 0.009 {v7,e4-7,e11-12} 0.154 {v1,v7,v9,e5-6} 0.154 {v7,e4-7,e13-14} 0.057 {v2,v10,v13,e5-6} 0.009 {v2,v14,e6-7,e9-10} 0.050 {v2,v13,e6-7,e9-10} 0.249 {v4,v6,e2-4,e7-13} 0.087 {v6,v14,e2-4,e7-13} 0.194 {v4,v6,e4-7,e6-10} 0.012 {v4,v5,v8,e4-7,e6-10} 0.011 {v5,v6,e2-4,e7-13} 0.117 {v4,v8,v10,v11,e5-6} 0.097 {v5,v6,e6-7,e9-10} 0.011 {v4,v8,e4-7,e6-7,e13-14} 0.007 {v6,e4-7,e6-7,e13-14} 0.039 {v4,e5-6,e4-7,e7-8,e6-10} 0.008 {v2,v4,v8,e4-7,e13-14} 0.006 {v5,v8,v11,e4-7,e6-10} 0.118 {v4,v6,v8,e4-7,e13-14} 0.004 {v8,v9,v12,e2-4,e7-13} 0.050 {v4,v8,e4-7,e6-7,e13-14} 0.156 {v3,v9,v13,e4-7,e7-8,e12-14} 0.055 {v6,v8,v9,e6-7,e9-10} 0.096

[0115] Analysis revealed that the attacker chose the {v2,v13,e6-7,e9-10} and {v6,v14,e2-4,e7-13} strategies with probabilities of 0.249 and 0.194, respectively. The attack costs for nodes 2, 13, 6, and 14 are 3, 2, 3, and 2, respectively, placing them among the nodes with medium attack costs in the network. Furthermore, edges e6-7 and e7-13 are directly connected to the central node 7, indicating a strong preference for nodes with medium degrees and edges directly connected to critical nodes. Conversely, the defender chose the {v4,v7,e5-6} strategy with a probability of 0.211. The attack cost for node 7 is 6, making it a critical node in the network. This suggests that the defender prioritizes protecting critical nodes and does not show a significant preference in edge selection.

[0116] Further calculations revealed that when the attacker and defender have the same number of resources, in equilibrium, the attacker's resource allocation for nodes and edges is 0.59:0.41, while the defender's allocation is 0.48:0.52. Clearly, the attacker's resource allocation is more balanced, while the defender allocates more resources to nodes. To investigate the impact of different resource allocation ratios on the equilibrium outcome, we designed the following experiment: Let the attacker's cost constraint coefficient α... A ∈{0.15,0.25,0.35,0.45}, the cost constraint coefficient α of the defender. D ∈{0.15,0.25,0.35,0.45}. Meanwhile, the resource allocation ratio θ between the attacker and defender for nodes and edges is... A ,θD ∈[0.1,0.9], for example, when θ A When θ = 0.3, it means the attacker uses 30% of their attack resources to attack nodes and the remaining 70% to attack edges. D Related explanations and θ A The same. Solving for the Nash equilibrium value under different conditions yields the following results: Figure 3 As shown.

[0117] To further study the characteristics of attack and defense resource allocation under different resource constraints, we map the hybrid strategy Nash equilibrium into a value that reflects whether attack and defense resources are evenly distributed across nodes and edges, according to the following formula.

[0118]

[0119] in, For a mixed-strategy Nash equilibrium, the probability θ is not zero. A value, The corresponding probability; For a mixed-strategy Nash equilibrium, the probability θ is not zero. D value, This represents the corresponding probability. Therefore, T A or T D A larger value indicates a more uneven distribution of resources among nodes and edges, a strategy known as resource concentration; conversely, a smaller value indicates a more even distribution of resources among nodes and edges, a strategy known as resource equalization. We calculate T. A -T D The value is obtained as follows Figure 4 The results are shown.

[0120] Analysis of the results reveals that when the attacker's resources are not superior—that is, when the attacker has few resources or is relatively inferior to the defender's resources—they tend to adopt a resource concentration strategy. In military strategy, when facing a superior enemy force, one should concentrate superior forces to defeat the enemy piecemeal; this aligns with the above conclusion. For the defender, when defensive resources are far less than attacking resources, they tend to adopt a resource concentration strategy to defend key components in the critical infrastructure network. However, in most cases, the defender employs a resource averaging strategy to defend as many components in the critical infrastructure network as possible.

[0121] 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 network game optimization strategies based on a hybrid node and edge strategy, characterized in that, The method includes: Step 1: Obtain the network topology, determine the strategy sets of the attacker and defender, and construct the basic model of network game. Step 2: Based on the network topology and network metrics, calculate the attack and defense costs of nodes and edges in the network, and determine the cost constraints for the attacker and defender. Step 3: Consider the hybrid attack and defense strategies of nodes and edges to obtain the attacking side's strategy set and the defending side's strategy set. Step 4: Using the maximum connected component size as the index to represent network performance, calculate the payoffs of the attacker and defender under each policy profile to obtain the payoff matrix of the network game model. Step 5: Replace the basic model of the network game with a linear programming problem to obtain the mixed strategy Nash equilibrium solution for the attacker and defender. The network in question is a critical infrastructure network, represented as a simple undirected graph G(V,E), where V represents the set of all nodes in the network, N = |V| represents the number of nodes in the network, and E represents the set of connections. The hybrid attack and defense strategy of nodes and edges refers to the fact that both the attacker and the defender can simultaneously choose nodes and edges as game strategies. The attacker's attack strategy is defined as follows: And it satisfies: Where vector X represents an attack strategy of the attacker, and S A Let N be the set of all attack strategies, and C be the total number of nodes in the critical infrastructure network. A For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, The defender's defense strategy is defined as follows: And it satisfies: Where vector Y represents a defense strategy of the defender, and S D C is the set of all defense strategies. D For the defender's available defense resources, if the defender selects defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise, The attack and defense cost of a node is determined by measuring its importance based on its degree value, which in turn determines the attack and defense cost of that node. The attack and defense costs of a node are as follows: Among them, Y i ND and Y i NA Representing node v respectively i The cost of defense versus the cost of attack, k i Representative node v i The degree of θ∈(-∞,∞) is the node cost sensitivity coefficient. When θ<0, the attack and defense cost of a node is inversely proportional to the importance of the node; when θ=0, the attack and defense cost of a node is independent of the importance of the node, that is, the attack and defense cost of all nodes is the same and is 1; when θ>0, the attack and defense cost of a node is directly proportional to the importance of the node. The attack and defense costs of the aforementioned edge are: in, and Representing edge e respectively ij The cost of defense versus the cost of attack, v i and v j For edge e ij The two endpoints, k i and k j Ω represents the degree at both endpoints. i Indicates the relationship with node v i There exists a set of nodes connected by edges, Ω. j Indicates the relationship with node v j There exists a set of nodes connected by edges, where 'a' represents the node connected to node 'v'. j Connected nodes, b represents the node v i Connected nodes.

2. The method for generating network game optimization strategies based on a hybrid node and edge strategy according to claim 1, characterized in that, The cost constraints for both attackers and defenders refer to the fact that both sides have limited resources and can only protect or attack a subset of nodes and edges in the critical infrastructure network. Based on the definition of the attack and defense costs for nodes and edges, the following equation is obtained: In other words, given a total of C available resources, the attacker and defender can achieve a complete attack or defense on a node or edge in one go. Therefore, the attack resources available to the attacker are: C A =α A C Where, α A ∈[0,1] represents the attack cost constraint coefficient for the attacker; similarly, the available defense resources for the defender are: C D =α D C Where, α D ∈[0,1] represents the defense cost constraint coefficient of the defender.

3. The method for generating network game optimization strategies based on a hybrid node and edge strategy according to claim 2, characterized in that, The aforementioned profit matrix is ​​divided into the attacker's profit matrix and the defender's profit matrix, U A (X,Y) represents the attacker's gain, U D (X,Y) represents the defender's gains: Define U A (X,Y) is the attacker's payoff function, where X is the attacker's chosen attack strategy and Y is the defender's chosen defense strategy. The calculation formula is as follows: Similarly, define U D (X,Y) is the payoff function for the defender, calculated as follows: Where Γ is the network performance evaluation function, which monotonically does not increase as the number of network nodes decreases, and G is the initial target network. For the target network after one round of attack and defense, the largest connected component size is chosen as the network performance evaluation function to reflect the loss of network performance. At this point, the attacker's payoff function is: The defender's payoff function is: Among them, L max The maximum connected component size of the initial network. This represents the largest interconnected segment size of the network after a round of competition. C represents the network connectivity scale after one round of game play. i This indicates the size of the network connectivity segment before the game.

4. The method for generating network game optimization strategies based on a hybrid node and edge strategy according to claim 1, characterized in that, The linear programming problem in step 5 is: For the attacker, the goal is to destroy critical infrastructure network nodes and edges with a limited number of available resources, thereby minimizing the connectivity of the critical infrastructure network. Therefore, the optimization problem is defined as follows: Where max represents maximization, U A (X,Y) represents the attacker's gain, Σ represents the summation symbol, and C A For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise, For the defender, the goal is to effectively protect critical infrastructure network nodes and edges with a limited number of available resources, and maintain the normal functioning of the critical infrastructure network. Therefore, its optimization problem is defined as follows: Where max represents maximization, U D (X,Y) represents the attacker's gain, Σ represents the summation symbol, and C D For the attack resources available to the attacker, if the attacker chooses to attack node v i ,So otherwise, Similarly, if the attacker chooses to attack edge e ij ,So otherwise, If the defender selects the defense node v i ,So otherwise, Similarly, if the defending side chooses to defend edge e ij ,So otherwise, By solving the two optimization problems mentioned above, we can obtain the optimal strategy choices for both the attacker and defender in the game.

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