Network game strategy generation method based on node average path constraint

Abstract

The application discloses a network game strategy generation method based on node average path constraint, and the method comprises the following steps: acquiring the topological structure of a network, determining the strategy set of an attack party and a defense party, and constructing a basic model of the network game; calculating the shortest path between nodes in each attack strategy and each defense strategy, taking the average value, and obtaining the average path corresponding to each attack strategy and each defense strategy; setting a constraint function based on the average path, so that the selection probability of each attack strategy and each defense strategy is less than the corresponding function value, and obtaining the strategy constraint set of the attack party and the strategy constraint set of the defense party; representing the network performance by a maximum connected piece scale index, calculating the income of the attack party and the defense party under each strategy profile, and obtaining the income matrix of the network game model; and replacing the basic model of the network game with a linear programming problem for solving, and obtaining the mixed strategy Nash equilibrium solution of the attack party and the defense party.

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 strategies based on node average path constraints. Background Technology

[0002] In the current field of game theory research, there is a special type of network game. In this type of game, the network is not a physical network such as computer systems, but rather critical infrastructure, such as train stations and airports, is abstracted as nodes in a network topology. The connections between different sites are abstracted as edges, creating a complex network of infrastructure. Due to their importance and irreplaceability, some infrastructures are vulnerable to attacks or threats. Applying network game models to analyze and study the optimal strategies of the players in their interactions helps in developing the best defense strategies.

[0003] Existing research on infrastructure network game theory primarily focuses on unconstrained game models, failing to consider the varying difficulty of implementing different strategies and thus failing to align with real-world situations. In reality, due to limited resources, decision-makers tend to choose easier-to-implement strategies rather than more difficult ones, resulting in different constraints on the probability of choosing strategies of varying difficulty. Therefore, existing methods do not account for the different implementation difficulties of strategies in reality and cannot comprehensively analyze actual game problems.

[0004] Scholars have proposed a constrained bimatrix game framework with practical applications in various fields, such as packet congestion modeling in wireless networks. Other scholars have used linear programming duality theory to study the existence of solutions for two-player zero-sum constrained matrix games. This theory provides insights for solving more complex game problems. Currently, there is limited research on incorporating the implementation difficulty of strategies into complex infrastructure network games, making this area of ​​research significant. Summary of the Invention

[0005] This invention aims to address at least one of the technical problems existing in the prior art. To this end, this invention discloses a method for generating network game strategies based on node average path constraints. The method is based on a two-player zero-sum game, using the node average path to set constraints on the probability of choosing each strategy, generating a set of strategy constraints. A feasible algorithm is then used to solve for the Nash equilibrium solution, ultimately obtaining the network game strategy generation method.

[0006] The objective of this invention is achieved through the following technical solution: a method for generating network game strategies based on node average path constraints, the method comprising:

[0007] Step 1: Obtain the network topology, determine the attacker's attack strategy set and the defender's defense strategy set, and construct the basic model of network game.

[0008] Step 2: Calculate the shortest path between nodes in each attack strategy and each defense strategy and take the average value to obtain the average path corresponding to each attack strategy and each defense strategy.

[0009] Step 3: Set constraint functions based on the average path to make the selection probability of each attack strategy and each defense strategy less than the corresponding function value, thus obtaining the strategy constraint set of the attacker and the strategy constraint set of the defender.

[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 network is a critical infrastructure network, the nodes are critical infrastructures, including railway stations, and the connections are links between critical infrastructures.

[0014] Specifically, the network game strategy generation method based on node average path constraints is characterized in that the attacker's attack strategy set is S. A ={X1,X2,...,X m The set of defense strategies of the defending side is S. D ={Y1,Y2,...,Y n}, for node V i If node V is attacked but not defended against, then... i The edges connected to it will be removed.

[0015] Specifically, the network game strategy generation method based on node average path constraints is characterized in that the average path in step 2 refers to the average of the shortest paths between each pair of nodes in the node set. For the attacker, each attack strategy X... i The average path value refers to the average of the shortest paths between each pair of nodes in the attack node set, which can be expressed as: For the defender, each defense strategy Y i The average path value refers to the average of the shortest paths for each pair of nodes in the set of defense nodes, and can be expressed as:

[0016] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the constraint function in step 3 is divided into an attacker's constraint function and a defender's constraint function, wherein the attacker's constraint function is:

[0017]

[0018]

[0019] in, Let exp represent the unnormalized constraint coefficients corresponding to the i-th attack strategy of the attacker, and let exp represent the logarithmic function. X represents the attacker's i-th attack strategy. i Average path value, X represents the attacker's i-th attack strategy. i The normalized constraint coefficient, C A Let C represent the set of unnormalized constraint coefficients of the attacker, min(C A ) represents set C A The minimum value in, max(C) A ) represents set C A The maximum value in.

[0020] The defender's constraint function is:

[0021]

[0022]

[0023] in, Let exp represent the unnormalized constraint coefficients corresponding to the i-th defense strategy of the attacker, and let exp represent the logarithmic function. Y represents the i-th defense strategy of the defender. i Average path value, Y represents the i-th defense strategy of the defender. i The normalized constraint coefficient, C D Let C represent the set of unnormalized constraint coefficients of the defending side, min(C D ) represents set C D The minimum value in, max(C) D ) represents set C D The maximum value in.

[0024] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the attacker's strategy constraint set is:

[0025] X i ∈S A

[0026] Where, p i This indicates that in the mixed strategy, the attacker chooses the i-th attack strategy X. i The probability, Let X represent the i-th attack strategy. i The attack constraint coefficient, X i Let S represent the i-th attack strategy, ∈ denotes belonging to the symbol. A This represents the set of attack strategies employed by the attacker.

[0027] The set of policy constraints for the defender is as follows:

[0028] Y j ∈S D

[0029] Where, q j This indicates that in the hybrid strategy, the defender chooses the j-th defense strategy Y. j The probability, Y represents the j-th defense strategy. j The defense constraint coefficient, Y j Let S represent the j-th defense strategy, ∈ denotes belonging to the symbol. D This represents the set of defensive strategies employed by the defender.

[0030] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the payoff matrix is ​​divided into the attacker's payoff matrix and the defender's payoff matrix, U A (X,Y) represents the attacker's gain, U D (X,Y) represents the defender's gains:

[0031]

[0032]

[0033] Where Γ(G) represents the maximum connected component size of the initial network G. The maximum connected component size of the network after one round of play is represented as follows: The attacker's payoff matrix and the defender's payoff matrix are respectively expressed as:

[0034]

[0035]

[0036] Among them, U A U represents the attacker's payoff matrix. D This represents the payoff matrix for the defending side. This indicates that the attacker has chosen attack strategy X. iThe defender chooses defense strategy Y. j The attacker's gains at that time This indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The benefits for the defending side.

[0037] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the linear programming problem in step 5 is:

[0038] max z

[0039]

[0040] maxω

[0041]

[0042] Where max represents maximization, z is the attacker's gain, ω is the defender's gain, and X... i Y represents the i-th attack strategy. j U represents the j-th defense strategy. A (X i ,Y j ) indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The attacker's gains, U D (X i ,Y j ) indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The defensive side's benefit, p i This indicates that in the mixed strategy, the attacker chooses the i-th attack strategy X. i The probability, q j This indicates that in the hybrid strategy, the defender chooses the j-th defense strategy Y. j The probability, S A S represents the set of attack strategies employed by the attacker. D This represents the set of defense strategies of the defender, and ∑ represents the summation symbol. Let represent any symbol, and ∈ denotes belonging to the symbol. Solving the linear programming problem, we obtain p = (p1, p2, ..., p...). m ) T , q=(q1,q2,...,q n ) T This is the Nash equilibrium solution of the game.

[0043] Compared with existing methods, the advantages of this invention are as follows: Network game theory has been a research hotspot in recent years; however, existing research has not considered the impact of the average path between nodes on the probability of strategy selection. This invention proposes a network game strategy generation method based on node average path constraints, provides specific constraint functions for each strategy, and finally establishes a nonlinear programming model to solve for the Nash equilibrium solution. Network game theory based on node average path constraints is more consistent with reality and facilitates the practical application of network game theory. Attached Figure Description

[0044] Figure 1 A flowchart illustrating an embodiment of the present invention is shown;

[0045] Figure 2 A schematic diagram of the network topology in an embodiment of the present invention is shown;

[0046] Figure 3 A schematic diagram of probability allocation according to an embodiment of the present invention is shown;

[0047] Figure 4 A schematic diagram illustrating the probability allocation for the defender in an embodiment of the present invention is shown. Detailed Implementation

[0048] 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.

[0049] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0050] In this embodiment, the attacker is the attacker and the defender is the defender. The network model is taken as a railway network, which 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;

[0051] Let A(G) denote the set to be joined, and 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 jIf there is no railway line between them, then a ij =a ji =0.

[0052] 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.

[0053] like Figure 1 As shown, a network game strategy generation method based on node average path constraints is described, the method comprising:

[0054] Step 1: Obtain the network topology, determine the attacker's attack strategy set and the defender's defense strategy set, and construct the basic model of network game.

[0055] Step 2: Calculate the shortest path between nodes in each attack strategy and each defense strategy and take the average value to obtain the average path corresponding to each attack strategy and each defense strategy.

[0056] Step 3: Set constraint functions based on the average path to make the selection probability of each attack strategy and each defense strategy less than the corresponding function value, thus obtaining the strategy constraint set of the attacker and the strategy constraint set of the defender.

[0057] 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.

[0058] 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.

[0059] 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.

[0060] Specifically, the network game strategy generation method based on node average path constraints is characterized in that the attacker's attack strategy set is S. A ={X1,X2,...,X m The set of defense strategies of the defending side is S. D ={Y1,Y2,...,Y n}, for node V i If node V is attacked but not defended against, then... i The edges connected to it will be removed.

[0061] Specifically, the network game strategy generation method based on node average path constraints is characterized in that the average path in step 2 refers to the average of the shortest paths between each pair of nodes in the node set. For the attacker, each attack strategy X... i The average path value refers to the average of the shortest paths between each pair of nodes in the attack node set, which can be expressed as: For the defender, each defense strategy Y i The average path value refers to the average of the shortest paths for each pair of nodes in the set of defense nodes, and can be expressed as:

[0062] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the constraint function in step 3 is divided into an attacker's constraint function and a defender's constraint function, wherein the attacker's constraint function is:

[0063]

[0064]

[0065] in, Let exp represent the unnormalized constraint coefficients corresponding to the i-th attack strategy of the attacker, and let exp represent the logarithmic function. X represents the attacker's i-th attack strategy. i Average path value, X represents the attacker's i-th attack strategy. i The normalized constraint coefficient, C A Let C represent the set of unnormalized constraint coefficients of the attacker, min(C A ) represents set C A The minimum value in, max(C) A ) represents set C A The maximum value in.

[0066] The defender's constraint function is:

[0067]

[0068]

[0069] in, Let exp represent the unnormalized constraint coefficients corresponding to the i-th defense strategy of the attacker, and let exp represent the logarithmic function. Y represents the i-th defense strategy of the defender. i Average path value, Y represents the i-th defense strategy of the defender. i The normalized constraint coefficient, C DLet C represent the set of unnormalized constraint coefficients of the defending side, min(C D ) represents set C D The minimum value in, max(C) D ) represents set C D The maximum value in.

[0070] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the attacker's strategy constraint set is:

[0071] X i ∈S A

[0072] Where, p i This indicates that in the mixed strategy, the attacker chooses the i-th attack strategy X. i The probability, Let X represent the i-th attack strategy. i The attack constraint coefficient, X i Let S represent the i-th attack strategy, ∈ denotes belonging to the symbol. A This represents the set of attack strategies employed by the attacker.

[0073] The set of policy constraints for the defender is as follows:

[0074] Y j ∈S D

[0075] Where, q j This indicates that in the hybrid strategy, the defender chooses the j-th defense strategy Y. j The probability, Y represents the j-th defense strategy. j The defense constraint coefficient, Y j Let S represent the j-th defense strategy, ∈ denotes belonging to the symbol. D This represents the set of defensive strategies employed by the defender.

[0076] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the payoff matrix is ​​divided into the attacker's payoff matrix and the defender's payoff matrix, U A (X,Y) represents the attacker's gain, U D (X,Y) represents the defender's gains:

[0077]

[0078]

[0079] Where Γ(G) represents the maximum connected component size of the initial network G. The maximum connected component size of the network after one round of play is represented as follows: The attacker's payoff matrix and the defender's payoff matrix are respectively expressed as:

[0080]

[0081]

[0082] Among them, U A U represents the attacker's payoff matrix. D This represents the payoff matrix for the defending side. This indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The attacker's gains at that time This indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The benefits for the defending side.

[0083] Furthermore, the network game strategy generation method based on node average path constraints is characterized in that the linear programming problem in step 5 is:

[0084] max z

[0085]

[0086] maxω

[0087]

[0088] Where max represents maximization, z is the attacker's gain, ω is the defender's gain, and X... i Y represents the i-th attack strategy. j U represents the j-th defense strategy. A (X i ,Y j ) indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The attacker's gains, U D (X i ,Y j ) indicates that the attacker has chosen attack strategy X. i The defender chooses defense strategy Y. j The defensive side's benefit, p i This indicates that in the mixed strategy, the attacker chooses the i-th attack strategy X. i The probability, q j This indicates that in the hybrid strategy, the defender chooses the j-th defense strategy Y. j The probability, S AS represents the set of attack strategies employed by the attacker. D This represents the set of defense strategies of the defender, and ∑ represents the summation symbol. Let represent any symbol, and ∈ denotes belonging to the symbol. Solving this model, we obtain p = (p1, p2, ..., p...). m ) T , q=(q1,q2,...,q n ) T This is the Nash equilibrium solution of the game.

[0089] In real life, infrastructure network structures vary widely. This experiment uses a 10-node network structure as an example. Figure 2 As shown, it is assumed that both the attacker and defender have limited resources and that the number of nodes that can be selected is 2.

[0090] In this embodiment, two sets of comparative experiments are conducted: one without considering the average path constraint of nodes and the other with considering the average path constraint of nodes.

[0091] Next, the payoff matrix is ​​calculated. After obtaining the payoff matrix, the model solution process yields the Nash equilibrium mixed policy solution. Since each policy corresponds to a choice probability, the probabilities corresponding to different attack and defense policies can be mapped to different nodes. The mapping method is as follows:

[0092]

[0093]

[0094] in and It represents the probability distribution of individual nodes for two participants. and It is the probability distribution of all attack and defense strategies.

[0095] Based on the above, a node probability allocation graph that does not consider the average path constraint of nodes can be drawn as follows: Figure 3 As shown, nodes V1 and V2 have the lowest probability of being attacked, and they have the highest degree centrality, proximity degree centrality, betweenness centrality, and eigenvector centrality. However, the defenders allocate the highest probability to protecting nodes V1 and V2. This indicates that, in general, nodes with higher attributes are more likely to be protected. As the number of nodes to be attacked or defended increases, the probability distribution of the nodes becomes more uniform.

[0096] The probability distribution graph of the node probability distribution graph considering the average path constraint of the nodes is as follows: Figure 4As shown, after considering the average path constraint of the nodes, the node probability distribution changes significantly. The node with the highest probability of being attacked is V1, while the attack probability of V6 and V8 becomes 0. The node with the highest probability of being defended is also V1, and the probabilities of the other nodes also fluctuate slightly.

[0097] 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 strategies based on node average path constraints, characterized in that, The method includes: Step 1: Obtain the topology of the infrastructure network, determine the attack strategy set of the attacker and the defense strategy set of the defender, and construct the basic model of infrastructure network game. Step 2: Calculate the shortest path between nodes in each attack strategy and each defense strategy and take the average value to obtain the average path corresponding to each attack strategy and each defense strategy. Step 3: Set constraint functions based on the average path to make the selection probability of each attack strategy and each defense strategy less than the corresponding function value, thus obtaining the strategy constraint set of the attacker and the strategy constraint set of the defender. Step 4: Using the maximum connected component size as the metric to represent the infrastructure network performance, calculate the payoffs for the attacker and defender under each strategy profile to obtain the payoff matrix of the infrastructure network game model. Step 5: Replace the basic model of the infrastructure network game with a linear programming problem to obtain the mixed strategy Nash equilibrium solution for the attacker and defender. The infrastructure network model is abstracted into a simple undirected graph. ,in, Represents the set of all nodes in the network. Indicates the number of nodes in the network. Indicates a join set; Represents a joined set; The constraint functions in step 3 are divided into the attacker's constraint functions and the defender's constraint functions. The attacker's constraint functions are as follows: in, Indicates the attacker's... The unnormalized constraint coefficients corresponding to each attack strategy Represents the logarithmic function. Indicates the attacker's... attack strategies Average path value, Indicates the attacker's... attack strategies The normalized constraint coefficients, This represents the set of unnormalized constraint coefficients of the attacker. Represents a set The minimum value in, Represents a set The maximum value in; The defender's constraint function is: in, Indicates the attacker's... The unnormalized constraint coefficients corresponding to each defense strategy Represents the logarithmic function. The first one represents the defending side One defense strategy Average path value, The first one represents the defending side One defense strategy The normalized constraint coefficients, This represents the set of unnormalized constraint coefficients of the defending side. Represents a set The minimum value in, Represents a set The maximum value in.

2. The network game strategy generation method based on node average path constraints according to claim 1, characterized in that, The attack strategy set of the attacker is as follows: The set of defense strategies of the defending side is as follows: For nodes If a node is attacked but not defended against, then... The edges connected to it will be removed.

3. The method for generating network game strategies based on node average path constraints according to claim 1 or 2, characterized in that, The average path in step 2 refers to the average of the shortest paths between each pair of nodes in the node set. For the attacker, each attack strategy... The average path value refers to the average of the shortest paths between each pair of nodes in the attack node set, expressed as: ; For the defender, each defense strategy The average path value refers to the average of the shortest paths for each pair of nodes in the set of defense nodes, expressed as: .

4. The network game strategy generation method based on node average path constraints according to claim 1, characterized in that, The attacker's policy constraint set is as follows: in, This indicates that in a hybrid strategy, the attacker chooses the first... attack strategies The probability, Indicates the first attack strategies Attack constraint coefficient, Indicates the first One attack strategy, Indicates that it belongs to the symbol. This represents the set of attack strategies employed by the attacker. The set of policy constraints for the defender is as follows: in, This indicates that in a hybrid strategy, the defender chooses the first... One defense strategy The probability, Indicates the first One defense strategy The defense constraint coefficient, Indicates the first One defense strategy, Indicates that it belongs to the symbol. This represents the set of defensive strategies employed by the defender.

5. The method for generating network game strategies based on node average path constraints 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. This represents the attacker's gains. Indicates the defender's gains: in, This represents the maximum connected component size of the initial network G. The maximum connected component size of the network after one round of play is represented as follows: The attacker's payoff matrix and the defender's payoff matrix are respectively expressed as: in, This represents the attacker's payoff matrix. This represents the payoff matrix for the defending side. This indicates that the attacker has chosen an attack strategy. The defender chooses a defense strategy. The attacker's gains at that time This indicates that the attacker has chosen an attack strategy. The defender chooses a defense strategy. The benefits for the defending side.

6. The method for generating network game strategies based on node average path constraints according to claim 1 or 5, characterized in that, The linear programming problem in step 5 is: in, To maximize, For the attacker's benefit, For the benefit of the defending side, Indicates the first One attack strategy, Indicates the first One defense strategy, This indicates that the attacker has chosen an attack strategy. The defender chooses a defense strategy. The attacker's gains at that time This indicates that the attacker has chosen an attack strategy. The defender chooses a defense strategy. The benefits for the defending side at that time. This indicates that in a hybrid strategy, the attacker chooses the first... attack strategies The probability, This indicates that in a hybrid strategy, the defender chooses the first... One defense strategy The probability, This represents the set of attack strategies employed by the attacker. This represents the set of defensive strategies of the defending side. Let represent the summation symbol, ∀ represent any symbol, and ∈ represent the symbol. Solving the linear programming problem yields the following results. This is the Nash equilibrium solution of the game.

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