Simulation study of enterprise intelligent transformation behavior based on complex network evolutionary game

Abstract

The accelerating global economic competition and the rapid development of intelligent technologies present both new opportunities and challenges for enterprises. Intelligent transformation has become an imperative trend for enhancing competitiveness, yet Chinese enterprises are still in the preliminary stages. Focusing on the supply-side (intelligent server providers) and the demand-side (adopting enterprises), this study develops a two-layer heterogeneous complex network model grounded in complex network and evolutionary game theories. We analyze the dynamic evolutionary mechanisms and key influencing factors of strategic choices for both types of firms under different scenarios. Python-based simulations reveal that increased government subsidies, reduced intelligent server costs, higher additional benefits from transformation, and appropriate pricing strategies all promote evolutionary cooperation between the two sides. Furthermore, the network structure significantly impacts strategic selection. The model’s parameters are calibrated using 2023 financial data from Foxconn Industrial Internet Co., Ltd. to anchor the simulation in a representative large-enterprise scenario. This research extends the study of intelligent transformation from a static perspective to a dynamic, spatial-relationship-aware view, and addresses the limitation of participant homogeneity by employing a two-layer heterogeneous network model, thereby providing theoretical support and context-specific insights for enterprise intelligent transformation.

Introduction

The escalating intensity of global economic competition, coupled with the swift evolution of intelligent technologies (such as Artificial Intelligence, Big Data, Cloud Computing, and Industrial Internet of Things), is profoundly reshaping production, management, and service paradigms¹. Driven by advancements in computing power and data resources, intelligent technologies have moved from foundational research to large-scale commercial application, becoming a critical catalyst for improving enterprise operational efficiency, optimizing decision-making processes, and fostering new business models². This context underscores the strategic necessity and urgency of intelligent transformation as governments and enterprises worldwide adopt it as a key approach to drive industrial upgrading and enhance competitiveness.

Specifically for Chinese enterprises—especially those at the supply and demand ends of the intelligent server value chain—intelligent transformation faces both opportunities and deficiencies³. On one hand, China has achieved remarkable progress in intelligent technology R&D and policy support, leading to tangible productivity and service quality improvements in several application areas. On the other hand, gaps persist in the depth of commercialization, industrial synergy, organizational execution, and upstream-downstream collaboration compared to international leading standards. Existing research on enterprise intelligent transformation predominantly focuses on macro-econometric or cross-sectional case analyses, examining the impact of smart technologies on productivity and industrial structure. However, these studies often overlook the crucial aspects of inter-firm strategic interaction, spatial/network dependence, and the dynamic evolutionary process, thereby limiting a comprehensive spatio-temporal understanding of transformation pathways and policy instruments⁴. To address these limitations, this study explicitly integrates network structure, strategic interaction, and dynamic evolution into a unified analytical framework.

To address these shortcomings, the academic community has made several attempts: introducing evolutionary game theory and complex network methods to capture strategy diffusion and evolution within networks⁵; conducting simulation studies to identify equilibrium states and strategic evolution trends under varying parameters⁶; and employing scenario analysis for policy variables and firm behavior⁷. Despite these efforts, several systemic deficiencies remain: First, most studies rely on static or homogenous assumptions, making it difficult to capture the heterogeneity and cross-layer coupling effects between the supply and demand sides. Second, single-layer network or homogenous node models fail to realistically reflect the bidirectional influence and strategy transmission paths between upstream and downstream sectors of the value chain. Third, many works lack parameter calibration based on real enterprise data and systematic robustness checks, thereby restricting the practical applicability and policy operability of the conclusions.

To bridge these gaps, this study proposes and implements a complete methodological framework. Specifically, the technical roadmap and steps are as follows: First, at the theoretical and modeling level, we construct a two-layer heterogeneous complex network, placing the intelligent server supply-side and demand-side in two separate layers to characterize heterogeneous node attributes, intra-layer topology, and inter-layer coupling relationships. Second, centering on the evolutionary game, we establish the strategic game framework for supply-side and demand-side enterprises, derive the replicator dynamic equations, perform local stability analysis via the Jacobian matrix, and compare the influence of different strategy update rules on evolutionary outcomes. Third, at the numerical and empirical level, large-scale simulations are conducted using Python, combined with Monte Carlo scenario experiments, sensitivity analysis, and parameter scanning, to systematically investigate the impact of government subsidies, server acquisition and setup costs, additional transformation benefits, server pricing, and network structure changes on strategic evolution and equilibrium results. To enhance the practical relevance of the simulation, key parameter ranges are calibrated using 2023 financial data from Foxconn Industrial Internet (FII) and its major clients as a representative large-enterprise scenario, followed by robustness testing and sensitivity analysis to examine the stability of results across parameter variations. Finally, based on the simulation results, scenario-based policy simulations and optimization recommendations are developed to identify the threshold conditions and key intervention means for promoting collaborative evolution between the supply and demand sides.

Based on the research methodology and technical roadmap described above, the main contributions of this study are threefold. First, on the theoretical level, this paper integrates evolutionary game theory with a two-layer heterogeneous complex network model to analyze the strategic evolutionary mechanism and equilibrium structure of supply-side and demand-side enterprises under network coupling, contributing insights that may help enrich the research perspective on enterprise intelligent transformation. Second, on the methodological level, we propose and implement a research framework that includes the construction of a two-layer heterogeneous network model, replicator dynamics and stability analysis, and numerical simulation, providing a replicable and extensible approach for studying multi-agent strategic evolution in complex systems. Third, on the practical level, through scenario analysis and threshold testing within the model, this study identifies key factors and simulation-based implications related to the coordinated evolution between supply and demand. These findings provide conditional and model-grounded insights that may inform policy discussions and enterprise decision-making, rather than constituting direct empirical evidence or implementation-ready recommendations.

The remainder of this paper is structured as follows: “Literature review” presents a literature review, systematically summarizing research progress in evolutionary games, complex networks, and enterprise intelligent transformation to clarify the research gaps and theoretical foundation. “Construction of a complex network game model for enterprise intelligent transformation” provides a status analysis, discussing the current state of enterprise intelligent transformation and delving into the internal and external factors influencing strategic choice. “Sensitivity analysis of parameters in the evolutionary game on complex networks for enterprise intelligent transformation” is dedicated to model construction, establishing the evolutionary game model, deriving the replicator dynamic equations, and constructing the two-layer complex network model. “Impact of dynamic rewards and dynamic network structure on game outcomes” covers simulation analysis, designing experiments to analyze the pattern of strategic evolution and propose policy recommendations for promoting transformation. “Discussion and conclusion” provides the discussion and conclusion, summarizing the findings, offering management and policy suggestions, elucidating research limitations, and providing an outlook for future research directions.

Literature review

Research progress on the evolution of intelligent technology and industrial empowerment

The evolution of intelligent technology has progressed from early symbolization and expert systems to the modern AI framework represented by statistical learning, deep learning, and large models⁸. The emergence of the Transformer architecture and large-scale pre-trained models in recent years has driven cross-industry applications in NLP, computer vision, and decision support systems, significantly lowering the barrier for intelligent technology deployment⁹. In terms of industrial empowerment, the literature concentrates on how intelligence improves total factor productivity and extends the industrial chain by optimizing production processes, enhancing forecasting capabilities, and reshaping service models¹⁰; it also highlights the crucial mediating role of the institutional environment, industrial clusters, talent, and platform ecosystems in technology diffusion and large-scale application¹¹. Existing studies further indicate that organizational change, process reengineering, and business model innovation brought by intelligence are key pathways to realizing technological value, noting that heterogeneity in returns arises from differences across industries and firm sizes¹². Therefore, research on intelligent technology and industrial empowerment requires the integration of macro-econometric evidence with micro-organizational behavior and application cases to explain the transformation mechanisms and boundary conditions under the interaction of technology, organization, and institution.

Drivers, pathways, and performance impact of enterprise intelligent transformation

Enterprise intelligent transformation is not merely an engineering process of technology adoption, but a systemic process of strategic, organizational, and capability reconstruction¹³. Studies based on the resource-based view and dynamic capabilities perspective suggest that internal resource endowments (e.g., data assets, R&D capability) and external environmental pressures (e.g., market competition, policy incentives) collectively drive enterprises to initiate intelligent strategies¹⁴. Econometric research widely finds that AI penetration and smart equipment investment enhance the marginal output of labor and capital, and promote product chain extension and innovation output, though these effects are moderated by firm size, industry nature, and human capital structure¹⁵. Micro-case studies further indicate that organizational governance, talent cultivation, and process restructuring are crucial intermediaries for translating technological benefits, while improper technology introduction may lead to a “silo effect” or resource mismatch, potentially compressing short-term performance¹⁶. In summary, the literature emphasizes systematic consideration from strategic formulation, capability building, to implementation pathways, especially the need to focus on the transmission mechanisms of benefits and costs between the supply and demand ends during transformation.

Research on strategic evolution and policy intervention from an evolutionary game perspective

Evolutionary game theory provides a natural framework for analyzing how boundedly rational agents adjust strategies in dynamic interaction, with update mechanisms like replicator dynamics and the Fermi rule being widely used to characterize the process of strategy diffusion¹⁷. Literature applying evolutionary games to corporate technology adoption and cooperation shows that local interaction structures, payoff matrix forms, and noise intensity significantly affect cooperation thresholds and steady-state distributions¹⁸; external incentives (such as subsidies, tax breaks) exhibit heterogeneous effects across different network topologies, sometimes demonstrating nonlinear thresholds or perverse incentives¹⁹. Recent research also combines evolutionary games with Stackelberg, public goods, and other models to analyze strategic coordination problems in multi-agent games involving government, enterprises, and consumers²⁰. For the intelligent transformation issue, evolutionary games can reveal the synergistic or antagonistic evolutionary paths of the supply side (service providers) and the demand side (adopters) under heterogeneous benefit and cost structures, offering dynamic insights on critical subsidy ranges, incentive combinations, and network intervention points for policy design.

Application progress of complex network methods and multi-layer/heterogeneous network modeling

Complex network theory emphasizes the profound impact of topological structure on propagation, robustness, and collective behavior, with small-world and scale-free network models explaining high clustering and hub effects in real systems²¹. Applying network methods to enterprise intelligence research, scholars focus on the influence of node heterogeneity (size, resources, integration capability) and positional effects (hub/peripheral) on technology diffusion and strategy imitation²². Multi-layer or two-layer network frameworks are utilized to differentiate the coupling relationships between upstream/downstream of the industrial chain or technology supply/demand interaction, thereby more realistically capturing cross-layer transmission and feedback mechanisms²³. Methodologically, research employs topological feature analysis, centrality metrics, propagation dynamics simulation, and agent-based simulation to explore how network structure amplifies or inhibits cooperative behavior²⁴. A recent trend is to combine network evolution with strategy update rules, allowing connection structures and strategies to co-evolve, to capture the impact of relationship restructuring on transformation pathways. This offers an important methodological tool for characterizing the mutual influence between the supply and demand sides in intelligent transformation²⁵.

Methodological framework, parameter calibration, and empirical validation based on agent-based simulation

To validate theory and evaluate policy effectiveness, the literature widely employs numerical methods such as agent-based simulation, Monte Carlo experiments, and sensitivity analysis²⁶. In terms of specific implementation, researchers mostly use tools like Python and MATLAB to construct reproducible simulation experiments, setting payoff matrices, strategy update rules, and various network topologies to evaluate evolutionary trajectories and steady-state distributions under different parameter combinations²⁷. For parameter calibration, studies utilize corporate financial data, industry statistics, or expert ratings to define cost, benefit, and subsidy ranges, thereby enhancing the reality of simulation conclusions and the operability of policy²⁸. Current methodological advances include parallelizing Monte Carlo for computational efficiency, parameter estimation based on Bayesian or least squares methods to enhance identifiability, and policy-oriented scenario simulation (e.g., subsidy gradients, price changes) to identify critical thresholds²⁹. However, the literature also points out that simulation results are sensitive to initial conditions and update rules, emphasizing the need for systematic robustness checks and comparison with micro-empirical data to enhance the credibility of conclusions³⁰.

Research gaps

Despite the rich progress in theoretical tools, methodological implementation, and empirical application, several systemic deficiencies persist concerning the intelligent transformation of enterprises at the supply and demand ends. First, timeliness and technical context lag: A large body of research is based on technology and policy scenarios prior to 2022–2023, failing to fully cover the impact of recent breakthroughs like large models and generative AI on value chain interaction. Second, methodologically, most studies rely on single-layer or homogenous assumptions, with few simultaneously characterizing the heterogeneous attributes of the supply and demand sides and their cross-layer coupling feedback within a two-layer heterogeneous network. Third, simulation and parameter calibration often rely on macro statistics or subjective expert assignment, lacking systematic calibration and external validity testing based on enterprise-level financial and behavioral data. Furthermore, existing policy analyses mostly concentrate on a single intervention variable (e.g., subsidies), lacking research on multi-strategy collaborative optimization and threshold identification under resource constraints. Finally, the linkage mechanism among network-strategy-policy has not formed a unified modeling and assessment paradigm, limiting the translation of research findings into policy formulation and enterprise practice. Based on these shortcomings, this study aims to fill the aforementioned research gaps by constructing an evolutionary game model on a two-layer heterogeneous complex network, using parameter calibration based on 2023 enterprise financial data, and combining Monte Carlo and threshold analysis, thereby enhancing theoretical explanatory power and policy applicability.

Construction of a complex network game model for enterprise intelligent transformation

Assumptions and parameter settings of the game model

Game model assumptions

It should be noted at the outset that this study employs a simulation-based modeling methodology. All agents in the model represent stylized, modeled firms constructed for the purposes of theoretical analysis and numerical simulation, rather than empirically recruited participants. The “population” of enterprises refers to the modeled nodes in the network, not to a sample drawn from a real organization survey. Accordingly, the findings should be evaluated against the standards appropriate for computational simulation studies, rather than those for participant-based empirical research.

Assumption 1

The total number of enterprises related to the intelligent server supply chain is , which can be divided into supply-side and demand-side enterprises. Supply-side enterprises are capable of constructing intelligent servers and can provide services for purchasing or leasing these servers to demand-side enterprises. Their strategy set is {cooperate, not cooperate}. The cooperation strategy indicates that the supply-side enterprises actively construct intelligent servers, whereas the non-cooperation strategy indicates that the supply-side enterprises do not construct intelligent servers.

In contrast, demand-side enterprises are technically disadvantaged and cannot construct intelligent servers themselves. They can only purchase or lease servers constructed by supply-side enterprises. Their strategy set is also {cooperate, not cooperate}. Cooperation implies that demand-side enterprises actively undergo intelligent transformation, modify traditional working methods, improve operational efficiency, reduce labor costs, and capture a larger market share. Non-cooperation indicates that enterprises operate in a traditional manner without undertaking intelligent transformation.

Assumption 2

The price of intelligent servers sold by supply-side enterprises is , and the cost of constructing an intelligent server is . When demand-side enterprises choose the cooperation strategy to undergo intelligent transformation, they must cooperate with supply-side enterprises that also choose cooperation. Intelligent transformation allows improved efficiency and reduced labor costs, enabling demand-side enterprises to save costs and additionally obtain profit . When demand-side enterprises choose not to cooperate, their profit from traditional operations is and the associated cost is .

Assumption 3

The timing of entry into the intelligent server market affects the profits of both supply-side and demand-side enterprises³¹. If demand-side enterprises undertake intelligent transformation earlier, they can capture a larger market share. The additional profit from early transformation is set as , where . If supply-side enterprises delay server construction, they can reduce construction costs under more mature technology conditions, with the server construction cost set as , where .

The symbols used in the game model are summarized in Table 1.

Table 1.

Symbol description.

Parameter	Meaning	Parameter	Meaning
	Total number of enterprises in the market		Construction cost of intelligent servers for supply-side enterprises
	Government subsidy for demand-side enterprises choosing intelligent transformation		Revenue from traditional operations for demand-side enterprises
	Government subsidy for supply-side enterprises choosing intelligent transformation		Revenue from traditional operations for supply-side enterprises
	Cost of purchasing or leasing intelligent servers		Maximum additional profit from intelligent transformation for demand-side enterprises
	Cost of traditional operations for demand-side enterprises		Market advantage coefficient for demand-side enterprises
	Cost savings from intelligent transformation for demand-side enterprises		Technology progress coefficient

Payoff matrix of the game model

Based on the above assumptions, the payoff matrix for the intelligent transformation game under the four possible decision combinations is shown in Table 2.

Table 2.

Payoff matrix of enterprise intelligent transformatio.

Demand-side	Cooperate ()	Not Cooperate ()
Supply-side
Cooperate ()
Not Cooperate ()

When both supply-side and demand-side enterprises choose cooperation, the supply-side enterprise’s payoff is the government subsidy plus the server price minus the construction cost: . The demand-side enterprise’s payoff is the government subsidy minus the server cost and traditional operation cost, plus cost savings from transformation, traditional revenue, and extra profit from transformation: .

When the supply-side chooses cooperation and the demand-side chooses not to cooperate, the supply-side’s payoff is , while the demand-side’s payoff is .

When the supply-side does not cooperate but the demand-side cooperates, the supply-side’s payoff is , and the demand-side’s payoff is .

When both choose not to cooperate, payoffs are for supply-side and for demand-side.

Analysis of the enterprise intelligent transformation game model

Replicator dynamics and phase diagram analysis

(1) Demand-side enterprises.

Let the expected payoff for the demand-side choosing “cooperate” be , choosing “not cooperate” be , and the average expected payoff be :

1

2

3

The replicator dynamics equation for the demand-side is:

4

The first derivative and the function are:

5

6

According to the requirements for stable points in evolutionary game theory, when the proportional relationship deviates from these stable points , the replicator dynamics will still drive it back to these levels. That is, for the probability of the demand-side enterprises choosing the cooperation strategy to be in a stable state, the following must be satisfied: and . Evidently, , thus is an increasing function of . When , and . At this point, the demand-side enterprise player’s choice of strategy, regardless of which one, is an equilibrium strategy, and it will not change over time.

1. When , . At this time, , and is the Evolutionarily Stable Strategy (ESS) for the demand-side enterprises, meaning that the demand-side enterprises’ choice of the “cooperation” strategy is the unique ESS;

2. When , . At this time, , and is the ESS for the demand side, meaning that the demand-side enterprises’ choice of the “non-cooperation” strategy is the unique ESS.The phase diagram of the demand-side enterprises’ replicator dynamics is obtained, as shown in Fig. 1.

Fig. 1.

Replicator dynamics phase diagram for demand-side enterprise strategy evolution.

Figure 1 indicates that the probability of the demand-side enterprises choosing the proactive governance strategy can be represented by the area in the phase diagram, calculated as:

7

Inference 1: The probability of demand-side enterprises choosing the “cooperation” strategy is positively correlated with the government subsidy for demand-side enterprises choosing intelligent transformation, , the cost saved by demand-side intelligent transformation, , and the maximum additional profit from demand-side enterprises choosing intelligent transformation, . It is negatively correlated with the cost of purchasing or leasing intelligent servers, , and the time of market entry, .

Proof: According to the expression for , the probability of demand-side enterprises choosing the “cooperation” strategy, we calculate the first-order partial derivatives with respect to each factor: , , , , and . This proves that an increase in , , or , or a decrease in or , will all lead to an increase in the probability of demand-side enterprises choosing the “cooperation” strategy.

Inference 1 indicates that government assistance in the form of subsidies to reduce enterprise transformation costs will help improve the enthusiasm of demand-side enterprises for choosing intelligent transformation. The government should further increase the subsidy intensity for demand-side enterprises choosing intelligent transformation to reduce enterprises’ initial investment costs, thereby encouraging more enterprises to adopt the cooperation strategy. This can include fiscal subsidies, tax incentives, and other forms that directly lower the financial barriers to enterprise transformation. Concurrently, if demand-side enterprises can achieve significant cost savings and additional profit growth through intelligent technology, they will also be more inclined to choose transformation. However, excessively high transformation costs or delayed market entry time will reduce enterprises’ willingness to cooperate. Demand-side enterprises should seize the opportunity to carry out intelligent transformation as early as possible. Early entrants will be able to gain greater market share and enhance competitiveness through intelligent technology. Therefore, enterprises should actively plan their deployment, avoid excessive observation or postponement of transformation, and thus prevent the loss of competitive advantage.

(2) Supply-side enterprises.

Let be the expected payoff for the supply-side enterprises choosing the “cooperation” strategy, be the expected payoff for choosing the “non-cooperation” strategy, and be the average expected payoff:

8

9

10

The replicator dynamics equation for the supply-side enterprises’ strategy selection is:

11

The first-order derivative of and the defined are, respectively:

12

13

According to the requirements for stable points in evolutionary game theory, when the proportional relationship deviates from these stable points, the replicator dynamics will still drive it back to these levels. That is, for the probability of the supply-side enterprises choosing the “cooperation” strategy to be in a stable state, the following must be satisfied: and . Evidently, , thus is an increasing function of . When , and . At this point, the supply-side enterprise player’s choice of strategy, regardless of which one, is an equilibrium strategy, and it will not change over time.

(a) When , . At this time, , and is the ESS for the supply-side enterprises (Note: The original text incorrectly said “government’s ESS” here, but based on the context of supply-side strategy, it should be the supply-side’s ESS), meaning that the supply-side enterprises’ choice of the “cooperation” strategy is the unique Evolutionarily Stable Strategy; (b) When , . At this time, , and is the ESS for the supply-side enterprises, meaning that the supply-side enterprises’ choice of the “non-cooperation” strategy is the unique Evolutionarily Stable Strategy.

The phase diagram of the supply-side enterprises’ replicator dynamics is obtained, as shown in Fig. 2.

Fig. 2.

Replicator dynamics phase diagram for supply-side enterprise strategy evolution.

Figure 2 indicates that the probability of the demand-side enterprises choosing the proactive governance strategy can be represented by the area in the phase diagram, calculated as:

14

Inference 2: The probability of supply-side enterprises choosing the “cooperation” strategy is positively correlated with the government subsidy for supply-side enterprises choosing intelligent transformation, , and the cost of purchasing or leasing intelligent servers, . It is negatively correlated with the cost of constructing intelligent servers for supply-side enterprises, .

Proof: According to the expression for , the probability of supply-side enterprises choosing the “cooperation” strategy, we calculate the first-order partial derivatives with respect to each factor: , , . This proves that an increase in or , or a decrease in , will all lead to an increase in the probability of supply-side enterprises choosing the “cooperation” strategy.

Inference 2 indicates that supply-side enterprises are more likely to choose the cooperation strategy and actively participate in intelligent transformation when the government provides subsidies and the cost of intelligent equipment decreases. The government should further increase fiscal support for supply-side enterprises, particularly subsidies for the construction of intelligent servers and equipment investment. By alleviating the investment pressure on supply-side enterprises, this can promote more enterprises to actively participate in the promotion and application of intelligent technology. Fiscal support can take various forms, such as direct subsidies, tax incentives, and low-interest loans, to reduce the capital burden on enterprises in intelligent server construction. Simultaneously, the government and the industry should promote the decline in the cost of intelligent equipment through policy incentives and market mechanisms. Supply-side enterprises can reduce the manufacturing and maintenance costs of server hardware through economies of scale, technological innovation, and industry cooperation. This will further reduce the economic burden of intelligent transformation on supply-side enterprises and enhance their willingness to cooperate. However, if the construction cost of intelligent servers is high, the supply-side enterprises’ willingness to cooperate will decrease. Supply-side enterprises should focus on optimizing technical solutions and adopting more efficient and cost-effective architectural schemes. Furthermore, the industry should encourage cooperation among supply-side enterprises to share resources for the construction and maintenance of intelligent servers, reduce the construction cost for individual enterprises, and enhance overall competitiveness.

Stability analysis of the replicator dynamic system

By combining Eqs. (4) and (10), the replicator dynamic system for supply-side and demand-side enterprises is obtained as:

15

Based on Eq. (15), five equilibrium points can be solved, including four pure strategy equilibrium points: (0, 0), (1, 0), (0, 1), and (1, 1), and one mixed strategy equilibrium point: . The research by Selten³² indicates that the ESS of an evolutionary game system must be a pure strategy Nash equilibrium; thus, the mixed strategy equilibrium among the five solved equilibrium points is certainly not an ESS. Consequently, only the four pure strategy equilibrium points need to be analyzed in the subsequent steps.

According to the method proposed by Friedman³³, the stability of the equilibrium points (ESS) is analyzed using the Jacobian matrix. The Jacobian matrix is derived from the above equations as follows:

16

Friedman’s research indicates that an equilibrium point inside the evolutionary game system is a stable point (ESS) only if it simultaneously satisfies the conditions that the determinant of the Jacobian matrix, , and the trace of the matrix, . Substituting the four pure strategy equilibrium points into the Jacobian matrix, the determinant and trace for the strategy combinations of supply-side and demand-side enterprises are shown in Table 3.

Table 3.

Strategy combination stability analysis results.

Pure strategy combination
(0, 0)
(1, 0)
(0, 1)
(1, 1)

From Table 3, the stability conditions for each strategy combination can be determined: The stability condition for the {non-cooperation, non-cooperation} strategy combination is: and . The stability condition for the {cooperation, non-cooperation} strategy combination is: and . The stability condition for the {non-cooperation, cooperation} strategy combination is: and . The stability condition for the {cooperation, cooperation} strategy combination is: and .

Under the current market environment, the cooperation strategy chosen by both supply-side and demand-side enterprises represents the most ideal outcome achievable by the dynamic game, primarily based on the following reasons. Firstly, cooperation can effectively enhance the efficiency and competitiveness of both parties. For supply-side enterprises, cooperation with demand-side enterprises allows them to accurately obtain market demand, optimize production processes, and increase profit margins through customized products and value-added services. For demand-side enterprises, cooperation helps them acquire products and services that meet their needs at lower costs while enabling them to respond promptly to technological and market changes, thus enhancing their competitive advantage. Secondly, cooperation facilitates the efficient allocation of resources. In the context of enterprise intelligent transformation, both supply and demand sides need to jointly address the complexity and uncertainty brought by technological innovation. Through cooperation, supply-side enterprises can better grasp market dynamics, reducing the risk of overproduction or resource waste, while demand-side enterprises can leverage the supply-side enterprises’ technological advantages and innovative capabilities to optimize production and operations, reducing the cost and risk associated with exploring new technologies. Finally, government policy guidance also promotes cooperation between the supply and demand sides. Many governments have introduced policies supporting enterprise intelligent transformation, including subsidies, tax incentives, and other stimulating measures, which provide policy support for supply-demand cooperation. By jointly developing and sharing technology and market resources, both parties can better obtain policy benefits and reduce market uncertainties in competition.

Therefore, {cooperation, cooperation} is the ideal evolutionary target for this game system, meaning that the equilibrium point (1, 1) is the target stable point that is beneficial to both parties. Based on the Jacobian matrix stability conditions, the conditions for the equilibrium point $(1, 1)$ to reach a stable state are: and . Once the stability conditions are met, the equilibrium point (1, 1) will exist in the evolutionary game system, and the dynamic evolutionary phase diagram of the system is shown in Fig. 3.

Fig. 3.

System dynamic evolutionary phase diagram.

Analysis of the evolutionary game model for enterprise intelligent transformation

As demonstrated in “Analysis of the enterprise intelligent transformation game model”, under bounded rationality, the optimal strategy profile for both the supply-side and demand-side enterprises is {cooperation, cooperation}. MATLAB R2021b is employed to conduct the evolutionary game simulation. To ensure that the simulation results are grounded in real-world conditions, this study calibrates the model parameters using revenue-related data obtained from the WIND database and enterprise research reports. Foxconn Industrial Internet (FII) is selected as a representative large-scale supply-side enterprise, while Microsoft, Dell, Amazon, and NVIDIA—its major clients—serve as representative demand-side enterprises. It should be noted that this calibration reflects a high-capacity market scenario rather than the full spectrum of enterprise types. The initial simulation parameters and data sources are shown in Table 4.

Table 4.

Variable descriptions and data sources.

Variable	Value	Description	Data source
	1.25 billion yuan	Government subsidies received by FII in 2023	WIND database
	15%	Government subsidy equal to 15% of intelligent equipment procurement cost	Chongqing Government Document No. 69 (2018)
	48.6 billion yuan	Average server procurement cost of demand-side enterprises in 2023	WIND database
	486.4 billion yuan	NVIDIA’s operating costs in 2023	Enterprise financial report
	20 billion yuan	Cost savings enabled by intelligent transformation	Existing literature34,35
	46.1 billion yuan	Average operational cost of FII-related businesses in 2023	WIND database
	636.7 billion yuan	NVIDIA’s operating revenue in 2023	Enterprise financial report
		NVIDIA’s maximum annual revenue level exceeding 50% industry profitability	Essence securities research report

In evolutionary game theory, participants dynamically adjust their strategies through learning and imitation. The final equilibrium is shaped not only by external environmental conditions but also by the initial state of the game³⁶. The following analysis examines how variations in key parameters influence strategic evolution, thereby identifying conditions under which the desired cooperative equilibrium {cooperate, cooperate} is achieved. Notably, the model focuses on the directional impact and threshold effects of parameters rather than their absolute magnitudes, allowing the results to remain informative under different enterprise scales.

(1) Government subsidy .

Simulation parameters are set as: . The evolutionary trajectories of both players after changes in government subsidies for demand-side enterprises are shown in Fig. 4.

Fig. 4.

Changes in government subsidy .

Figure 4 shows that an increase in accelerates the demand-side enterprise’s transition toward the cooperation strategy. When , the demand-side enterprise’s cooperation probability evolves toward 1. According to Replicator Dynamic Eq. (4), the coefficient of is positive, indicating that the willingness of demand-side enterprises to cooperate is positively correlated with the subsidy level. Hence, higher additional returns from cooperation result in a stronger incentive to cooperate. Correspondingly, the supply-side enterprise also becomes more inclined toward cooperation, and this trend strengthens as increases. Although does not directly affect the supply-side decision in Eq. (10), the increased cooperation probability of demand-side enterprises significantly boosts the supply-side’s cooperative tendency.

(2) Government reward .

Simulation parameters are set as: . The evolutionary trajectories after changes in subsidies to supply-side enterprises are presented in Fig. 5.

Fig. 5.

Changes in government subsidy .

Figure 5 illustrates that higher accelerates the supply-side enterprise’s shift toward cooperation. According to Eq. (10), the coefficient of is positive, implying that the supply-side enterprise’s cooperative willingness increases proportionally with government rewards. When cooperation yields higher extra returns, the supply-side enterprise becomes more likely to adopt cooperation. Meanwhile, the demand-side enterprise also shows an increasing cooperative tendency as rises. Although is not directly involved in Eq. (4), increases in the supply-side’s cooperation probability have positive spillover effects on the demand-side enterprise.

(3) Server leasing cost .

Simulation parameters are set as: . Figure 6 shows the evolutionary outcomes after the server leasing cost changes.

Fig. 6.

Changes in server leasing cost .

Figure 6 demonstrates that higher significantly reduces the demand-side enterprise’s cooperation willingness. According to Eq. (4), the coefficient of is negative, indicating an inverse relationship between server leasing cost and cooperation probability. When cooperation results in a higher cost burden, demand-side enterprises reduce cooperative behavior. Conversely, the supply-side enterprise’s cooperation probability increases as rises. In Eq. (10), has a positive coefficient, suggesting that higher server leasing prices enhance the supply-side’s relative benefit from cooperation.

(4) Cost savings from intelligent transformation .

Simulation parameters are set as: . Figure 7 presents the evolutionary outcomes under varying levels of cost savings.

Fig. 7.

Cost savings from intelligent transformation undergo changes.

Figure 7 shows that increases in significantly enhance the demand-side enterprise’s cooperation probability. When , the cooperation probability converges to 1. In Eq. (4), the coefficient of is positive, meaning that cost savings positively boost cooperative incentives. The supply-side enterprise is also affected, showing stronger cooperative inclination as increases. Although does not directly influence Eq. (10), a higher demand-side cooperation probability promotes cooperative behavior on the supply side as well.

(5) Server infrastructure cost .

Simulation parameters are: . The results are illustrated in Fig. 8.

Fig. 8.

Server setup cost .

Figure 8 indicates that increases in reduce the supply-side’s cooperation probability. In Eq. (10), the coefficient of is negative, showing that higher infrastructure costs discourage cooperation. As the supply-side enterprise becomes less willing to cooperate, the demand-side enterprise also reduces its cooperation probability. Although does not appear directly in Eq. (4), the decline in the supply-side’s cooperation exerts a negative spillover effect on the demand-side enterprise.

(6) Maximum additional profit .

Simulation parameters are: . Figure 9 illustrates the strategic evolution after adjusting .

Fig. 9.

Change in the maximum additional profit .

Figure 9 shows that increases in enhance the demand-side enterprise’s cooperative tendency. When reaches 270, the cooperation probability converges to 1. In Eq. (4), the coefficient of is positive, indicating that greater additional returns strengthen the demand-side enterprise’s incentive to adopt cooperation. This also positively influences the supply-side enterprise, promoting a higher cooperation probability. Although does not directly enter Eq. (10), a higher cooperation level on the demand-side drives greater cooperative behavior on the supply-side.

Sensitivity analysis of parameters in the evolutionary game on complex networks for enterprise intelligent transformation

Construction of the complex network model and analysis of topological properties

Currently, the most widely applied models in the field of complex networks are the Watts–Strogatz (WS) small-world network and the Barabási–Albert (BA) scale-free network. Existing studies indicate that WS networks are characterized by short average path lengths and high clustering coefficients. However, real markets exhibit far more heterogeneous and intricate structural properties, making the WS model overly simplified and unsuitable for capturing the complexity of supply–demand interactions³⁷. In contrast, scale-free networks display a power-law degree distribution in which a small number of nodes possess a large number of connections, while the majority maintain only a few. This structural feature better reflects how key enterprises exert disproportionate influence on market dynamics, thereby allowing a more accurate simulation of supply–demand relationships³⁸. Therefore, this study adopts a scale-free network to model spatial relationships within the context of enterprise intelligent transformation.

Construction logic of the two-layer complex network

In the two-layer complex network framework applied to enterprise intelligent transformation, supply-side and demand-side enterprises constitute two distinct groups. These groups engage in strategic interactions, while individual entities within each group decide whether to imitate their neighbors based on pairwise comparison and strategy-update rules. In this study, enterprises supplying intelligent servers are represented as the upper-layer network, whereas demand-side enterprises comprise the lower-layer network. Each node in the network represents either a supply-side or demand-side enterprise, and edges denote supply–demand relations or competitive interactions among enterprises.

A complex network based on the Barabási–Albert (BA) scale-free model is constructed to simulate the intricate relationships within the supply–demand system. The BA model is a classical method for generating scale-free networks and is characterized by two key mechanisms: incremental growth and preferential attachment. Incremental growth means that the network evolves from a small initial complete graph by continuously adding new nodes. Preferential attachment implies that newly added nodes are more likely to connect to existing nodes with higher degrees, reflecting a “rich-get-richer” dynamic. As a result, the BA model produces a network with a power-law degree distribution , where is typically close to 3. In such networks, a small number of nodes serve as hubs with high degrees, while most nodes maintain low-degree connections, forming a long-tailed distribution.

In constructing this network, supply and demand nodes are first defined, yielding a total of 500 nodes, including 50 supply-side and 450 demand-side enterprises. The network is generated using the BA model, with each new node establishing three connections upon entering the network, subsequently forming a dynamically growing scale-free structure. The first 50 nodes are designated as supply-side enterprises, while the remaining nodes represent demand-side enterprises, producing a heterogeneous network structure. The probability that a new node connects to an existing node is proportional to the latter’s degree, expressed as

where denotes the degree of node . This preferential attachment mechanism enables higher-degree nodes to acquire new links more easily, thus forming hub nodes and enhancing overall network robustness.

In constructing the two-layer network using Python, the networkx library’s function barabasi_albert_graph(n, m) is employed, where n is the total number of nodes and m is the number of edges each new node forms upon entering the network. Assuming that the intelligent server supply chain includes 500 market entities, the proportion of supply-side nodes is predefined in the code, which determines the specific numbers of supply and demand nodes. Each node is then assigned an attribute label based on its type. Following this logic, a two-layer complex network for supply-side and demand-side enterprises in the intelligent server market is successfully constructed (see Fig. 10).

Fig. 10.

Dual-layer complex network of supply-side and demand-side enterprises in the smart server industry.

This BA-based network construction approach effectively captures the structural characteristics observed in real-world systems such as supply chains and social networks, particularly the dominance of a few hub nodes. By explicitly distinguishing supply-side from demand-side nodes, the model better reflects the heterogeneity of actual market environments and facilitates deeper analysis of the complex strategic interactions between enterprises during intelligent transformation.

Network topological analysis

(1) Degree centrality.

Centrality measures the position and importance of individual nodes within a network. Degree centrality directly quantifies the number of connections a node possesses. Nodes with high degree centrality serve as pivotal hubs within the network. The calculation formula is:

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In the formula, is the degree centrality of node , is the degree of node , and is the total number of nodes in the network.

The histogram of the node degree centrality distribution for the nodes in the complex network constructed in this paper is shown in Fig. 11a. The degree centrality of most nodes is relatively low, falling between 0.006 and 0.01; however, the degree centrality of a small number of nodes reaches 0.14, which implies that there are a few key nodes in the network connected to a relatively large number of other nodes.

Fig. 11.

Distribution histograms of complex network centrality metrics. (a) Histogram of degree centrality distribution, (b) Histogram of closeness centrality distribution, (c) Betweenness centrality distribution histogram.

(2) Closeness centrality.

The closeness centrality metric refers to the average shortest path length from a node to all other nodes. It can reflect the efficiency of information transmission of that node. The calculation formula is:

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In the formula, is the closeness centrality of node , is the shortest path between node and node , and is the total number of nodes in the network.

The closeness centrality of the nodes in this network is shown in Fig. 11b, which conforms to a normal distribution. That is, the closeness centrality of most nodes is at a moderate level, falling between 0.29 and 0.336. Only a small number of nodes have significantly large or small closeness centrality, below 0.27 or above 0.35.

(3) Betweenness centrality.

The betweenness centrality metric describes the frequency with which a node acts as an ‘intermediary’ on the paths between other pairs of nodes, reflecting the node’s role in the overall network connectivity. The calculation formula is:

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In the formula, is the betweenness centrality of node , and is the number of shortest paths between node and node that pass through node .

The betweenness centrality of the nodes in this network is shown in Fig. 11c. Clearly, the capacity of most nodes in the network to act as intermediaries for information transfer is not significant. Only a few nodes have a betweenness centrality reaching 22,915, which demonstrates a critical bridging role capable of influencing the information flow across the entire network.

Complex network evolutionary game rules setting

Game rules

Based on the constructed supply-side–demand-side enterprise scale-free network and the developed evolutionary game model for intelligent transformation, the following hypotheses are proposed:

Hypothesis 1

The constructed supply-side–demand-side enterprise scale-free network with nodes can be represented as , where is the set of enterprise nodes, , and is the proportion of supply-side enterprises among all enterprises. is the set of edges between enterprise nodes, . If , then node and node are neighbors, meaning the two enterprises have a supply-demand relationship or a competitive relationship; if , there is no direct connection between the two nodes.

Hypothesis 2

During the process of intelligent transformation, due to the limited information acquisition, market inefficiency, and geographical constraints, enterprises cannot establish contact with all other enterprises. Among different types of enterprises, supply-side enterprises only play the game with the demand-side enterprises they are connected with. Among enterprises of the same type, only connected enterprises will compare payoffs with each other.

Hypothesis 3

Enterprises are boundedly rational. In the process of intelligent transformation, whether an enterprise chooses the intelligent transformation strategy is related to the payoff, but there is still a possibility of not choosing the optimal strategy due to risk aversion. Therefore, after the end of each round of the game, enterprises follow a strategy update rule and adjust their strategy with a certain probability.

Strategy update rules

Common strategy update rules for evolutionary games on complex networks include unconditional imitation, replicator dynamics, and the Fermi rule.

(1) Unconditional imitation: Under this rule, agents simply copy the strategy of the most successful neighbor without evaluating any payoff differences or other factors³⁹.

(2) Fermi rule: The Fermi process models bounded rationality under incomplete information and uncertainty. Rather than representing blind imitation of neighbors, this rule captures the probabilistic nature of strategic adjustment under real-world decision constraints: the probability that a firm updates its strategy is a smooth, continuous function of the payoff difference between itself and a reference neighbor, scaled by a noise parameter . This formulation reflects the fact that firms cannot perfectly evaluate all investment outcomes—such as long-term ROI, technological uncertainty, and market volatility—and therefore adjust strategies probabilistically based on observed payoff comparisons rather than mechanically copying the highest-performing peer. When tends to infinity, the update rule approaches a fully deterministic selection process; when tends to zero, payoff differences cease to influence strategic choice and the updating process becomes essentially random, reflecting extreme decision-making uncertainty⁴⁰.

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(3) Disconnection mechanism: A node randomly selects a neighbor with the same strategy, calculates the cumulative payoff difference between them, and uses the Fermi rule to calculate the probability of the node changing its strategy. If the strategy is not updated, the connection between the current node and the neighbor node is broken.

To select the most suitable strategy update rule for the subsequent simulation experiments, a Monte Carlo simulation experiment for strategy update and cooperation strategy evolution on a complex network is designed. In each round of evolution, the payoff for each node under cooperation and defection is first calculated, adjusted according to the node type and its strategy. Then, the node strategy is updated through the Fermi rule, the unconditional imitation rule, or the disconnection mechanism, respectively. The Fermi rule determines the probability of strategy change by calculating the payoff difference between the neighbor node and itself; the unconditional imitation rule simply mimics the strategy of the neighbor node with the highest payoff; and the disconnection mechanism disconnects the connection with the neighbor node having a lower payoff when the strategy is not changed. The Monte Carlo simulation experiment is repeated 100 times to ensure the stability and reliability of the results. Finally, a scatter plot and the average cooperation rate curve are plotted, as shown in Fig. 12.

Fig. 12.

Impact of different strategy update rules on evolutionary outcomes.

In the simulation experiment, the three different strategy update rules led to significantly different evolutionary outcomes. The unconditional imitation rule, by directly mimicking the strategy of the highest-payoff neighbor, resulted in the rapid spread of the cooperation strategy in the network, making the overall strategy quickly converge towards full cooperation. However, this direct imitation mechanism ignores strategy diversity and randomness, easily leading to strategy homogenization, which is unsuitable for a complex and volatile market environment⁴¹. The disconnection mechanism adjusted the network structure by severing connections with lower-payoff neighbors when individuals did not change their strategy. Although this mechanism can promote cooperation, the frequent disconnection and reconnection operations slow down the strategy propagation, and the final cooperation rate can only stabilize at a high but not complete level, failing to reach the state of full cooperation⁴².

By introducing stochasticity, the Fermi rule implies that enterprises, when adjusting their strategies, consider not only the existence of payoff differences with neighboring enterprises but also their magnitude. This is reflected in the model as the probability of strategy updating being a smooth function of payoff differences rather than a deterministic switch. Importantly, this mechanism does not represent blind imitation; instead, it captures bounded rationality under incomplete information, where enterprises cannot perfectly evaluate all investment outcomes (e.g., long-term ROI, technological uncertainty, and market volatility) and therefore adjust strategies probabilistically. Consequently, the Fermi rule provides a tractable approximation of decision-making that integrates both payoff-driven rationality and real-world uncertainty, making it suitable for modeling strategic adaptation in complex and dynamic market environments⁴³.

In summary, the Fermi rule demonstrates better adaptability and realism when simulating market conditions and the game behavior among enterprises. This method not only provides a reasonable theoretical framework but also offers a powerful tool for understanding and predicting behavioral dynamics in complex markets. Based on these considerations, this study selects the scale-free network as the network framework for the evolutionary game and uses the Fermi rule as the strategy update rule for game participants for the parameter sensitivity analysis in the next section.

Simulation experiment

Initial parameter setting

After constructing the two-layer complex network, Python is used for the numerical simulation of the game process. To more intuitively demonstrate the influence of different factors on the dynamic decisions of supply-side and demand-side enterprises, this paper selects key factors such as government subsidies, intelligent server purchase cost, intelligent server construction cost, and the maximum additional profit from enterprise intelligent transformation. Python is used to analyze the strategy evolution trajectory over a period (Rounds) of data simulation.

The initial parameter settings reference the actual data in Table 3.4 and draw on the research results of Rong et al.⁴⁴: , , , , , , , , , , . Furthermore, to reduce the randomness of the simulation results, the Monte Carlo simulation method is used to simulate each data point 100 times, and the average value is taken⁴⁵.

Parameter sensitivity analysis

(1) Impact of Government Subsidies for Demand-Side Enterprises.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 13.

Fig. 13.

Impact of on Evolutionarily Stable Strategies.

Keeping other parameters constant, Fig. 13 shows the impact of the government subsidy for demand-side enterprises, , on the evolutionarily stable strategies of the two groups: intelligent server supply-side enterprises and demand-side enterprises. Overall, the increase in can increase the proportion of enterprises choosing the cooperation strategy in both the supply-side and demand-side groups, driving both groups to evolve towards cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy slightly increases, but the increase is not significant. In contrast, demand-side enterprises are more sensitive to the subsidy amount, and the proportion of demand-side enterprises choosing the cooperation strategy increases significantly. When increases to 73, the evolutionarily stable strategy of the demand-side enterprises converges toward the cooperation strategy, with the proportion choosing cooperation approaching 1.

(2) Impact of government subsidies for supply-side enterprises.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 14.

Fig. 14.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 14 shows the impact of the government subsidy for supply-side enterprises, , on the evolutionarily stable strategies of the two groups: intelligent server supply-side enterprises and demand-side enterprises. Overall, the increase in can increase the proportion of enterprises choosing the cooperation strategy in both the supply-side and demand-side groups, driving both groups to evolve towards cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy rises, and the proportion of the entire group choosing cooperation tends toward 1. Furthermore, the proportion of demand-side enterprises choosing the cooperation strategy also increases, but the increase is not significant and is unable to drive the entire demand-side group to evolve towards the cooperation strategy.

(3) Impact of intelligent server purchase or rental cost.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 15.Keeping other parameters constant, Fig. 15 shows the impact of the intelligent server purchase or rental cost, , on the evolutionarily stable strategies of the two groups: intelligent server supply-side enterprises and demand-side enterprises. Overall, only a reasonable intelligent server purchase price can maintain a high proportion of cooperation for both supply-side and demand-side enterprises, driving both groups to evolve toward cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy significantly increases, and the proportion of the entire group choosing cooperation tends toward 1. Notably, when increases from 286 to 486, the overall cooperation proportion of supply-side enterprises evolves from tending toward 0 to tending toward 1. However, when the cost further increases to 683, the cooperation proportion slightly decreases, tending toward $0.8$. In contrast, as increases, the proportion of demand-side enterprises choosing the cooperation strategy gradually decreases. When , the cooperation proportion of demand-side enterprises tends toward $1$. However, as increases to 683, the overall cooperation proportion of demand-side enterprises gradually decreases and tends toward $0$.

Fig. 15.

Impact of on Evolutionarily Stable Strategies.

(4) Impact of server construction cost.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 16.

Fig. 16.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 16 shows the impact of the cost of constructing intelligent servers for supply-side enterprises, , on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises. Overall, only when the server construction cost is controlled within a reasonable range do supply-side and demand-side enterprises maintain a high cooperation proportion, driving both groups to evolve toward cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy significantly decreases, and the proportion of the entire group choosing cooperation tends toward 0. When increases from 461 to 661, the overall cooperation proportion of supply-side enterprises rapidly decreases from 0.9 to 0. Meanwhile, demand-side enterprises have lower sensitivity to changes in server construction cost, and the increase in has little impact on the strategy choice of demand-side enterprises. Despite the significant drop in the cooperation proportion of supply-side enterprises, the cooperation proportion of demand-side enterprises fluctuates between 0.8 and 0.9.

(5) Impact of maximum additional profit.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 17.

Fig. 17.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 17 shows the impact of the maximum additional profit gained by demand-side enterprises choosing intelligent transformation, , on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises. Overall, a moderate increase in can increase the proportion of enterprises choosing the cooperation strategy in both the intelligent server supply-side and demand-side groups, driving both groups to evolve toward cooperation. Specifically, the increase in has little impact on the strategy choice of supply-side enterprises, with their cooperation proportion consistently fluctuating between 0.8 and 0.9. In contrast, demand-side enterprises are very sensitive to the additional profit brought by intelligent transformation. As increases, the proportion of demand-side enterprises choosing the cooperation strategy significantly rises, and the proportion of the entire group choosing cooperation tends toward 1. Specifically, when increases from 220 to 320, the overall cooperation proportion of demand-side enterprises rapidly increases from tending toward $0$ to tending toward 0.8; when increases to 420, the overall cooperation proportion further increases and tends toward 1.

(6) Impact of node degree.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 18.

Fig. 18.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 18 shows the impact of the node degree on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises. Overall, only a complex network structure with a reasonable value of allows supply-side and demand-side enterprises to maintain a high cooperation proportion, driving both groups to evolve toward cooperation. Specifically, the increase in has little impact on the strategy choice of supply-side enterprises, with their cooperation proportion consistently fluctuating between 0.8 and 1. In contrast, the strategy choice of demand-side enterprises is more sensitive to changes in network structure. As increases, the proportion of demand-side enterprises choosing the cooperation strategy significantly rises, and the proportion of the entire group choosing cooperation tends toward 1. Specifically, when is 1, the cooperation proportion of demand-side enterprises tends toward 0; when rises to 10, the overall cooperation proportion of demand-side enterprises tends toward 1.

Impact of dynamic rewards and dynamic network structure on game outcomes

Impact of time-sensitive dynamic reward model on game outcomes

In the field of evolutionary games on complex networks, promoting the widespread adoption of cooperative behavior is one of the core research objectives. In various social scenarios, such as policymaking, business operations, and ecological protection, collaboration among individuals can significantly improve the overall system efficiency, thereby driving sustained socio-economic development^46,47. However, in reality, influenced by short-term gains, individuals tend to choose self-interested non-cooperative behavior, making long-term cooperative relationships difficult to maintain. For instance, in the Prisoner’s Dilemma model, the optimal solution for a rational participant is usually to adopt the defection strategy; and in public goods games, although cooperation can generate collective benefits, the mechanism of equal benefit distribution often weakens the incentive for cooperation at the individual level, leading to a continuous decline in cooperation.

Taking business competition as an example, inter-firm cooperation can often achieve cost reduction and market expansion through resource sharing and collaborative innovation. However, if an enterprise violates the cooperation agreement in pursuit of its own interests, such as by adopting unfair competition tactics to disrupt market order, other enterprises may imitate this behavior, leading to a vicious competition scenario⁴⁸. This not only harms the overall industry interest but also weakens market stability. To address such issues, a strategy incentive mechanism combining rewards and punishments can be designed: providing rewards such as tax breaks and financial subsidies to enterprises that adhere to cooperation agreements, and imposing penalties such as fines or restrictions on market activities for those that violate the agreements. By optimizing the incentive mechanism, the motivation for cooperation among enterprises can be significantly enhanced, thereby preventing industry degradation caused by defection.

In environmental governance, dynamic incentive mechanisms are equally important. Governments often need to use specific policy tools to regulate the environmental behavior of enterprises, for example, by providing financial support or tax incentives to enterprises that meet pollution emission standards, or encouraging their development of eco-friendly technologies; conversely, severe penalties, such as fines or production halts, are imposed on non-compliant enterprises. This dynamic reward-and-punishment mechanism, based on the achievement of goals, can significantly improve enterprises’ decision-making behavior, guiding them to adopt green production methods, thereby reducing ecological damage and promoting sustainable development.

Against this background, introducing a dynamically adjusted payoff distribution mechanism can significantly enhance the level of cooperation in complex network games. This mechanism abstracts the dynamic adjustment of incentives based on observed behavioral changes of participants, where cooperative behavior receives additional rewards and defection behavior is subject to penalties. It does not imply strict real-time monitoring, but rather represents a stylized approximation of responsive policy adjustment processes. By strengthening the benefit of cooperation and lowering the attractiveness of non-cooperative behavior, this mechanism provides effective support for the sustained evolution of group cooperation in complex networks. Especially in multi-agent interaction scenarios, this mechanism can provide a more scientific theoretical basis for optimizing resource allocation and maximizing long-term payoffs.

Therefore, this study hypothesizes that introducing a dynamic payoff mechanism on the basis of traditional complex network evolutionary games will help explore a new path to encourage game participants to be actively involved and promote the improvement of the overall system cooperation rate.

Construction of the time-sensitive dynamic reward model

The construction of the time-sensitive dynamic reward model is based on the evolution of game behavior in complex networks, aiming to incentivize cooperative behavior and suppress defection through a dynamic reward and punishment mechanism, thereby enhancing the overall network cooperation level. The model achieves this by identifying changes in node strategies between decision periods, providing rewards to nodes moving towards cooperation, and imposing penalties on nodes moving away from cooperation, while incorporating a time-sensitivity parameter to capture the persistence of incentives. This formulation can also approximate delayed or periodic policy responses in practice.

In the model, each node represents a game participant and possesses two core attributes: strategy and time-sensitivity parameter. The strategy describes the node’s choice in a given round of the game, which can be cooperation () or defection (). The time-sensitivity parameter records the duration of the reward or punishment, initialized to zero. The node’s payoff is dynamically adjusted based on its strategy change and interaction with neighboring nodes.

The reward and punishment mechanism of the model is realized by identifying changes in the node’s strategy between two consecutive rounds of the game. When a node’s strategy changes as follows between two rounds, the reward or punishment mechanism is triggered:

(1) Cooperators proclivity: When a node’s strategy changes from defection to cooperation in the current round, it is categorized as exhibiting a proclivity towards cooperation. In this case, the enterprise node receives an additional payoff bonus, with its payoff multiplied by a factor greater than 1, . Simultaneously, the time-sensitivity parameter is set to an integer value , ensuring the reward remains effective for the subsequent rounds of the game.

(2) Defectors proclivity: When a node’s strategy changes from cooperation to defection in the current round, it is categorized as exhibiting a proclivity away from cooperation. In this case, the enterprise node is penalized, with its payoff multiplied by a factor less than 1, . The time-sensitivity parameter is also set to to maintain the persistence of the punishment.

To reflect the time-sensitive nature of the rewards and punishments, the model introduces a dynamic diminishing mechanism. When the node’s time-sensitivity parameter is greater than zero, it decreases by 1 after each round of the game until it reaches zero. As long as the time-sensitivity parameter remains active, the node’s payoff continues to be affected by the reward or punishment. Once the time-sensitivity parameter is zero, the node’s payoff returns to its normal state and is no longer subject to the reward or punishment. If the node’s strategy remains unchanged between the two rounds of the game, its payoff remains unchanged. The model design allows for flexible control over the intensity and duration of the rewards and punishments through the reward factor , the punishment factor , and the time-sensitivity parameter . Mathematically, the payoff of node in round is expressed as:

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Simulation experiment

To more intuitively demonstrate the influence of different factors on the dynamic decisions of supply-side and demand-side enterprises, Python is used for the numerical simulation of the game process. This section selects key factors such as the reward factor, punishment factor, reward/punishment period, and reward mechanism for simulation analysis, with particular attention to how different parameter settings can reflect varying policy response speeds and intensities. Furthermore, to reduce the randomness of the simulation results, the Monte Carlo simulation method is also used to simulate each data point 100 times, and the average value is taken⁴⁵.

(1) Impact of the reward factor.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 19.

Fig. 19.

Effect of on the evolutionary stable strategy.

Keeping other parameters constant and fixing the punishment factor at 0, Fig. 19 shows the impact of the time-sensitive reward factor on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises in the context of intelligent transformation. Overall, the increase in can increase the proportion of enterprises choosing the cooperation strategy in both the intelligent server supply-side and demand-side groups, driving both groups to evolve towards cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy continuously rises, and when increases to 0.9, the supply-side enterprise group converges more quickly toward the {cooperation} strategy. Similarly, as increases, the proportion of demand-side enterprises choosing the cooperation strategy also rises. When is increased from 0.1 to 0.5, the evolutionary outcome of the demand-side enterprise strategy choice shifts from tending toward the {non-cooperation} strategy to tending toward the {cooperation} strategy.

(2) Impact of the punishment factor.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 20.

Fig. 20.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 20 displays the impact of the time-sensitive punishment factor on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises in the context of intelligent transformation. Overall, the increase in has a significant differential impact on the strategy evolution of the supply-side and demand-side groups.

Specifically, for Supply-Side Enterprises, as increases, the proportion of supply-side enterprises choosing the cooperation strategy slightly increases, with the cooperation rate for all three values tending to stabilize after reaching approximately 0.65. This indicates that supply-side enterprises are not highly sensitive to changes in the punishment factor, and the punishment mechanism does not significantly affect the final cooperation rate of the entire group.

Conversely, for Demand-Side Enterprises, the impact of is more critical. As increases, the evolutionarily stable strategies of demand-side enterprises significantly diverge. When , the cooperation rate rapidly rises to a peak of approximately 0.85 in the short term, but then sharply declines and eventually tends toward 0, failing to drive the entire demand-side group toward the cooperation strategy. When and , the cooperation rate, after reaching its peak, successfully maintains a high level, stabilizing above 0.8, which drives the demand-side group to evolve toward the cooperation strategy. Furthermore, the stable cooperation rate when is slightly higher than when .

(3) Impact of the reward and punishment period.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 21.

Fig. 21.

Impact of on evolutionarily stable strategies.

Keeping other parameters constant, Fig. 21 displays the impact of the time-sensitivity parameter on the evolutionarily stable strategies of the two groups: supply-side and demand-side enterprises in the context of intelligent transformation. Overall, the increase in significantly enhances the final cooperation rate of supply-side enterprises, and also has a positive but relatively smaller impact on the cooperation rate of demand-side enterprises.

Specifically, for Supply-Side Enterprises, as the reward and punishment period increases, the proportion of supply-side enterprises choosing the cooperation strategy shows a stepwise and significant increase. When , the cooperation rate stabilizes at approximately 0.4; as increases to 10, 15, and 20, the cooperation rate stabilizes at approximately 0.6, 0.8, and 1.0, respectively. This indicates that a longer reward and punishment period strengthens the persistence of the incentive and penalty mechanisms, leading to a more significant promotion of the cooperation strategy for supply-side enterprises, with the entire group eventually tending toward full cooperation when .

Conversely, for Demand-Side Enterprises, as increases, the proportion of demand-side enterprises choosing the cooperation strategy also rises, but the difference in the final cooperation rate across all values is small, stabilizing between 0.85 and 0.95. This suggests that demand-side enterprises are less sensitive to changes in the reward and punishment period than supply-side enterprises; a relatively short period (e.g., ) is already sufficient to stabilize the cooperation rate at a high level. Although the cooperation rate slightly increases with a further increase in , the magnitude is small, and the group’s evolutionarily stable strategy consistently converges toward high cooperation levels.

(4) Impact of the reward mechanism.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 22.

Fig. 22.

Impact of the reward mechanism on evolutionarily stable strategies. (a) Fixed reward mechanism, (b) Time-based reward mechanism.

Keeping other parameters constant, Fig. 22 displays the impact of the government subsidy for supply-side enterprises, on the evolutionarily stable strategies of the two groups: intelligent server supply-side and demand-side enterprises. Overall, the increase in can raise the proportion of enterprises choosing the cooperation strategy in both the intelligent server supply-side and demand-side enterprises, driving both groups to evolve towards cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy rises, and the proportion of the entire group choosing cooperation tends toward 1. Furthermore, the proportion of demand-side enterprises choosing the cooperation strategy also increases, but the increase is not significant and is unable to drive the entire demand-side group to evolve toward the cooperation strategy.

Impact of the dynamic network topology structure model on game outcomes

In the study of evolutionary games on complex networks, the influence of network topology structure on game outcomes has consistently been a focal point of academic attention. Traditional complex network research usually assumes a static network structure; however, in the real world, network structures are often dynamic, changing as enterprise cooperation networks reorganize due to market environments and competitive landscapes, and social interaction networks adjust due to changes in individual behavior and information flow. Dynamic network topology structure models, by making the connection relationships between network nodes dynamic, can more realistically reflect the changes in interactions between nodes in the system and their influence on game behavior.

Inter-firm cooperation networks are not immutable but constantly adjust in response to market opportunities, policy incentives, and technological changes. A stable cooperative relationship in one phase may be dissolved due to defection or external shocks, while new connections may form due to common interests. This dynamic change significantly affects the willingness of enterprises to cooperate and their strategy choices. Similarly, in the field of social governance, the interaction relationships of individuals within social networks may also reorganize due to changes in policy implementation, social opinion, or group interests, and this change in topological structure can have a profound impact on overall cooperative behavior. Therefore, introducing a dynamic network topology structure model into game analysis can provide a deeper understanding of the evolution process of network relationships and their impact on cooperative behavior.

To make the model in this study more closely approximate reality, this subsection introduces a dynamic network topology structure model based on the original model, to analyze its impact on the game outcomes and explore the influence of spatial relationships on evolutionary results.

Construction of the dynamic network model

The dynamic network model classifies the nodes in the network into two types: Cooperative strategy nodes () and Defecting strategy nodes (), and achieves the dynamic evolution of the network structure through a link-breaking and link-adding mechanism. After the end of each round of the game, every node in the network adjusts its topology based on the strategies and payoffs of its neighbors.

Firstly, the Link-Breaking Mechanism: Each node iterates through all its neighbors and checks for the existence of defecting neighbors. If node ‘s neighbor adopts a defection strategy, node may break the link with that neighbor with a certain probability . Here, can be interpreted as an effective adjustment rate that implicitly captures real-world frictions such as switching costs, contractual constraints, and coordination barriers, meaning that only a limited proportion of links can be reconfigured in each round. However, to prevent the appearance of isolated nodes in the network, if either the node or its neighbor is left with only one connection after the link-breaking operation, the link-breaking operation is canceled. This mechanism ensures that the network’s connectivity is not destroyed, while reflecting the tendency of individuals to reduce interaction with non-cooperative individuals.

After the link-breaking mechanism is completed, the model further adjusts node connections through the Link-Adding Mechanism: after node breaks the link with a defecting neighbor , it screens all its neighbors holding the cooperation strategy and selects a cooperative neighbor with the highest payoff as the preferred reference. Node then randomly selects one node connected to node h and establishes a new link with it. This link-adding method emphasizes the clustering effect among cooperative nodes, meaning cooperative individuals are more inclined to establish connections with other cooperative individuals, forming cooperation-centric local structures. Given the presence of implicit adjustment frictions, this rewiring process represents gradual and partial network adaptation rather than instantaneous partner replacement. If a node has no cooperative neighbors, the link-adding operation is automatically skipped.

Simulation experiment

To more intuitively demonstrate the influence of different factors on the dynamic decisions of supply-side and demand-side enterprises, Python is used for the numerical simulation of the game process. This section selects key factors such as the reward factor, punishment factor, reward/punishment period, and reward mechanism for data simulation analysis. Furthermore, to reduce the randomness of the simulation results, the Monte Carlo simulation method is also used to simulate each data point 100 times, and the average value is taken.

The specific parameter changes are: . The evolution of the game strategies for supply-side and demand-side enterprises is shown in Fig. 23.Keeping other parameters constant, and fixing the punishment factor at 0, Fig. 19 displays the impact of the time-sensitive reward factor on the evolutionarily stable strategies of the two groups: intelligent server supply-side and demand-side enterprises in the context of intelligent transformation. Overall, the increase in can raise the proportion of enterprises choosing the cooperation strategy in both the intelligent server supply-side and demand-side enterprises, driving both groups to evolve towards cooperation. Specifically, as increases, the proportion of supply-side enterprises choosing the cooperation strategy continuously rises, and when increases to 0.9, the supply-side enterprise group more quickly converges toward the {cooperation} strategy. Similarly, as increases, the proportion of demand-side enterprises choosing the cooperation strategy also rises. When is increased from 0.1 to 0.5, the evolutionary outcome of the demand-side enterprise strategy choice shifts from tending toward the {non-cooperation} strategy to tending toward the {cooperation} strategy.

Fig. 23.

Effect of p on evolutionarily stable strategies.

Discussion and conclusion

Discussion

This study focuses on the strategic choices of supply-side and demand-side enterprises during the process of intelligent transformation, constructing a two-layer complex network evolutionary game model. Through theoretical analysis and simulation, we systematically investigated the role of key influencing factors on corporate strategy evolution. Within the static game framework, we first established the game model for supply-side and demand-side firms, clarifying the relationship between game parameters and strategic equilibrium. By calculating the payoff matrix and the replicator dynamic equations, we found that the evolutionarily stable equilibrium (ESS) corresponding to the ideal market state is one where both parties choose the cooperative strategy. The derived conditions for the ESS indicate that government subsidies, the cost of intelligent servers, and the additional profit gained by enterprises from adopting intelligent transformation are the core factors influencing the formation of cooperative strategies.

The research findings show that demand-side enterprises are sensitive to government subsidies and additional transformation profits, and their willingness to cooperate is easily influenced by policy and profit incentives. In contrast, supply-side enterprises are relatively less sensitive to subsidies, relying more on their own technological capabilities and market demand. Moderate costs for purchasing and setting up intelligent servers are critical for promoting mutual cooperation; high costs inhibit the demand-side’s willingness to cooperate, while a reasonable cost setting can increase the supply-side’s revenue and attract the demand-side to participate in the transformation. More importantly, within the two-layer heterogeneous network structure, these effects exhibit structural heterogeneity and nonlinear diffusion characteristics in the simulation: changes in key parameters do not lead to uniform responses across all enterprises, but instead propagate through network connections, where highly connected nodes (hubs) and inter-layer linkages may amplify or dampen the overall cooperation dynamics. This suggests that policy effectiveness may depend not only on parameter magnitude but also on network position and connectivity, although such observations remain conditional on the model setting.

Beyond parameter-level analysis, the two-layer heterogeneous structure enables the model to capture interaction patterns that are not observable in single-layer or homogeneous network settings, within the scope of the simulation framework. In a homogeneous single-layer setting, all firms share the same degree distribution and interaction rules, which masks the systematic asymmetry between supply-side and demand-side enterprises—particularly their differential sensitivities to subsidies, cost shocks, and network connectivity. The two-layer framework explicitly captures cross-layer structural asymmetry: cooperative behavior and policy incentives diffuse at different rates through the supply and demand layers, and inter-layer linkages function as critical channels through which supply-side hub nodes can trigger cascading cooperation effects on the demand side, while the reverse channel is comparatively weaker. This uneven policy transmission has theoretical implications: it suggests that targeting high-connectivity nodes in the supply-side layer may be disproportionately effective in promoting system-wide cooperation—an insight that only emerges from the structural heterogeneity introduced by the two-layer design and cannot be derived from models that treat all agents as equivalent.

By introducing a dynamic reward mechanism, this paper optimizes strategy evolution by adjusting node payoffs in a responsive manner to incentivize cooperation and inhibit defection. It should be noted that such responsiveness does not require strict real-time implementation; rather, it can be interpreted as periodic or delayed policy adjustment. The simulation results indicate that as long as incentives are sufficiently persistent (e.g., through the time-sensitivity parameter), the cooperation-promoting effect remains robust even under non-instantaneous responses. Simulation analysis demonstrates that as the reward and punishment factors increase, the proportion of both supply-side and demand-side enterprises choosing the cooperative strategy rises significantly, leading to an increase in the overall network cooperation level. Appropriate reward and punishment cycles can enhance the long-term stability of the reward mechanism, maintaining a high cooperation rate in the system across multiple game rounds. Compared to a fixed reward mechanism, the time-sensitive dynamic reward mechanism proves more effective in stimulating cooperation between the supply and demand sides, particularly significantly altering the strategy evolution trend among demand-side enterprises. This result indicates that, in the process of enterprise intelligent transformation, designing a scientifically sound dynamic incentive mechanism can effectively overcome non-cooperative behaviors driven by short-term interests, thereby pushing the system toward the ideal equilibrium.

The introduction of a dynamic network topology further enhanced the realism of the model. Through mechanisms of edge breaking and edge adding, nodes can dynamically adjust their connection relationships based on neighbors’ strategies and payoffs, forming localized network structures centered on cooperation. Simulation results show that dynamic networks enhance the clustering effect of cooperative nodes, increasing the overall network cooperation level, and revealing the profound impact of network structure changes on strategic evolution. Combining the dynamic network topology mechanism with the dynamic reward mechanism, the simulation results indicate that their synergistic effect can further increase the proportion of both supply-side and demand-side enterprises choosing the cooperative strategy, making it easier for the system to reach the ideal evolutionarily stable state. This finding suggests that, in the practical management of enterprise intelligent transformation, simultaneous attention to the inter-firm relationship network and the reward/incentive mechanism can significantly enhance the cooperation rate and reduce the risk of market degradation.

Based on the simulation results within the model framework, this paper offers several tentative policy implications. Because the model is stylized and parameter calibration is based on a limited set of large enterprises, these implications should be understood as conditional and model-based insights rather than directly operational prescriptions. First, the model suggests that government support—such as fiscal subsidies and tax incentives—may help encourage firms to choose cooperative strategies, particularly for demand-side enterprises whose cooperation probability proves more sensitive to subsidy levels in the simulation. Second, the results indicate that the cost of intelligent servers is a critical mediating factor: only within a moderate cost range do both supply-side and demand-side enterprises sustain high cooperation rates. This implies that policies aimed at lowering initial investment thresholds, such as R&D subsidies, may be worth further exploration, though their effectiveness in heterogeneous real-world contexts will depend on firm-specific conditions. Furthermore, the model suggests that enhancing the expected economic benefits of intelligent transformation is important for stimulating demand-side cooperation. Finally, the simulation indicates that combining dynamic incentive mechanisms with network structure adjustment produces synergistic cooperation-promoting effects, suggesting that the design of persistent and responsive incentive policies merits further empirical investigation.

Beyond these practical implications, this study also contributes to the theoretical development of evolutionary game models on complex networks. By integrating a two-layer heterogeneous network structure with a time-sensitive dynamic reward mechanism, the model extends traditional evolutionary game frameworks from homogeneous and static settings to a more realistic environment characterized by cross-layer interactions and adaptive incentives. The findings highlight that cooperation dynamics are not only determined by payoff parameters but are also shaped by network topology, node heterogeneity, and dynamic adjustment processes. These insights are not limited to the intelligent server industry, but can also be generalized to other sectors involving supply–demand interactions and technology adoption, such as digital platforms, manufacturing ecosystems, and green innovation networks.

Conclusion

Within the scope of the proposed simulation model, the results suggest that the adoption rate of cooperative strategies in enterprise intelligent transformation may be influenced by corporate revenue and cost structures, the introduction of dynamic reward mechanisms, and the configuration of inter-firm network structures. The simulation results suggest that government fiscal support and subsidies may significantly incentivize the transformation enthusiasm of demand-side enterprises, while supply-side firms appear to rely more on technological maturity and market demand. The model indicates that the proper configuration of intelligent server costs and additional transformation profits is important for the formation of cooperative strategies on both sides. Moreover, the combined effect of the dynamic reward mechanism and the dynamic network topology mechanism is observed to increase the level of corporate cooperation within the simulation, providing model-based insights that may inform further discussion of intelligent transformation management. These conclusions are conditional on the model’s stylized assumptions and the specific calibration context; their applicability to diverse real-world enterprise settings requires further empirical investigation.

However, this research has several limitations. First, the model parameters and simulation analysis are primarily based on theoretical assumptions and calibrated using data from large-scale enterprises, which may not fully capture the characteristics of SMEs. Although the model emphasizes parameter sensitivity and evolutionary trends rather than specific numerical values, its external applicability to heterogeneous enterprise groups still requires further empirical validation. Second, the parameter choices in the dynamic reward and network topology models (such as reward factor, punishment factor, edge breaking probability, etc.) may perform differently in various market environments, requiring fine-tuned adjustments in practical application. Additionally, this paper mainly focuses on the strategic evolution of supply-side and demand-side enterprises, and assumes stylized incentive adjustment processes; in reality, policy implementation may involve delays and information constraints, which should be further incorporated into future model extensions. Finally, the study primarily focuses on short-term strategy evolution, and issues concerning long-cycle cooperation stability and sustainability require further exploration.

Future research can be expanded in the following directions. First, by collecting actual data on enterprise intelligent transformation, the model can be empirically validated to improve its predictive capability and the reliability of policy recommendations. Second, the model can be extended to multi-level, multi-industry, and multi-regional complex networks to explore multi-dimensional strategic interactions and evolutionary patterns. Third, the dynamic reward mechanism and network structure adjustment strategies can be optimized in conjunction with macro-policies and market dynamics to achieve long-term stability and sustainable development of enterprise intelligent transformation. Fourth, future work should focus on long-term strategic evolution and the stability of game equilibrium, investigating adaptive pathways for enterprise intelligent transformation under uncertainty.

Author contributions

L.S. wrote the original draft of the manuscript. C.X., Y.J., D.B., and J.L. reviewed and edited the manuscript. All authors have read and approved the final manuscript.

Funding

Not applicable.

Data availability

Data can be obtained from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Institutional review board statement

Not applicable.