A method for constructing a network attack and defense game model based on node importance

Abstract

The present invention provides a method for constructing a network attack and defense game model based on node importance. The method adopts multi-attribute decision-making theory to identify key nodes, and obtains a comprehensive node evaluation index by using an ideal solution distance method of a set entropy weight method through four commonly used node evaluation indexes. The index is used in an attack and defense model of a complex network, and defensive resources are deployed by ranking the importance of the nodes to achieve the purpose of network defense. Based on game theory, an attack and defense game model in a complex network is established. Key nodes can be found through the proposed network attack and defense game model based on node importance, and resources are invested in nodes with high importance. The optimal resource deployment is found through the solution of game theory, thereby achieving effective resource utilization and network protection.

Description

Technical Field

[0001] The present invention relates to the field of network vulnerability and security game, and in particular to a method for constructing a network attack and defense game model based on node importance. Background Art

[0002] Critical infrastructure systems, such as communications, power, railways, and energy systems, play a vital role in maintaining social stability. Damage to these facilities can severely impact the safety of life and property. Consequently, the protection of critical infrastructure networks has become a hot topic in the field of system security in recent years. Critical infrastructure typically comprises multiple system components and exhibits a distinct networked structure. Current research primarily analyzes these structures by abstracting them into complex networks. The heterogeneous topology of complex networks results in significant variation in the importance of nodes within the network, significantly impacting the network's structure and functionality. When these nodes fail, their impact rapidly spreads throughout the entire network. Therefore, accurately quantifying the importance of nodes within complex networks and identifying critical nodes is of great theoretical and practical significance. Furthermore, there is an urgent need for methods to identify critical nodes within a network and proactively invest resources to mitigate performance degradation during attacks. Summary of the Invention

[0003] In order to comprehensively solve the above problems, especially to address the shortcomings of the existing technology, the present invention provides a method for constructing a network attack and defense game model based on node importance, which can comprehensively solve the above problems.

[0004] To achieve the above objectives, the present invention adopts the following technical means:

[0005] A method for constructing a network attack and defense game model based on node importance includes the following steps:

[0006] Step S1: Modeling the attack and defense model of complex networks:

[0007] The network infrastructure system is represented by a simple undirected graph G(V,E), where V is a set of nodes, is an edge set;

[0008] The number of nodes |V| is represented by n, assuming A(G)=(a ij ) N×N is the adjacency matrix of G, if node v i and v j adjacent, then there is a ij =a ji =1, otherwise a ij =a ji =0;

[0009] Node v i The degree is is equal to the number of edges connected to it;

[0010] Step S2: Identification of key nodes:

[0011] The initial evaluation matrix is obtained by using four node evaluation indicators: degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality. The importance of the node is calculated using the ideal distance method based on the entropy weight method to obtain a comprehensive evaluation indicator.

[0012] The comprehensive evaluation index is used in the attack and defense model of complex networks, and defensive resources are deployed based on the importance ranking of nodes to achieve the goal of network defense.

[0013] Step S3: Modeling the attack and defense game model of complex networks:

[0014] Based on game theory, an attack and defense game model of complex networks is established. The attack and defense game model of complex networks includes the basic assumption model, cost model, strategy model and benefit model of the model.

[0015] A further preferred solution of the present invention: the degree centrality of the node in step S2 represents the direct influence of the node, which is expressed as follows:

[0016]

[0017] Among them, k i is the degree of node i;

[0018] For a connected network with n nodes, we can calculate the v of any node i The average shortest distance to other nodes in the network:

[0019]

[0020] A further preferred embodiment of the present invention: the betweenness centrality of the node in step S2 generally refers to the shortest path betweenness centrality, which describes the control power of the node on the network flow transmitted along the shortest path in the network; the node v i The betweenness of is defined as:

[0021]

[0022] Where N is the number of nodes in the network, σ st (i) represents the number of shortest paths from node s to node t through node i, σ st represents the total number of shortest paths from node s to node t.

[0023] A further preferred solution of the present invention is that the closeness centrality of the node in step S2 is calculated by averaging the distances between the node and all other nodes in the network to eliminate interference from special values. The smaller the average distance between a node and other nodes in the network, the greater the closeness centrality of the node. The closeness centrality of the node is expressed as:

[0024]

[0025] A further preferred embodiment of the present invention is as follows: the eigenvector centrality of the node in step S2 considers the linear relationship between the centrality index of a node and the centrality indexes of other surrounding nodes, which is the linear superposition of the centrality values of its adjacent nodes. The larger the eigenvector centrality value of the node, the more important the node. The eigenvector centrality measure is defined as follows:

[0026]

[0027] Where EC(i) represents the eigenvector centrality of node i, λ is a constant, and A ij is the adjacency matrix of the network, x j is the value of the jth item of the normalized largest eigenvector.

[0028] A further preferred embodiment of the present invention is as follows: the initial evaluation matrix in step S2 is:

[0029]

[0030] There are m evaluation indicators and n evaluation objects. To study the importance of nodes in complex networks, the evaluation objects are n nodes in the network, and the evaluation indicator is degree centrality x i1 , eigenvector centrality x i2 , betweenness centrality x i3 , closeness centrality x i4 ;

[0031] Normalized to:

[0032]

[0033] The normalized evaluation matrix is:

[0034]

[0035] Determine the best and worst solutions:

[0036] The optimal solution (consisting of the maximum value of each column in Z):

[0037]

[0038] The worst solution (consisting of the minimum value of each column in Z):

[0039]

[0040] According to the information entropy theory, the smaller the information entropy of an indicator is, the more information it contributes, and the greater its weight is. Therefore, the information entropy value of an indicator is set to E. j (0≤E j ≤1):

[0041]

[0042] The weights are:

[0043]

[0044] where ω j is the weight (importance) of the jth attribute;

[0045] Calculate the degree of proximity between each evaluation index object and the optimal solution and the worst solution:

[0046]

[0047] Calculate the closeness C between the evaluation object and the optimal solution i :

[0048]

[0049] Where 0≤C i ≤1, C i →1 indicates that the evaluation object is better.

[0050] A further preferred embodiment of the present invention is as follows: the basic assumption model in step S3 is:

[0051] Only one attacker and one defender are considered, and both attacker and defender have complete information about the network;

[0052] All attack and defense actions are targeted at nodes in the network. When a node is successfully attacked, the edges connected to the node are also removed.

[0053] When a node is attacked by an attacker but not protected by a defender, it is considered that the node has been successfully attacked and is removed from the network. When a node is protected by a defender, the attacker's attack on the node is considered invalid and the node still functions in the network.

[0054] A further preferred embodiment of the present invention is as follows: the cost model in step S3 is:

[0055] The attacker's goal is to maximize network damage by attacking nodes in the network. The defender minimizes resource consumption by investing resources in protecting nodes. However, whether attacking or defending, taking measures against nodes will inevitably consume certain resources or pay a certain price, namely:

[0056]

[0057] where r i ≥0, r i Represents the reference attributes of the node (degree, betweenness, etc.), r i is the importance of the node, Represents node v i The attack cost and defense cost of A ,q D (>0) indicates attack (defense) cost-sensitive parameter;

[0058] In actual scenarios, the use of resources is not unlimited. In order to achieve effective use of resources and accurate and efficient attack (defense) of nodes, it is defined as:

[0059]

[0060] Where α∈[0,1] is the attack cost constraint parameter, and β∈[0,1] is the defense cost constraint parameter. Parameters α and β represent how much resources a player can invest in their actions. As α increases, the attacker can attack more targets when adopting the same attack strategy. In the extreme case of α=1, the attacker can attack all targets.

[0061] A further preferred embodiment of the present invention is as follows: the strategy model in step S3 is:

[0062] Assumptions is the set of nodes to be attacked, and the attack strategy is defined as X = [x1, x2, L, x n ]∈S A , where S A is the attacker’s strategy set. If node v i If attacked, then x i =1, that is, v i ∈V A , otherwise x i =0, let C X The total cost of the attacker adopting strategy X is defined as:

[0063]

[0064] Therefore, the resource budget is limited to:

[0065]

[0066] For a rational attacker, an attack is definitely not a one-time event. The best outcome is naturally a successful attack. However, based on the rational defense of the actual defender, this extreme situation is basically unlikely to occur. Therefore, in order to ensure that each attack achieves effective resource utilization and consider that there are still resources left for the next attack after the attack fails, the number of attack nodes of the attacker is defined as follows:

[0067]

[0068] Similarly, for the defender in strategy Y (Y=[y1,y2,L,y n ]∈S D ,y i =1 means the node is defended) the total cost is:

[0069]

[0070] The resource budget is limited to:

[0071]

[0072] In addition to resource budget constraints, for defenders who are in a weak position and need to ensure the normal operation of the network, some critical goals must consider resource investment and the necessary number of nodes to be protected. Therefore, the minimum number of nodes to be protected is defined as:

[0073]

[0074] A further preferred embodiment of the present invention: the profit model in step S3 is:

[0075] According to the assumptions of the game, v i When defended (y i =1), it will not be removed when an attacker attacks it, however, when an undefended node is attacked, it will be removed from the network, that is, when x i =1,y i = 0, the node is removed. Assume that the set of removed nodes is , so the network after removing the node is , then the attacker’s profit is:

[0076]

[0077] Where Γ represents the network performance measurement function, usually the maximum clique ratio or network efficiency;

[0078] Similarly, when all attacked nodes are removed without defense, the network can be expressed as The defender's payoff is:

[0079]

[0080] Compared with the prior art, the present invention has the following beneficial effects:

[0081] The present invention provides a method for constructing a network attack and defense game model based on node importance, which adopts multi-attribute decision-making theory to identify key nodes, and obtains a comprehensive node evaluation index using the ideal solution distance method of the set entropy weight method through four commonly used node evaluation indexes; the index is used in the attack and defense model of a complex network, and defensive resources are deployed by ranking the importance of the nodes to achieve the purpose of network defense; based on game theory, an attack and defense game model in a complex network is established; the proposed network attack and defense game model based on node importance can find key nodes, and resources are invested in high-importance nodes to find the optimal resource deployment through the solution of game theory, thereby achieving effective resource utilization and network protection. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] Figure 1 This is a flow chart of a method for constructing a network attack and defense game model based on node importance according to the present invention;

[0083] Figure 2 This is a graph showing the change in the maximum clique size of the scale-free network under different attack modes in an embodiment of the present invention. DETAILED DESCRIPTION

[0084] For ease of understanding, the following will describe the technical solutions in the embodiments of the present invention in detail with reference to the accompanying drawings. The described embodiments are only a part of the embodiments of the present invention.

[0085] Example 1

[0086] like Figure 1 As shown, an embodiment of the present invention provides a method for constructing a network attack and defense game model based on node importance, comprising the following steps:

[0087] Step S1: Modeling the attack and defense model of complex networks:

[0088] The network infrastructure system is represented by a simple undirected graph G(V,E), where V is a set of nodes, is an edge set;

[0089] The number of nodes |V| is represented by n, assuming A(G)=(a ij ) N×N is the adjacency matrix of G, if node v i and v j adjacent, then there is a ij =a ji =1, otherwise aij =a ji =0;

[0090] Node v i The degree is , equal to the number of edges connected to it;

[0091] Step S2: Identification of key nodes:

[0092] The initial evaluation matrix is obtained by using four node evaluation indicators: degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality. The importance of the node is calculated using the ideal distance method based on the entropy weight method to obtain a comprehensive evaluation indicator.

[0093] The comprehensive evaluation index is used in the attack and defense model of complex networks, and defensive resources are deployed based on the importance ranking of nodes to achieve the goal of network defense.

[0094] Step S3: Modeling the attack and defense game model of complex networks:

[0095] Based on game theory, an attack and defense game model of complex networks is established. The attack and defense game model of complex networks includes the basic assumption model, cost model, strategy model and benefit model of the model.

[0096] The identification of key nodes in step S2 utilizes multiple commonly used evaluation indicators using a multi-attribute decision-making method (ideal distance method) to obtain a comprehensive indicator. This allows for a comprehensive assessment of node importance and better describes the importance of nodes in the network. Furthermore, the use of the entropy weight method makes the weight of each evaluation indicator in describing node importance more clearly.

[0097] The complex network attack and defense game model in step S3 clearly states that the cost is related to the importance of its nodes, and decision-making and deployment are made based on their importance. The strategies adopted by both the attacker and the defender are rational participation from the perspective of game theory, and the optimal resource deployment is obtained through the Nash equilibrium solution, which is conducive to the protection of the actual network and the effective use of resources.

[0098] Example 2

[0099] like Figure 1 As shown, an embodiment of the present invention provides a method for constructing a network attack and defense game model based on node importance. The method extracts the degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality of all nodes in the network and uses a multi-attribute decision-making method to establish a comprehensive key node identification index - the EWM-TOPSIS method (the ideal solution distance method based on the entropy weight method). The attack and defense game model is constructed based on the node importance identified by the EWM-TOPSIS method.

[0100] To build a network model, the network infrastructure system is represented by a simple undirected graph G(V,E), where V is a set of nodes. is an edge set. The number of nodes |V| is represented by n. Assume that A(G)=(a ij ) N×N is the adjacency matrix of G. If node v i and v j adjacent, then there is a ij =a ji =1, otherwise a ij =a ji = 0. Node v i The degree is , which is equal to the number of edges connected to it.

[0101] Many mechanisms and functions within a network are significantly impacted by a small number of nodes. These nodes are generally considered critical. If a node experiences a malicious attack or unexpected event that significantly degrades network performance, it is considered critical and is often referred to as a key node. A single node evaluation metric often cannot fully assess node importance, so the TOPSIS method is used to calculate a comprehensive node evaluation metric.

[0102] Degree Centrality (DC): Degree centrality is expressed as the proportion of network nodes that have direct connections with the node, reflecting the connection with other nodes, where k i is the degree of node i;

[0103]

[0104] For a connected network with n nodes, we can calculate the v of any node i The average shortest distance to other nodes in the network:

[0105]

[0106] The betweenness centrality of a node generally refers to the shortest path betweenness centrality, which describes the control power of the node on the network flow transmitted along the shortest path in the network; the node v i The betweenness of is defined as:

[0107]

[0108] Where N is the number of nodes in the network, σ st (i) represents the number of shortest paths from node s to node t through node i, σ st represents the total number of shortest paths from node s to node t.

[0109] The closeness centrality of a node eliminates the interference of special values by calculating the average distance between the node and all other nodes in the network. The smaller the average distance between a node and other nodes in the network, the greater the closeness centrality of the node. The closeness centrality of a node is expressed as:

[0110]

[0111] The eigenvector centrality of a node considers the linear relationship between the centrality index of a node and the centrality indexes of other nodes around it. It is the linear superposition of the centrality values of its adjacent nodes. The larger the eigenvector centrality value of a node, the more important the node is. The eigenvector centrality measure is defined as follows:

[0112]

[0113] Where EC(i) represents the eigenvector centrality of node i, λ is a constant, and A ij is the adjacency matrix of the network, x j is the value of the jth item of the normalized largest eigenvector.

[0114] Example 3

[0115] like Figure 1 As shown, an embodiment of the present invention provides a method for constructing a network attack and defense game model based on node importance, comprising the following steps:

[0116] Step S1: Modeling the attack and defense model of complex networks:

[0117] The network infrastructure system is represented by a simple undirected graph G(V,E), where V is a set of nodes, is an edge set;

[0118] The number of nodes |V| is represented by n, assuming A(G)=(a ij ) N×N is the adjacency matrix of G, if node v i and v j adjacent, then there is a ij =a ji =1, otherwise a ij =a ji =0;

[0119] Node v i The degree is , equal to the number of edges connected to it;

[0120] Step S2: Identification of key nodes:

[0121] The initial evaluation matrix is obtained by using four node evaluation indicators: degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality. The importance of the node is calculated using the ideal distance method based on the entropy weight method to obtain a comprehensive evaluation indicator.

[0122] The comprehensive evaluation index is used in the attack and defense model of complex networks, and defensive resources are deployed based on the importance ranking of nodes to achieve the goal of network defense.

[0123] Step S3: Modeling the attack and defense game model of complex networks:

[0124] Based on game theory, an attack and defense game model of complex networks is established. The attack and defense game model of complex networks includes the basic assumption model, cost model, strategy model and benefit model of the model.

[0125] The degree centrality of the node in step S2 represents the direct influence of the node, which is expressed as follows:

[0126]

[0127] Among them, k i is the degree of node i;

[0128] For a connected network with n nodes, we can calculate the v of any node i The average shortest distance to other nodes in the network:

[0129]

[0130] The betweenness centrality of a node in step S2 generally refers to the shortest path betweenness centrality, which describes the control power of a node on the network flow transmitted along the shortest path in the network; i The betweenness of is defined as:

[0131]

[0132] Where N is the number of nodes in the network, σ st (i) represents the number of shortest paths from node s to node t through node i, σ st represents the total number of shortest paths from node s to node t.

[0133] The closeness centrality of a node in step S2 is calculated by averaging the distances between the node and all other nodes in the network to eliminate the interference of special values. The smaller the average distance between a node and other nodes in the network, the greater the closeness centrality of the node. The closeness centrality of a node is expressed as:

[0134]

[0135] The eigenvector centrality of the node in step S2 considers the linear relationship between the centrality index of a node and the centrality indexes of other surrounding nodes. It is the linear superposition of the centrality values of its adjacent nodes. The larger the eigenvector centrality value of the node, the more important the node. The eigenvector centrality measure is defined as follows:

[0136]

[0137] Where EC(i) represents the eigenvector centrality of node i, λ is a constant, and A ij is the adjacency matrix of the network, x j is the value of the jth item of the normalized largest eigenvector.

[0138] The initial evaluation matrix in step S2 is:

[0139]

[0140] There are m evaluation indicators and n evaluation objects. To study the importance of nodes in complex networks, the evaluation objects are n nodes in the network, and the evaluation indicator is degree centrality x i1 , eigenvector centrality x i2 , betweenness centrality x i3 , closeness centrality x i4 ;

[0141] Normalized to:

[0142]

[0143] The normalized evaluation matrix is:

[0144]

[0145] Determine the best and worst solutions:

[0146] The optimal solution (consisting of the maximum value of each column in Z):

[0147]

[0148] The worst solution (consisting of the minimum value of each column in Z):

[0149]

[0150] According to the information entropy theory, the smaller the information entropy of an indicator is, the more information it contributes, and the greater its weight is. Therefore, the information entropy value of an indicator is set to E. j (0≤E j ≤1):

[0151]

[0152] The weights are:

[0153]

[0154] where ω j is the weight (importance) of the jth attribute;

[0155] Calculate the degree of proximity between each evaluation index object and the optimal solution and the worst solution:

[0156]

[0157] Calculate the closeness C between the evaluation object and the optimal solution i :

[0158]

[0159] Where 0≤C i ≤1, C i →1 indicates that the evaluation object is better.

[0160] The basic assumption model in step S3 is:

[0161] Only one attacker and one defender are considered, and both attacker and defender have complete information about the network;

[0162] All attack and defense actions are targeted at nodes in the network. When a node is successfully attacked, the edges connected to the node are also removed.

[0163] When a node is attacked by an attacker but not protected by a defender, it is considered that the node has been successfully attacked and is removed from the network. When a node is protected by a defender, the attacker's attack on the node is considered invalid and the node still functions in the network.

[0164] The cost model in step S3 is:

[0165] The attacker's goal is to maximize network damage by attacking nodes in the network, while the defender's goal is to minimize resource consumption by investing resources in protecting nodes. However, whether attacking or defending, taking measures against nodes will inevitably consume certain resources or pay a certain price, namely:

[0166]

[0167] where r i ≥0, r i Represents the reference attributes of the node (degree, betweenness, etc.), r i is the importance of the node, Represents node v i The attack cost and defense cost of A ,qD (>0) indicates attack (defense) cost-sensitive parameter;

[0168] In actual scenarios, the use of resources is not unlimited. In order to achieve effective use of resources and accurate and efficient attack (defense) of nodes, it is defined as:

[0169]

[0170] Where α∈[0,1] is the attack cost constraint parameter, and β∈[0,1] is the defense cost constraint parameter. Parameters α and β indicate how much resources a player can invest in their actions. As α increases, the attacker can attack more targets while adopting the same attack strategy. In the extreme case of α=1, the attacker can attack all targets.

[0171] The policy model in step S3 is:

[0172] Assumptions is the set of nodes to be attacked, and the attack strategy is defined as X = [x1, x2, L, x n ]∈S A , where S A is the attacker’s strategy set. If node v i If attacked, then x i =1, that is, v i ∈V A , otherwise x i =0, let C X The total cost of the attacker adopting strategy X is defined as:

[0173]

[0174] Therefore, the resource budget is limited to:

[0175]

[0176] For a rational attacker, an attack is definitely not a one-time event. The best outcome is naturally a successful attack. However, based on the rational defense of the actual defender, this extreme situation is basically unlikely to occur. Therefore, in order to ensure that each attack achieves effective resource utilization and consider that there are still resources left for the next attack after the attack fails, the number of attack nodes of the attacker is defined as follows:

[0177]

[0178] Similarly, for the defender in strategy Y (Y=[y1,y2,L,y n ]∈S D ,y i =1 means the node is defended) the total cost is:

[0179]

[0180] The resource budget is limited to:

[0181]

[0182] In addition to resource budget constraints, for defenders who are in a weak position and need to ensure the normal operation of the network, some critical goals must consider resource investment and the necessary number of nodes to be protected. Therefore, the minimum number of nodes to be protected is defined as:

[0183]

[0184] The profit model in step S3 is:

[0185] According to the assumptions of the game, v i When defended (y i =1), it will not be removed when an attacker attacks it, however, when an undefended node is attacked, it will be removed from the network, that is, when x i =1,y i = 0, the node is removed. Assume that the set of removed nodes is , so the network after removing the node is , then the attacker’s profit is:

[0186]

[0187] Where Γ represents the network performance measurement function, usually the maximum clique ratio or network efficiency;

[0188] Similarly, when all attacked nodes are removed without defense, the network can be expressed as The defender's payoff is:

[0189]

[0190] The application effect of the present invention is described in detail below with reference to simulation.

[0191] The network of the present invention is a scale-free network, wherein G(V, E)=G(100, 200). Table 1 shows the top ten node attributes of the key node identification module in the network.

[0192] Table 1 Comprehensive indicators of the top ten nodes in the scale-free network

[0193]

[0194]

[0195] Figure 2 It can be seen that degree centrality, closeness centrality, and EWM-TOPSIS attacks are more damaging to the scale-free network. Before the number of attacks reaches 17, closeness centrality attacks are slightly more damaging to the network than degree centrality and EWM-TOPSIS attacks. However, when the number of attacks reaches 17, the network under EWM-TOPSIS suffers even more severe damage. The network damage is particularly significant when the number of attacks reaches 20.

[0196] In the description of the present invention, it should be understood that the terms "upper", "lower", "front", "back", "left", "right", "top", "bottom", "inside", "outside", etc., indicating directions or positional relationships, are based on the directions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific direction, be constructed and operated in a specific direction, and therefore should not be understood as limiting the present invention.

[0197] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art will be able to modify the technical solutions described in the aforementioned embodiments or substitute equivalents for some of the technical features. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A method for constructing a network attack and defense game model based on node importance, characterized in that: The following steps are involved: Step S1: Modeling the attack and defense model of complex networks: The network infrastructure system uses a simple undirected graph To indicate that is a node set, is an edge set; Number of nodes use Indicates that, assuming yes The adjacency matrix of node and Adjacent, then there is ,otherwise ; node The degree is , equal to the number of edges connected to it; Step S2: Identification of key nodes: The initial evaluation matrix is obtained by using four node evaluation indicators: degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality. The importance of the node is then calculated using the ideal distance method based on the entropy weight method to obtain a comprehensive evaluation indicator. The comprehensive evaluation index is used in the attack and defense model of complex networks, and defensive resources are deployed based on the importance ranking of nodes to achieve the goal of network defense. Step S3: Modeling the attack and defense game model of complex networks: Based on game theory, establish an attack and defense game model for complex networks. The attack and defense game model for complex networks includes the basic assumption model, cost model, strategy model, and benefit model. The cost model in step S3 is: The attacker's goal is to maximize network damage by attacking nodes in the network. The defender minimizes resource consumption by investing resources in protecting nodes. However, whether attacking or defending, taking measures against nodes will inevitably consume certain resources or pay a certain price, namely: （15） in , Represents the reference property of a node, is the importance of the node, 、 Representation node The attack cost and defense cost of Represents attack and defense cost-sensitive parameters; In actual scenarios, resource utilization is not unlimited. To achieve effective resource utilization and accurate and efficient attack and defense of nodes, it is defined as: （16） in is the attack cost constraint parameter, is the defense cost constraint parameter, parameter and Indicates how much resources players can invest in their actions, as With the increase of In the extreme case, the attacker can attack all targets; The strategy model in step S3 is: Assumptions is the set of nodes to be attacked, and the attack strategy definition ,in is the attacker's strategy set, if node If attacked, ,Right now ,otherwise ,set up Adopting strategies for the attacker The total cost at time , is defined as: （17） Therefore, the resource budget is limited to: （18） For a rational attacker, an attack is definitely not a one-time event. The best outcome is naturally a successful attack. However, based on the rational defense of the actual defender, this extreme situation is basically unlikely to occur. Therefore, in order to ensure that each attack achieves effective resource utilization and consider that there are still resources left for the next attack after the attack fails, the number of attack nodes of the attacker is defined as follows: （19） Similarly, Indicates that the node is defended. For the defender, The total cost under （20） The resource budget is limited to: （21） In addition to resource budget constraints, for defenders who are in a weak position and need to ensure the normal operation of the network, some critical goals must consider resource investment and the necessary number of nodes to be protected. Therefore, the minimum number of nodes to be protected is defined as: （22）。 2. A method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The degree centrality of the node in step S2 represents the direct influence of the node, which is expressed as follows: （1） in, For nodes degree; For A connected network of nodes can calculate any node The average shortest distance to other nodes in the network: （2）。 3. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The betweenness centrality of the node in step S2 generally refers to the shortest path betweenness centrality, which describes the control power of the node on the network flow transmitted along the shortest path in the network; The betweenness of is defined as: （3） Where N represents the number of nodes in the network, Represents the node Slave nodes To Node The number of shortest paths, Represents the total number of shortest paths from the node To Node .

4. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The closeness centrality of the node in step S2 is calculated by calculating the average value of the distance between the node and all other nodes in the network to eliminate the interference of special values. The smaller the average distance between a node and other nodes in the network, the greater the closeness centrality of the node. The closeness centrality of the node is expressed as: （4）。 5. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The eigenvector centrality of the node in step S2 considers the linear relationship between the centrality index of a node and the centrality indexes of other surrounding nodes, which is the linear superposition of the centrality values of its adjacent nodes. The larger the eigenvector centrality value of the node, the more important the node. The eigenvector centrality measure is defined as follows: （5） in Representation node The eigenvector centrality of is a constant, is the adjacency matrix of the network, is the normalized largest eigenvector The value of the item.

6. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The initial evaluation matrix in step S2 is: （6） in evaluation indicators, evaluation objects, and the importance of nodes in complex networks is studied. The evaluation object is the nodes, and the evaluation index is degree centrality , eigenvector centrality , betweenness centrality , closeness centrality ; Normalized to: （7） The normalized evaluation matrix is: （8） Determine the best and worst solutions: Optimal solution: （9） Worst solution: （10） According to the information entropy theory, the smaller the information entropy of an indicator is, the more information it contributes, and the greater its weight is. Therefore, the information entropy value of an indicator is set to : （11） The weights are: （12） in For the The weight of each attribute; Calculate the degree of proximity between each evaluation index object and the optimal solution and the worst solution: （13） Calculate the degree of closeness between the evaluation object and the optimal solution : （14） in , The better the evaluation object.

7. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The basic assumption model in step S3 is: Only one attacker and one defender are considered, and both attacker and defender have complete information about the network; All attack and defense actions are targeted at nodes in the network. When a node is successfully attacked, the edges connected to the node are also removed. When a node is attacked by an attacker but not protected by a defender, it is considered that the node has been successfully attacked and is removed from the network. When a node is protected by a defender, the attacker's attack on the node is considered invalid and the node still functions in the network.

8. The method for constructing a network attack and defense game model based on node importance according to claim 1, characterized in that: The profit model in step S3 is: According to the assumptions of the game, When defended, it will not be removed when an attacker attacks it, however, when an undefended node is attacked, it will be removed from the network, that is, when When , the node is removed, assuming that the set of removed nodes is , so the network after removing the node is , then the attacker’s profit is: （23） in Represents the network performance measurement function, usually the maximum clique ratio or network efficiency; Similarly, when all attacked nodes are removed without defense, the network can be expressed as , then the defender's payoff is: （24）。

Images