Multi-agent cooperative control method based on evolutionary game theory under zero-trust architecture

Abstract

The invention relates to the technical field of zero-trust and evolutionary game, in particular to a multi-agent cooperative control method based on an evolutionary game theory under a zero-trust architecture, and the method comprises the steps: S1, constructing a multi-agent system of which the population scale is n; defining a strategy set for each agent; s2, each agent randomly selects one of the cooperation strategy A and the back strategy B as an initial strategy, and games with all neighbors; s3, calculating a real-time trust degree based on the historical cooperation rate and the real-time zero trust verification score; s4, calculating the fitness based on the interaction income and the neighbor relationship in the interaction result; s5, calculating a comprehensive evaluation value based on the real-time credibility and the fitness; s6, updating the strategy of each agent based on the comprehensive evaluation value difference between each agent and the neighbor; and S7, continuously selecting a strategy for updating each agent until the whole multi-agent system reaches an evolution stable state. According to the invention, the security of the multi-agent cooperative control system is improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of zero trust and evolutionary game, and particularly relates to a multi-agent cooperative control method based on evolutionary game theory under a zero trust architecture. BACKGROUND

[0002] The generation and evolution of cooperative behavior in multi-agent systems is one of the core mechanisms that support the sustainability and development of complex social and technical systems. With the rapid development and continuous maturation of swarm intelligence technology, multi-agent systems have become an important frontier in distributed control research. Due to their scalability, robustness in task execution, adaptability to dynamic environments, and significant cost-effectiveness, they have attracted widespread attention across disciplines.

[0003] Current theoretical research on group coordination control mainly follows two paths. The first path focuses on using classical coordination control frameworks to explore self-organization mechanisms, such as consensus control, flocking control, and formation control. These methods achieve global cooperation through simple, distributed, and localized interaction rules, enabling agent groups to autonomously complete collective tasks without centralized supervision. The second path emphasizes the integration of interdisciplinary methods, particularly the combination of game theory and group dynamics, to address the inherent contradiction between individual rationality and group optimality. This paradigm systematically reveals the internal relationship between individual interaction and group macro behavior. By modeling strategic interactions at the micro level, researchers can gain a deeper understanding of cooperation norms, role division, and how group intelligence spontaneously emerges from individual decision-making processes. The two complementary research paths together provide a theoretical foundation for understanding group intelligence from both macro coordination and micro behavior regulation perspectives. They not only deepen our understanding of the adaptive mechanisms of agent groups but also support a variety of practical applications, including autonomous vehicle systems, distributed robot swarms, intelligent transportation networks, environmental monitoring platforms, and complex industrial processes.

[0004] Existing basic research mostly assumes idealized cooperative interactions between agents, ignoring the competitive dynamics prevalent in real-world social and biological systems. In response to this, some research has introduced evolutionary game theory into group behavior studies, revealing how agents evolve into evolutionary stable strategies (ESS) through iterative adaptive learning processes under bounded rationality. This perspective highlights the complexity of strategy selection in environments that contain both cooperation and competition. Despite the important progress made by the above research, evolutionary systems often encounter so-called "cooperation dilemmas": betrayal strategies can bring higher individual benefits in the short term but may undermine the overall stability of the group.

[0005] Currently, the focus of research has gradually shifted from passive observation and analysis of evolutionary dynamics to active strategic intervention to promote sustained cooperation. However, there is still a key research gap in existing technology: existing frameworks often separate trust assessment from control design, lacking a unified mathematical form to characterize the impact of trust dynamics on system behavior. Although Huang et al. demonstrated the potential of Zero-Trust Architecture (ZTA) in the security of Internet of Vehicles, their model failed to fully integrate the trust evolution mechanism. Traditional security protocols that rely on single-step authentication are still vulnerable to continuously evolving attacks. For example, De Cusatis et al. proposed to enhance access security mechanisms through "first packet authentication", but still did not consider the propagation effect of trust in multi-agent environments. While the existing proposed zero-trust principles establish architectural standards, they have not yet clearly defined the implementation path in group cooperative control. SUMMARY

[0006] In view of the above problems of the prior art, the technical problem to be solved by the present application is to provide a multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture, which deeply integrates the "continuous verification" concept of zero-trust architecture and the dynamic strategy adjustment mechanism of evolutionary game theory, breaking through the limitations of "static trust" and "fixed strategy" in traditional multi-agent cooperative control, thereby improving the security of the multi-agent cooperative control system and ensuring that the system quickly reaches a stable state in complex interactions.

[0007] To solve the above technical problems, the present application adopts the following technical solutions:

[0008] The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture comprises:

[0009] S1: Construct a multi-agent system with a population size of n; all agents (i.e. individuals) in the multi-agent system are distributed on the vertices of a communication topology graph and define a strategy set for each agent; wherein the strategy set includes cooperative strategy A and betrayal strategy B;

[0010] S2: Each agent randomly selects one of the cooperative strategy S and the betrayal strategy B as the initial strategy, and plays a game with all its neighbors respectively;

[0011] S3: Calculate the real-time trust degree of each agent based on the historical cooperation rate and the real-time zero-trust verification score;

[0012] S4: Calculate the fitness of each agent based on the interaction income and neighbor relationship in the interaction result;

[0013] S5: Calculate the comprehensive evaluation value of each agent based on the real-time trust degree and the fitness;

[0014] S6: determining whether to update the strategy of each agent based on the difference between the comprehensive evaluation values of each agent and its neighbors;

[0015] S7: repeating steps S2 to S6 to continuously select and update the strategy of each agent until the entire multi-agent system reaches an evolutionary stable state, and achieving stable system control of the agent group based on the strategy of each agent.

[0016] Preferably, in step S1, each agent Agent i is defined as a five-tuple, i.e.:

[0017] Agent i =(S i ,B i ,C i ,F i ,T i );

[0018] In the formula: S i represents the state of the agent Agent i , which is a two-dimensional strategy space S i ∈{A,B}, where the strategy set is composed of a cooperation strategy A and a betrayal strategy B; B i represents the behavior of the agent Agent i , which is used to update the strategy under a pre-defined static adjacency matrix If the agent Agent i interacts with the agent Agent j , then a ij =1, otherwise a ij =0; C i represents the neighbor set of the agent Agent i ; F i represents the fitness of the agent Agent i ; and T i represents the real-time trust degree of the agent Agent i .

[0019] Preferably, in step S3, the real-time trust degree of the agent is calculated through a trust evaluation model;

[0020] The formula of the trust evaluation model is:

[0021] T i (t)=w h Z i (t)+w z V i (t);

[0022] In the formula: T i(t) represents the agent. i Real-time trust level; Z i (t) represents the agent. i Historical cooperation rate; V i (t) represents the agent. i Real-time zero-trust verification score; w h w z Indicates the weighting coefficient;

[0023] in:

[0024]

[0025] In the formula: C i (t′) represents the agent. i The number of successful collaborations at time t′; N i (t′) represents the agent. i The total number of interactions at time t′; α∈(0,1) represents the time decay factor; v ij (t)∈[0,1] represents the agent. i The probability of passing the j-th verification step at time t; w j Let the weight of the j-th step be such that it satisfies

[0026] Preferably, in step S3, v is calculated. ij The verification steps at time (t) include: identity legitimacy, which is verified by using a unique identifier; and data integrity, which is verified using a cryptographic hash.

[0027] Preferably, in step S4, the fitness of the agent is calculated using the following formula:

[0028]

[0029] In the formula: F i Agent i fitness; k i Agent i The number of neighbors; π ij Agent i With Neighbor Agent j Interactive benefits; Neighbor set C i ={j∈N∣a ij >0},a ij =1 indicates an agent i With Neighbor Agent j There is an interactive relationship, a ij =0 indicates that there is no interaction relationship.

[0030] Preferably, in step S4, the interaction revenue is calculated using the following revenue matrix:

[0031]

[0032] In the formula: A and B represent cooperation strategy and betrayal strategy; b and c represent benefits and costs.

[0033] Preferably, in step S5, the comprehensive evaluation value of the agent is calculated using the following formula:

[0034] CEV i (t)=w1F i (t)+w2T i (t);

[0035] Where: CEV i (t) represents the agent. i The overall evaluation value; F i (t) represents the agent. i fitness; T i (t) represents the agent. i The real-time trust level; w1, w2∈[0,1] represents the weights of fitness and real-time trust level, and w1+w2=1.

[0036] Preferably, in step S6, based on the difference in comprehensive evaluation values ​​between each agent and its neighbors, a probabilistic decision is made on whether to adopt the neighbor's strategy, wherein the agent... i Adopt Neighbor Agent j The probability of the strategy is:

[0037]

[0038] In the formula: P(S) i →S j ) represents the intelligent agent. i Adopt Neighbor Agent j The probability of the strategy, S i S j They represent the intelligent agent. i Neighbor Agent j The current strategy; CEV i CEV j They represent the intelligent agent. i Neighbor Agent j The overall evaluation value; C i ={j∈N∣a ij ω > 0 represents the neighbor set; ω > 0 represents the selection strength.

[0039] Compared with existing technologies, the multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture in this invention has the following advantages:

[0040] This invention overcomes the limitations of "static trust" and "fixed policy" in traditional multi-agent cooperative control by deeply integrating the "continuous verification" concept of zero-trust architecture with the dynamic policy adjustment mechanism of evolutionary game theory. Zero-trust architecture requires each agent to undergo authentication, behavior auditing, and dynamic authorization in every interaction. Combined with policy replication, mutation, and selection mechanisms in evolutionary game theory, this enables the agent swarm to achieve "trust-policy" co-evolution in a dynamic network environment, thereby improving the security of the multi-agent cooperative control system and ensuring that the system quickly reaches a stable state in complex interactions.

[0041] This invention constructs a more accurate agent decision-making framework through a two-dimensional comprehensive evaluation of real-time trust (based on historical cooperation rate and zero-trust verification score) and fitness (based on interaction benefits and neighbor relationships). Real-time trust reflects the agent's historical reliability and current security status, while fitness quantifies its contribution to benefits in collaborative tasks. The combination of these two factors forms a dynamic evaluation value, ensuring that policy update decisions consider both security and trustworthiness as well as task efficiency. This two-factor driven mechanism avoids the one-sidedness of traditional single-dimensional evaluation. Furthermore, during agent group policy updates, decisions are made based on the difference in comprehensive evaluation values ​​with neighbors, effectively promoting the diffusion of high-quality policies and eliminating low-quality policies, accelerating system convergence to an evolutionarily stable policy, thereby improving the trustworthiness and robustness of the multi-agent collaborative control system.

[0042] This invention achieves a precise "spatiotemporal dual-dimensional" characterization of trust assessment by dynamically weighting the real-time trust level using historical cooperation rate (reflecting long-term behavioral patterns) and real-time zero-trust verification score (reflecting the current security status). The historical cooperation rate quantifies the long-term trustworthiness of the agent by recording data such as the frequency of cooperation and the number of breaches in past interactions; the zero-trust verification score dynamically assesses the current security status through real-time identity authentication, behavioral analysis, and anomaly detection technologies. This dynamic trust level calculation mechanism enables the system to more sensitively identify changes in node status and adjust the trust level in a timely manner, improving the system's flexibility and adaptability while ensuring security.

[0043] This invention incorporates interaction rewards (reflecting an agent's actual contribution to a task) and neighbor relationships (reflecting its connection strength and position in the network) into fitness calculation, constructing a reward evaluation model that better reflects real-world collaborative scenarios. Interaction rewards directly reflect an agent's task contribution by quantifying the agent's gains and costs in task execution; neighbor relationships indirectly reflect its role in the collaborative network by analyzing its connection density and information flow efficiency with neighboring nodes. This combination allows fitness evaluation to consider both individual task performance and its structural role in the network, thus more comprehensively reflecting the agent's overall value. Furthermore, it enables agents to actively maintain good neighbor relationships while pursuing their own gains, promoting the connectivity and stability of the collaborative network, ultimately improving the task execution efficiency of the entire agent swarm and the overall system performance. Attached Figure Description

[0044] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0045] Figure 1 This is a logical block diagram of a multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture.

[0046] Figure 2 The evolution curve of the proportion of cooperative strategies. Detailed Implementation

[0047] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but only to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0048] It should be noted that similar reference numerals and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the figures, or the orientation or positional relationship commonly used when the product is in use. They are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third," etc., are only used to distinguish descriptions and should not be construed as indicating or implying relative importance. In addition, the terms "horizontal," "vertical," etc., do not mean that the component is required to be absolutely horizontal or suspended, but can be slightly tilted. For example, "horizontal" only means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted. In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0049] The following detailed explanation illustrates the specific implementation methods:

[0050] Example:

[0051] This embodiment discloses a multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture.

[0052] like Figure 1 As shown, a multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture includes:

[0053] S1: Construct a multi-agent system with a population size of n; all agents (individuals) in the multi-agent system are distributed on the vertices of the communication topology graph and each agent has a set of policies defined; the set of policies includes cooperation policy A and betrayal policy B;

[0054] In this embodiment, by distributing all agents in the multi-agent system on the vertices of the communication topology graph, the multi-agent cooperative optimization problem is abstracted into a mathematical graph theory problem, which is beneficial for better subsequent game theory analysis and proof.

[0055] S2: Each agent chooses either the cooperative strategy A or the defective strategy B as its initial strategy and plays against all its neighbors separately.

[0056] S3: Calculate the real-time trust level of each agent based on historical cooperation rate and real-time zero-trust verification score;

[0057] S4: Calculate the fitness of each agent based on the interaction benefits and neighbor relationships in the interaction results;

[0058] S5: Calculate the comprehensive evaluation value of each agent based on real-time trust and fitness;

[0059] S6: Determine whether to update the policies of each agent based on the differences in the comprehensive evaluation values ​​of each agent and its neighbors;

[0060] S7: Repeat steps S2 to S6, continuously selecting and updating the policies of each agent until the entire multi-agent system reaches an evolutionary stable state, and then achieve stable system control of the agent swarm based on the policies of each agent.

[0061] This invention overcomes the limitations of "static trust" and "fixed policy" in traditional multi-agent cooperative control by deeply integrating the "continuous verification" concept of zero-trust architecture with the dynamic policy adjustment mechanism of evolutionary game theory. Zero-trust architecture requires each agent to undergo authentication, behavior auditing, and dynamic authorization in every interaction. Combined with policy replication, mutation, and selection mechanisms in evolutionary game theory, this enables the agent swarm to achieve "trust-policy" co-evolution in a dynamic network environment, thereby improving the security of the multi-agent cooperative control system and ensuring that the system quickly reaches a stable state in complex interactions.

[0062] This invention constructs a more accurate agent decision-making framework through a two-dimensional comprehensive evaluation of real-time trust (based on historical cooperation rate and zero-trust verification score) and fitness (based on interaction benefits and neighbor relationships). Real-time trust reflects the agent's historical reliability and current security status, while fitness quantifies its contribution to benefits in collaborative tasks. The combination of these two factors forms a dynamic evaluation value, ensuring that policy update decisions consider both security and trustworthiness as well as task efficiency. This two-factor driven mechanism avoids the one-sidedness of traditional single-dimensional evaluation. Furthermore, during agent group policy updates, decisions are made based on the difference in comprehensive evaluation values ​​with neighbors, effectively promoting the diffusion of high-quality policies and eliminating low-quality policies, accelerating system convergence to an evolutionarily stable policy, thereby improving the trustworthiness and robustness of the multi-agent collaborative control system.

[0063] In summary, this invention introduces a novel comprehensive evaluation mechanism that combines fitness with dynamically updated trust. This trust model integrates historical cooperation success rates with exponential decay effects with real-time verification results, aligning with the principles of continuous authentication and dynamic authorization in zero-trust security. Through theoretical analysis and weak selection approximation, this invention derives the system's evolutionary dynamics equations and obtains explicit expressions for equilibrium conditions and equilibrium points. Results show that this model not only promotes stable policy evolution but also effectively adapts to dynamic environments and heterogeneous agent behaviors. This research provides a feasible theoretical framework and practical reference for secure and adaptive cooperation in multi-agent systems under zero-trust constraints.

[0064] To better illustrate the technical solution of the present invention, this embodiment is described in the following parts.

[0065] I. Interaction Model of Multi-Agent Systems

[0066] In this embodiment, the interaction model and trust evaluation mechanism of a multi-agent system under a zero-trust architecture are formally defined. To address the challenges of collaborative control in zero-trust scenarios, the states and behaviors of agents are defined as follows:

[0067] Consider a population of n agents (N = {1, 2, ..., n}), where each agent is... i Defined as a quintuple, namely: Agent i =(S i B i C i ,F i ,T i );

[0068] in:

[0069] S i The state is represented by a binary policy space S. i ∈{A,B}, where the set of strategies consists of cooperative strategy A and betrayal strategy B, and the agent dynamically selects strategy A or B based on local benefit evaluation;

[0070] B i The behavior is represented by a probabilistic decision rule used in a predefined static adjacency matrix. Next update strategy, if Agent i With Agent j Interaction, then a ij =1, otherwise a ij =0;

[0071] C i This represents communication, specifically the neighbor set C. i ={j∈N∣a ij>0} defines the interaction range of the intelligent agent;

[0072] F i Fitness is a competitive metric calculated by weighting interaction rewards and neighbor relationships, reflecting the agent's adaptability and performance in the system.

[0073] T i It represents the real-time trust level, a dynamic evaluation value that combines historical cooperation success rates with real-time security verification.

[0074] II. Trust Assessment Model

[0075] In a zero-trust architecture, trust between agents is dynamically quantified through a two-layer evaluation mechanism.

[0076] Specifically, the trust assessment model is defined as:

[0077] T i (t)=w h T i (t)+w z V i (t);

[0078] In the formula: T i (t) represents the agent. i Real-time trust level; Z i (t) represents the agent. i The historical cooperation rate reflects its long-term reliability; V i (t) represents the agent. i The real-time zero-trust authentication score is used to assess the security of the current interaction; h w z Denotes the weighting coefficients, satisfying w h +w z =1, used to balance historical credibility with immediate security risks.

[0079] 1) Historical cooperation rate: Historical cooperation rate Z i (t) is calculated using an exponentially weighted moving average (EWMA) with time decay:

[0080]

[0081] In the formula: C i (t′) represents the agent. i The number of successful collaborations at time t′ (i.e., interactions using complementary strategies without conflict); N i (t′) represents the agent. iThe total number of interactions at time t′; α∈(0,1) represents the time decay factor, which controls the decay rate of the influence of historical data (the larger α is, the more recent behavior is emphasized).

[0082] This invention ensures that early interactions gradually lose their influence through this model, enabling the system to better adapt to changes in the agent's behavior in dynamic environments.

[0083] 2) Real-time zero-trust verification: Real-time verification score V i (t) is a weighted fusion of multi-step security detection results:

[0084]

[0085] In the formula: v ij (t)∈[0,1] represents the agent. i The probability of passing the j-th verification step at time t; w j Let the weight of the j-th step be such that it satisfies The verification steps include: identity legitimacy, which is verified through a unique identifier (such as a digital certificate or hash fingerprint); and data integrity, which is verified using cryptographic hashes (such as SHA-256) to detect tampering.

[0086] Finally, combining historical cooperation with real-time verification, the complete trust model can be expressed as:

[0087]

[0088] This invention provides a rigorous and scalable mathematical framework for trust-aware collaboration in adversarial multi-agent environments. Simultaneously, it achieves a precise "spatiotemporal dual-dimensional" characterization of trust assessment by dynamically weighting the real-time trust level using historical cooperation rate (reflecting long-term behavioral patterns) and real-time zero-trust verification score (reflecting the current security state). The historical cooperation rate quantifies the long-term trustworthiness of agents by recording data such as the frequency of collaboration and the number of breaches in past interactions; the zero-trust verification score dynamically assesses the current security state through real-time identity authentication, behavioral analysis, and anomaly detection techniques. This dynamic trust level calculation mechanism enables the system to more sensitively identify changes in node states and adjust trust levels promptly, improving system flexibility and adaptability while ensuring security.

[0089] III. Evolutionary Game Theory Framework and Division of Labor Control Objectives

[0090] In recent years, interdisciplinary research has increasingly combined evolutionary game theory with advanced dynamic control methods, directly embedding complex trust assessment mechanisms and real-time adaptive strategies into the interaction process of intelligent agents. These integrated approaches can construct more robust, context-aware decision-making models that more closely resemble real-world operating conditions, thereby enhancing the system's resilience to internal uncertainties and external disturbances. By dynamically quantifying and updating trust levels based on agents' historical cooperative performance, reliability, and real-time behavioral verification, these frameworks can accurately adapt to rapidly changing network conditions and agent behavior.

[0091] Furthermore, coupling evolutionary game dynamics with adaptive trust management significantly enhances the ability of multi-agent systems to maintain stable cooperation, effectively mitigates the impact of untrusted agents, and rapidly responds to emerging competitive pressures. This fusion not only optimizes resource allocation and task execution efficiency but also ensures system robustness, particularly in adversarial or unpredictable environments. Therefore, this interdisciplinary approach has significant value in numerous practical applications, including: cooperative control of autonomous vehicle networks, task allocation in distributed robot swarms under uncertain terrain, adaptive assessment of information reliability in large-scale intelligent sensor deployments, trust-driven power grid load allocation, and governance of complex socio-technical systems accompanied by dynamic trust evolution.

[0092] In summary, the integration of evolutionary game theory, adaptive control strategies, and dynamic trust assessment provides a powerful theoretical and practical tool for bridging the gap between idealized agent interaction models and the complexities of real-world operation. This lays the foundation for the design of future intelligent, adaptive, and trustworthy multi-agent systems.

[0093] The "Snowpile Game" is a modeling approach for classic social dilemmas: two drivers are stranded on opposite sides of a snow-blocked road and must independently decide whether to cooperate (e.g., jointly clearing the snow and incurring costs) or betray (relying on the other's efforts). The payoff structure is as follows: if both cooperate, the cost of clearing the snow is shared; if one cooperates and the other betrays, the betrayer benefits more, while the cooperating party incurs a greater cost; if both betray, the system fails. The parameters are the payoff (b, the benefit gained from successfully clearing the snow) and the cost (c, the labor input), satisfying the condition b > c > 0. This relationship guarantees the existence of a Nash equilibrium, where agents naturally choose the opposite strategy.

[0094] In this embodiment, when extended to multi-agent networks, this dynamic mechanism drives spontaneous role differentiation among neighboring agents, leading to complementary task allocation—such as resource scheduling and direct task execution—ultimately achieving stable division of labor and cooperation. Furthermore, the evolutionary process introduces a trust metric based on historical cooperation success rates and real-time security verification. By dynamically adjusting the trust assessment, the framework effectively punishes unreliable or potentially malicious agents while incentivizing trustworthy behavior. Integrating trust assessment into evolutionary policy updates not only improves task execution and operational efficiency but also significantly enhances the system's resilience in adversarial or uncertain environments. This approach bridges the gap between theoretical game theory models and practical applications, providing scalable and robust solutions for various applications such as distributed sensor networks and collaborative autonomous systems.

[0095] Specifically, the payoff matrix is ​​shown below:

[0096]

[0097] When b > c, classical game theory analysis shows that the Nash equilibrium points to a strategy dichotomy: betrayal is the optimal choice when the opponent cooperates; cooperation brings greater benefits when the opponent betrays. In the context of a division of labor system, the snowplow game transcends the simple binary opposition of cooperation and betrayal; its essence lies in strategic differentiation—that is, assigning agents different functional roles (such as resource acquisition and processing) to achieve system collaboration. Here, strategic choice directly maps to operational specialization, rather than merely conflict resolution. To formalize this role-based collaboration, this embodiment uses the aforementioned 2×2 payoff matrix under the condition b > c.

[0098] In this embodiment, the control objective requires the agent to obtain a higher gain (i.e., S) when adopting complementary strategies with its neighbors. i ≠S j ,j∈C i The fitness F of the agent i This refers to local competitiveness, calculated through a weighted average of interactive benefits.

[0099] Therefore, this invention calculates the fitness of the agent using the following formula:

[0100]

[0101] In the formula: F i Agent i fitness; k i Agent i The number of neighbors (i.e., the Agent) i (degree); π ij Agent i With Neighbor Agentj Interactive benefits; Neighbor set C i ={j∈nF∣a ij >0},a ij =1 indicates an agent i With Neighbor Agent j There is an interactive relationship, a ij =0 indicates that there is no interaction relationship.

[0102] This invention ensures that agents who prioritize interacting with neighbors whose roles are complementary achieve higher fitness, thus driving the system towards a stable task division. Simultaneously, this invention incorporates interaction rewards (reflecting the agent's actual contribution to the task) and neighbor relationships (reflecting its connection strength and position in the network) into fitness calculation, constructing a reward evaluation model that better reflects real-world collaborative scenarios. Interaction rewards directly reflect an agent's task contribution by quantifying the agent's gains and costs in task execution; neighbor relationships indirectly reflect its role in the collaborative network by analyzing its connection density and information flow efficiency with neighboring nodes. This combination allows fitness evaluation to consider both individual task performance and its structural role in the network, thus more comprehensively reflecting the agent's overall value. Furthermore, it enables agents to actively maintain good neighbor relationships while pursuing their own gains, promoting the connectivity and stability of the collaborative network, ultimately improving the task execution efficiency of the entire agent swarm and the overall system performance.

[0103] IV. Strategy Evolution Mechanism

[0104] To dynamically balance task efficiency and safety in multi-agent cooperative systems, this invention proposes a Comprehensive Evaluation Value (CEV) policy evolution mechanism, supplemented by a probabilistic policy update method. CEV quantifies the overall performance of the agent, integrating fitness (task efficiency) and trust (safety and compliance), thereby supporting adaptive decision-making that considers both historical interactions and real-time verification.

[0105] Therefore, this invention calculates the comprehensive evaluation value of the agent using the following formula:

[0106] CEV i (t)=w1F i (t)+w2T i (t);

[0107] Where: CEV i (t) represents the agent. i The overall evaluation value; F i (t) represents the agent. ifitness; T i (t) represents the agent. i The real-time trust level; w1, w2∈[0,1] represents the weights of fitness and real-time trust level, and w1+w2=1.

[0108] Policy Update Rule (Based on Fermi Process): The agent updates its policy through a stochastic learning rule, namely the Fermi process. In this process, the agent decides whether to adopt the neighbor's policy probabilistically based on the difference in CEV between itself and its neighbors.

[0109] Therefore, this invention determines whether to adopt the neighbor's strategy based on the difference in comprehensive evaluation values ​​between each agent and its neighbors, using a probabilistic approach. The agent... i Adopt Neighbor Agent j The probability of the strategy is:

[0110]

[0111] In the formula: P(S) i →S j ) represents the intelligent agent. i Adopt Neighbor Agent j The probability of the strategy, S i S j They represent the intelligent agent. i Neighbor Agent j The current strategy; CEV i CEV j They represent the intelligent agent. i Neighbor Agent j The overall evaluation value; C i ={j∈N∣a ij ω>0 represents the neighbor set; ω>0 represents the selection strength, which is used to adjust the impact of the difference in the overall evaluation value on the decision. A larger ω value will amplify the sensitivity to the performance gap, thereby accelerating the convergence to the dominant strategy.

[0112] This invention ensures that the system can dynamically balance efficiency and safety during the evolution process through this mechanism, thereby achieving more resilient collaborative control.

[0113] V. Theoretical Analysis of Equilibrium Point

[0114] In this embodiment, the reasons for performing equilibrium point analysis include: 1) determining whether the parameter size relationship within the selected payoff matrix in this type of game satisfies the background setting; and 2) determining whether an evolutionary stable state can eventually be reached in this system, i.e., whether the evolutionary outcome is that individuals with the two strategies coexist in roughly equal proportions.

[0115] Specifically, the steps for equilibrium point theory analysis are as follows:

[0116] 1) Analyze a finite structured community distributed on a static undirected graph. For simplicity, the adjacency matrix A = [a ij All non-zero elements in the array are set to a. ij =1. The system consists of N nodes, where vertices represent agents and edges represent paired interactions. Each agent engages in a game of division of labor with its neighbors, choosing either strategy A or B. Key variables are defined as follows:

[0117] p B =1-p A (The proportion of agents for role B, where role B represents the agent that chooses strategy B); where p A and p B This represents the frequency of strategy A and strategy B in a multi-agent population.

[0118] The proportion of paired strategies: p AA ,p AB ,p BA ,p BB ; where p AA p represents the frequency of policy pairs (AA) formed by two adjacent agents; AB and p BA p represents the frequency of policy pairs AB or BA formed by two adjacent agents; BB This represents the frequency of the policy pair BB formed by two adjacent agents;

[0119] conditional probability q i|j The probability that a neighbor of an agent with policy j adopts policy i.

[0120] 2) The variables satisfy the following relationship:

[0121] p AB =p BA =p A (1-q A|A );

[0122] p BB =p B ·q B|B =1-2p A +p A q A|A ;

[0123]

[0124] p AA +p AB +p BA +p BB =pA q A|A +2p A (1-q A|A )+(1-2p A +p A q A|A ) = 1;

[0125] 3) Probability of change in strategy proportion:

[0126] Probability of change in strategy ratio: Character A's ratio increases The probability is:

[0127]

[0128] In the formula: k A and k B k represents the number of individuals among the agent's neighbors who use policy A and policy B. A +k B =k;

[0129] Character A's proportion decreased The probability is:

[0130]

[0131] Proportion to p AA Increase The probability is:

[0132]

[0133] p AA reduce The probability is:

[0134]

[0135] 4) Under the assumption that only one policy update occurs per unit time, the proportion p A The evolution equation is:

[0136] in:

[0137]

[0138] Substitute:

[0139]

[0140] Under symmetric equilibrium conditions The equation simplifies to:

[0141]

[0142] Considering disturbances Substituting the values, we obtain the first-order correction:

[0143]

[0144] Therefore, the equilibrium point is:

[0145]

[0146] 5) Further analysis of paired proportions p AA Evolution:

[0147]

[0148] When ω=0, the equation simplifies to:

[0149]

[0150] Under symmetry conditions, the equilibrium point is:

[0151]

[0152] VI. Experimental Instructions

[0153] In this embodiment, a multi-agent cooperative control model is proposed under a Zero-Trust Architecture (ZTA), incorporating evolutionary game theory to achieve a balance between cooperative efficiency and security. The model's performance is validated through MATLAB simulations on three classic network topologies: random networks, scale-free networks, and small-world networks. Experimental analysis focuses on the dynamic evolution of policies in these different structures and, combined with a security verification mechanism, systematically examines the trade-off between cooperative efficiency and robustness. Key evaluation metrics include: the convergence speed of role differentiation, the probability of trust-driven policy adoption, and resilience under simulated adversarial perturbations. Results show that the framework can achieve adaptive task allocation while adhering to the zero-trust principle; among them, due to its "highly clustered yet decentralized" connection mode, small-world networks exhibit a better balance between rapid consensus and attack resistance.

[0154] The experiment used a network of 200 agents. Initially, agents were randomly assigned either cooperation (policy A) or betrayal (policy B) at a ratio of approximately 1:1. Key parameters included: fitness weight w1 = 1.0, trust weight w2 = 0.8, and zero-trust verification weight w... z=0.3, and snowplow game parameters b=2, c=1 (to ensure that cooperation has an advantage in evolution). By adjusting the network generating function, three topologies are constructed: (1) small-world network (Watts–Strogatz model, reconnection probability β=0.3): simulating the balance between local cooperation and global coordination; (2) scale-free network (Barabási–Albert model, preferred connection number m=2): reflecting the centralized characteristics of hub nodes in real systems; (3) random network ( Model (with edge connection probability p = 0.05): serves as a control group for unstructured guidance.

[0155] Figure 2 The proportion of cooperative strategies p in three types of networks is shown. A The evolution trajectory of (t). Under the conditions of no external interference (proportion of malicious nodes = 0) and symmetric parameters, all networks eventually converge to the theoretical equilibrium point. The theoretical predictions of the model were verified. In the small-world network (blue curve), due to its high clustering properties, the policy feedback between neighbors forms a local positive loop, which enables the system to quickly approach the equilibrium point within 150 steps, and the fluctuations after convergence are minimal.

[0156] In scale-free networks (red curve), leveraging the strategic propagation advantage of hub nodes, the system approaches equilibrium in approximately 120 steps. However, subsequently at p A The network exhibits persistent small oscillations within the range of 0.48 to 0.52, stemming from random defection behavior at edge nodes, highlighting the dual impact of hub nodes on global stability. In the random network (green curve), the lack of structural guidance necessitates policy updates relying on long-range random interactions, extending the convergence time to 250 steps. Path redundancy further contributes to temporary imbalances, emphasizing the crucial role of network topology in regulating cooperative efficiency.

[0157] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture, characterized in that, include: S1: Construct a multi-agent system with a population size of n; Define a set of policies for each agent in a multi-agent system; the set of policies includes a cooperative policy A and a defective policy B. S2: Each agent chooses either the cooperative strategy A or the betrayal strategy V as its initial strategy and plays against all its neighbors separately. S3: Calculate the real-time trust level of each agent based on historical cooperation rate and real-time zero-trust verification score; S4: Calculate the fitness of each agent based on the interaction benefits and neighbor relationships in the interaction results; S5: Calculate the comprehensive evaluation value of each agent based on real-time trust and fitness; S6: Determine whether to update the policies of each agent based on the differences in the comprehensive evaluation values ​​of each agent and its neighbors; S7: Repeat steps S2 to S6, continuously selecting and updating the policies of each agent until the entire multi-agent system reaches an evolutionary stable state, and then achieve stable system control of the agent swarm based on the policies of each agent.

2. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 1, characterized in that: In step S1, each agent... i Defined as a quintuple, that is: Agent i =(S i ,B i ,C i ,F i ,T i ); In the formula: S i Agent i The state is a binary policy space S i ∈{A,B}, where the set of strategies consists of cooperative strategy A and defective strategy B; B i Agent i The behavior used in a predefined static adjacency matrix Next update strategy, if Agent i With Agent j Interaction, then a ij =1, otherwise a ij =0; C i Agent i The set of neighbors; F i Agent i fitness; T i Agent i Real-time trust level.

3. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 1, characterized in that: In step S3, the agent's real-time trust level is calculated using a trust assessment model; The formula for the trust assessment model is expressed as follows: T i (t)=w h Z i (t)+w z V i (t); In the formula: T i (t) represents the agent. i Real-time trust level; Z i (t) represents the agent. i Historical cooperation rate; V i (t) represents the agent. i Real-time zero-trust verification score; w h w z Indicates the weighting coefficient; in: In the formula: C i (t′) represents the agent. i The number of successful collaborations at time t′; N i (t′) represents the agent. i The total number of interactions at time t′; α∈(0,1) represents the time decay factor; v ij (t)∈[0,1] represents the agent. i The probability of passing the j-th verification step at time t; w j Let the weight of the j-th step be such that it satisfies 4. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 3, characterized in that: In step S3, calculate v ij The verification steps at time (t) include: identity legitimacy, which is verified by using a unique identifier; and data integrity, which is verified using a cryptographic hash.

5. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 1, characterized in that: In step S4, the agent's fitness is calculated using the following formula: In the formula: F i Agent i fitness; k i Agent i The number of neighbors; π ij Agent i With Neighbor Agent j Interactive benefits; Neighbor set C i ={j∈N∣a ij >0},a ij =1 indicates an agent i With Neighbor Agent j There is an interactive relationship, a ij =0 indicates that there is no interaction relationship.

6. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 5, characterized in that: In step S4, the interaction revenue is calculated using the following revenue matrix: In the formula: A and B represent cooperation strategy and betrayal strategy; b and c represent benefits and costs.

7. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 1, characterized in that: In step S5, the overall evaluation value of the agent is calculated using the following formula: CEV i (t)=w1F i (t)+w2T i (t); Where: CEV i (t) represents the agent. i The overall evaluation value; F i (t) represents the agent. i fitness; T i (t) represents the agent. i The real-time trust level; w1, w2∈[0,1] represents the weights of fitness and real-time trust level, and w1+w2=1.

8. The multi-agent cooperative control method based on evolutionary game theory under a zero-trust architecture as described in claim 1, characterized in that: In step S6, based on the differences in comprehensive evaluation values ​​between each agent and its neighbors, a probabilistic decision is made on whether to adopt the neighbor's strategy. The agent... i Adopt Neighbor Agent j The probability of the strategy is: In the formula: P(S) i →S j ) represents the intelligent agent. i Adopt Neighbor Agent j The probability of the strategy, S i S j They represent the intelligent agent. i Neighbor Agent j The current strategy; CEV i CEV j They represent the intelligent agent. i Neighbor Agent j The overall evaluation value; C i ={j∈N∣a ij ω > 0 represents the neighbor set; ω > 0 represents the selection strength.

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