Frontiers in Business, Economics and Management ISSN: 2766-824X | Vol. 19, No. 1, 2025 55 Research on Industrial Structure Dynamic Evolution Based on Multi-objective Optimization and Complex Networks Kexin Song 1, *, Lingwei Kong 1, Yijing Pan 1, Zhe Cheng 1, Hongyang Wang 1, Bozhi Yang 2, Xinyu Gu 3 1 School of medical information engineering, Shandong University of Traditional Chinese Medicine, Shandong, 250355, China 2 School of mathematics and statistics, Anyang Normal University, Anyang, 455000, China 3 School of mathematics and statistics, Changzhou Institute of Technology, Changzhou, 213001, China * Corresponding author: 18678853819@139.com Abstract: In the context of economic globalization, the interdependencies among industries have become increasingly complex, and traditional linear analysis methods are insufficient to meet the demands of optimizing industrial structures while balancing multiple objectives, such as employment and environmental sustainability. This study develops an innovative model that integrates dynamic systems, complex network theory, and multi-objective optimization techniques to analyze and optimize the industrial structure. The model accounts for the evolution of industrial interactions over time and seeks to maximize GDP, employment, and sustainability. Experimental results show that the optimal investment allocation across industries can significantly enhance GDP, with key sectors such as Finance and Technology contributing the most to the economic output. The model's dynamic evolution highlights differences in industry growth trajectories, and the analysis suggests that Finance, Manufacturing, and Energy are central to economic development. Visualizations reveal that optimal investment allocation can lead to a 15% increase in overall GDP while improving employment by 12% and reducing environmental costs by 8%. These findings offer valuable insights for policy-making and strategic investments aimed at promoting sustainable economic growth. Keywords: Industrial Structure, Multi-Objective Optimization, Dynamic Systems, Economic Growth, Sustainability. 1. Introduction In the context of economic globalization and industrial upgrading, the relationships between industries are becoming increasingly intricate. This complexity renders traditional linear analysis methods insufficient to fully capture the dynamic nature of industrial structure evolution. Many existing studies have concentrated either on optimizing industrial structures from a singular perspective or have been limited to static analyses [1]. These approaches fail to account for the broader patterns of industrial structure evolution and the changing interactions among industries, which are essential for understanding the ongoing transformations in the economy. Furthermore, while economic growth remains a primary goal, factors such as employment generation and environmental sustainability have also become critical objectives in the optimization of industrial structures. These additional goals add complexity to the process, highlighting the need for new, more comprehensive analytical frameworks. This study proposes to address these challenges by developing an innovative model that integrates dynamic systems theory, complex network theory, and advanced optimization techniques. This approach aims to analyze the evolution of industrial structures over time, while simultaneously addressing multiple objectives such as economic growth, employment, and environmental sustainability [2]. By considering these multifaceted factors, the model will offer a more holistic view of how industries interact and evolve. The significance of this research lies in its potential to provide a deeper understanding of industrial dynamics and its ability to inform policy decisions that promote long-term sustainable economic development. The innovation of this study lies in its integrated approach, which transcends traditional models and offers new insights into the complexities of industrial optimization in the modern globalized economy. 2. Analysis and Multi-objective Optimization Modeling of Industrial Structure Correlation 2.1. Data Sources The data sources for this study are listed in Table 1. The analysis utilizes several authoritative datasets, including the Input-Output tables from the Organisation for Economic Co- operation and Development (OECD), GDP contribution data from the World Bank, employment statistics from Eurostat, and policy and environmental information from the United Nations Environment Programme (UNEP). These diverse datasets provide a robust foundation for exploring the interdependencies among major industries and their collective impact on economic development. Table 1. Data sources Data Name Website Input-Output (IO) Tables https://www.oecd.org/en/data/datasets/in put-output-tables.html GDP Contribution Data https://data.worldbank.org/indicator/NY. GDP.MKTP.CD Employment Data https://ec.europa.eu/eurostat/web/lfs Policy and Environmental Data https://www.unep.org/data-resources 56 Prior to model construction, the dataset was meticulously refined. Missing entries in the employment impact metrics were imputed using the column mean, while outliers in variables such as final demand and total output were identified and capped via the interquartile range (IQR) method. Visualization toolsβ€”including violin plots, boxplotsβ€”confirmed that the preprocessing resulted in a consistent and reliable dataset for subsequent analysis. The specific process is shown in the Figure 1. Figure 1. Data preprocessing 2.2. Model Development The model formulation integrates concepts from economic theory, network analysis, and optimization. The economic interdependencies among industries can be examined through an Input-Output framework. In this model, the total output is represented by a vector X = [π‘₯1, π‘₯2, … , π‘₯𝑛 ] ⊀, while the final demand is captured by Y = [𝑦1 , 𝑦2, … , 𝑦𝑛 ] ⊀. The inter-industry relationships are encapsulated in the input coefficient matrix A = [π‘Žπ‘–π‘— ], where each coefficient is defined as π‘Žπ‘–π‘— = 𝑧𝑖𝑗 π‘₯𝑗 .The relationship between total output and final demand is expressed by the equation X = (I βˆ’ A)βˆ’1Y (1) With (I βˆ’ A)βˆ’1 being the Leontief inverse, which captures both the direct and indirect effects of changes in final demand. In this framework, the contribution of each industry to GDP is split into two components. The direct contribution of an industry is given by its final demand 𝑦𝑖 . The indirect contribution is determined by the sum of the impacts from all industries, computed as Ic 𝑖 = βˆ‘ π‘Žπ‘–π‘— 𝑛 𝑗=1 β‹… π‘₯𝑗 (2) Combining these yields the total contribution of industry 𝑖: 𝑐𝑖 total = 𝑦𝑖 + βˆ‘ π‘Žπ‘–π‘— 𝑛 𝑗=1 β‹… π‘₯𝑗 (3) The GDP contribution rate for each industry is then obtained by taking the ratio of its total contribution to the overall GDP. Industries can also be modeled as nodes within a directed weighted network, where the weights on the edges correspond to the input coefficients π‘Žπ‘–π‘— .This network is characterized by the graph 𝐺 = (𝑉, 𝐸, π‘Š ), with π‘Š = [π‘Žπ‘–π‘— ] . Several metrics help quantify the roles of industries within this network. For example, degree centrality is calculated by summing the input coefficients for each industry: DC(𝑣𝑖 ) = βˆ‘ π‘Žπ‘–π‘— 𝑛 𝑗=1 (4) Which reflects the direct influence of an industry on others. Betweenness centrality is another useful metric and is defined as BC(𝑣𝑖 ) = βˆ‘ πœŽπ‘ π‘‘(𝑖) πœŽπ‘ π‘‘ 𝑠≠𝑖≠𝑑 (5) Where πœŽπ‘ π‘‘ is the total number of shortest paths between nodes 𝑠 and 𝑑,and πœŽπ‘  𝑑(𝑖) is the number of those paths that pass through industry 𝑖. In addition, the PageRank algorithm is applied to gauge industry importance, with its formulation given by PR(𝑣𝑖 ) = 1βˆ’π‘‘ 𝑛 + 𝑑 β‹… βˆ‘ PR(𝑣𝑗) OutDegree(𝑣𝑗) π‘—βˆˆIn(𝑣𝑖) (6) Where 𝑑 is the damping factor (commonly set to 0.85). The relationships between industries are further explored using statistical tools. The Pearson correlation coefficient measures the linear relationship between the outputs of any two industries: πœŒπ‘–π‘— = Cov(𝑋𝑖,𝑋𝑗) πœŽπ‘‹π‘– β‹…πœŽπ‘‹π‘— (7) Where Cov (𝑋𝑖 , 𝑋𝑗 ) represents the covariance between industries 𝑖 and 𝑗, and πœŽπ‘‹π‘– , πœŽπ‘‹π‘— are their standard deviations [3]. To determine if one industry has a predictive influence on another, a Granger causality test is conducted; a significant F- test result implies that the output of industry 𝑗 can be said to Granger-cause the output of industry 𝑖. To address multiple policy goals such as GDP growth, employment, and sustainability, a Mult objective optimization model is developed. The objective function to maximize is formulated as max𝑍 = 𝛼1 βˆ‘ 𝑔 𝑛 𝑖=1 (π‘₯𝑖 ) + 𝛼2 βˆ‘ 𝑒 𝑛 𝑖=1 (π‘₯𝑖 ) + 𝛼3 βˆ‘ 𝑠 𝑛 𝑖=1 (π‘₯𝑖 )(8) Where 𝑔(π‘₯𝑖 ) represents the marginal GDP contribution, 𝑒(π‘₯𝑖 ) indicates the employment impact, and 𝑠(π‘₯𝑖 ) denotes the sustainability score for industry 𝑖. The coefficients 𝛼1, 𝛼2, and 𝛼3 are weights reflecting the importance of each factor. βˆ‘ π‘₯𝑖 𝑛 𝑖=1 = 1 (9) This optimization is subject to the constraints that the total investment sums to one and that each investment π‘₯𝑖 is non- negative. 57 A dynamic system further enriches the analysis by modeling the evolution of industry interactions over time. The growth of each industry’s GDP contribution 𝑋𝑖 (𝑑) is described by the differential equation [4]. 𝑑𝑋𝑖(𝑑) 𝑑𝑑 = 𝛽𝑖 β‹… 𝑋𝑖 (𝑑) + βˆ‘ 𝛾𝑖𝑗 𝑛 𝑗=1 β‹… 𝑋𝑗 (𝑑) (10) Where 𝛽𝑖 represents the intrinsic growth rate of industry 𝑖, and 𝛾𝑖𝑗 quantifies the influence exerted by industry 𝑗 on industry 𝑖. This formulation provides insights into both the immediate and cascading effects of economic activities over time. 3. Dynamic Modeling and Prediction of Government Investment Structure Optimization 3.1. Model Development To capture the evolution of GDP in response to investment, we adopt a system of nonlinear differential equations. The GDP of industry 𝑖 at time 𝑑, denoted by 𝐺𝑖 (𝑑), evolves as: 𝑑𝐺𝑖(𝑑) 𝑑𝑑 = 𝑓(𝐼𝑖 (𝑑)) + βˆ‘ π‘Ÿπ‘–π‘— 𝑛 𝑗=1 𝐺𝑗 (𝑑) βˆ’ πœ‡π‘– 𝐺𝑖 (𝑑) (11) Where: 𝑓(𝐼𝑖 (𝑑)) is the direct contribution of investment to GDP, modeled ass 𝑓(𝐼𝑖 (𝑑)) = 𝛼𝑖 (1 βˆ’ 𝑒 βˆ’π›½π‘–πΌπ‘–(𝑑)) (12) Where 𝛼𝑖 and 𝛽𝑖 are industry-specific parameters representing the growth potential and rate of saturation. π‘Ÿπ‘–π‘— represents the spillover effect from industry 𝑗 to 𝑖. πœ‡π‘– is the natural decay rate of GDP for industry 𝑖, capturing external shocks and competitive pressures. The total GDP across all industries is: 𝐺(𝑑) = βˆ‘ 𝐺𝑖 𝑛 𝑖=1 (𝑑) (13) The interdependencies between industries are modeled using a complex network. Let 𝑅 = {π‘Ÿπ‘–π‘— } denote the adjacency matrix of the network, where: π‘Ÿπ‘–π‘— = Correlation(𝐺𝑖 , 𝐺𝑗 ) (14) This correlation is computed from historical data, capturing both positive and negative Relationships: π‘Ÿπ‘–π‘— > 0 : Positive spillover effect; π‘Ÿπ‘–π‘— < 0 : Negative constraint or competition. Analyzed the network using graph-theoretic methods to identify key industries (nodes) with high centrality measures, indicating their importance in driving overall ecnomid growth [5]. Aimed to maximize GDP while considering employment growth and minimizing environmental costs. The multi- objective optimization problem is formulated as: max{𝐺, 𝐸, βˆ’πΆ} (15) Subject to: βˆ‘ 𝐼𝑖 𝑛 𝑖=1 ≀ 𝑇,   𝐼𝑖 β‰₯ 0,   βˆ€π‘– (16) Where: 𝐺 = βˆ‘ 𝐺𝑖 𝑛 𝑖=1 is the total GDP. 𝐸 = βˆ‘ 𝐸𝑖 𝑛 𝑖=1 (𝐼𝑖 ) is the total employment effect, modeled as: 𝐸𝑖 (𝐼𝑖 ) = πœ™π‘– ln(𝐼𝑖 + 𝛿𝑖) (17) Where πœ™π‘– and 𝛿𝑖 are industry-specific employment parameters. 𝐢 = βˆ‘ 𝐢𝑖 𝑛 𝑖=1 (𝐼𝑖 ) is the total environmental cost, modeled as: 𝐢𝑖 (𝐼𝑖 ) = πœ“π‘– 𝐼𝑖 2 (18) Where πœ“π‘– represents the environmental sensitivity of industry 𝑖. To improve the accuracy of predictions, use machine learning models such as Long Short-Term Memory (LSTM) networks. The LSTM model predicts future GDP for each industry based on historical data: �̂�𝑖 (𝑑 + 1) = LSTM(𝐼𝑖 (𝑑), 𝐺𝑖 (𝑑), 𝑅) (19) Where the inputs include: Current investment (𝐼𝑖 (𝑑)) , Current GDP (𝐺𝑖 (𝑑)) , Industry network matrix (R). The predicted GDP values (�̂�𝑖(𝑑 + 1)) are integrated into the dynamic model to refine the optimization process [6]. 3.2. Dynamic Investment Allocation via Slime Mold Algorithm The SMA (Slime Mold Algorithm) can enhance the ability to avoid getting trapped in local optima because it simulates the behavior of slime molds, which continue to explore other potential solutions even after finding a good solution. The specific steps of the algorithm are shown in the Figure 2 [7]. 58 Figure 2. Flowchart of the Slime Mold Algorithm The parameter settings for the Slime Mold Algorithm are shown in Table 2: Table 2. Slime Mold Algorithm Parameters Parameter Name Value Population Size 30 Maximum Number of Iterations 500 Sensitivity Initial Value 1 Vibration Parameter Initial Value 0.5 Random Disturbance Parameter 0.5 Positive Feedback Adjustment Weight Initial Value 0.7 4. Result 4.1. The Establishment of Simulation Model Figure 3 provides a comprehensive visualization of industry metrics and relationships. The radar chart highlights the relative strengths of each industry across GDP contribution, employment impact, and sustainability scores, with Finance and Technology excelling across metrics. The chord diagram reveals strong inter-industry dependencies, particularly between Manufacturing and Energy. Together, these visualizations offer insights into sectoral performance and relationships to guide strategic investments. Figure 3. Visualizing Industry Metrics and Relationships 4.2. Analysis of Experimental Results When applying the SMA to the problem, observe the performance changes of the SMA over multiple iterations and visualize the results. The results are shown in the Figure 4: 59 Figure 4. SMA Iteration Process As can be seen from the figure, the model performs well. Applying the model to solve the problem yields: Figure 5. Visualization of Investment and GDP Dynamics Figure 5 provides a comprehensive and innovative visualization of the relationship between investment allocation and GDP dynamics across different industries. It includes the following components. This plot illustrates the temporal evolution of GDP across various industries, highlighting the differences in growth trajectories because of optimal investment allocation. The figure captures the dynamic interaction between investment and industrial growth, providing insights into the changing contributions of each sector over time. This polar chart shows the final GDP contribution of each industry as a percentage of the total GDP. It facilitates a clear comparison of the relative impact of different industries on overall economic output, helping to identify key sectors that drive economic growth. Together, these visualizations offer a detailed understanding of the interdependencies between sectors and the effects of investment on industrial growth, serving as a critical tool for identifying sectoral priorities and informing policy decisions aimed at optimizing economic development. 5. Conclusion This study develops a novel model combining dynamic systems, complex network theory, and multi-objective optimization techniques to optimize industrial structures. The key findings of the research demonstrate that optimal investment allocation can significantly enhance GDP, employment, and environmental sustainability. By capturing the interdependencies between industries, the study identifies critical sectors such as Finance, Technology, and Manufacturing that drive economic growth. The model shows that targeted investments in these industries can lead to a 15% increase in GDP, a 12% improvement in employment, and an 8% reduction in environmental costs, offering a comprehensive approach to optimizing industrial structures for sustainable economic development. The results provide valuable insights for policymakers and business leaders in making data-driven decisions on investment allocation. 60 However, the model has certain limitations. Future research should focus on improving the model's adaptability by incorporating more granular and real-time data, exploring the impact of external shocks, and validating the model's applicability across different regions or sectors. By addressing these limitations, future studies can enhance the robustness and flexibility of this framework for more accurate industrial structure optimization. References [1] CAIRNS R, MARTINET V. Growth and long-run sustainability [J]. Environment and Development Economics, 2021, 26: 381-402. 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