Research on the Statistical Properties and Stability of Complex Interindustrial Networks

. Tis study consolidates input-output data from 42 sectors across 31 provinces and regions in China into a unifed dataset for 42 industrial sectors within eight major economic zones. Leveraging the maximum entropy method, we identify signifcant interindustrial relationships, subsequently forming a directed, weighted, complex network of these ties. Building upon this intricate network, we analyze its foundational statistical attributes. Te stability of the network’s structure is further assessed through simulations of varied network attacks. Our fndings demonstrate that the maximum entropy method is adept at extracting notable relationships between industrial sectors, facilitating the creation of a cogent complex interindustrial network. Although this established network exhibits high stability, it calls for targeted policy interventions and risk management, especially for industries with pronounced degree centrality and betweenness centrality. Tese pivotal industry nodes play a decisive role in the overall stability of the network. Te insights derived from our examination of complex interindustrial networks illuminate the structure and function of industrial networks, bearing profound implications for policymaking and propelling sustainable, balanced economic progress.


Introduction
With the advent of globalization and the digital age, the interplay between industries has evolved into unprecedented complexity.Tis intricate nature challenges conventional analytical methods, rendering them inadequate for in-depth analyses of complex industrial networks within the economic domain.By the close of the 20th century, the rise of complex network theory not only garnered extensive academic attention but also paved the way for a series of groundbreaking research endeavors.Seminal theories such as Watts and Strogatz's small-world theory [1], as well as the scale-free network theory posited by Barabási et al. [2], continue to exert profound infuence in contemporary discourse [3,4].
Te growing body of literature on network disruption and resilience, such as Iyer et al.'s work on attack robustness and centrality [5], Casali and Heinimann's study on the road network robustness [6], and Ficara et al.'s investigation into strategies for disrupting criminal networks [7,8], further enriches our understanding of the complex interplays and resilience mechanisms inherent within various network systems.
Industrial interactions have transcended mere singular linkages, gradually morphing into a sophisticated network system.Tis structure elucidates the realities of industries under the twin forces of globalization and technological advancement, ofering a renewed analytical lens for both theoretical exploration and practical application.Notably, within the realm of economics, the introduction of complex network theory has rejuvenated input-output analysis.Leontief's input-output table stands as a pivotal tool to unveil resource fows and dependencies between varying industries [9].Methods grounded in this paradigm have been employed by scholars like Serrano et al. [10], ofering insights into the collaborative actions, competitive relations, and stability inherent within industries, particularly in their analyses of world trade networks and China's energy fow networks.
Yet, the evolution of network science marches on.Many scholars are shifting their focus toward the dynamic behaviors within networks, especially concerning stability issues.Tis concern spans from natural ecosystems to manmade realms such as transportation and supply chain networks, addressing challenges of holistic stability and potential collapses triggered by localized node failures or attacks.As a result, network resilience and defense against attacks have gradually ascended to the forefront of complex network research.Te stability and resilience of vital infrastructures, like fnance and energy, have garnered widespread attention [11,12].Recent studies, through simulations and experiments, have unveiled the latent ramifcations of network attacks while probing novel strategies to bolster network stability [13][14][15].
In summary, despite signifcant advancements in the study of complex industrial networks, numerous areas remain untouched and present challenges.It is especially salient to highlight that the stability analysis of input-output correlation within these networks has been somewhat overshadowed.Predominant studies tend to zoom in on specifc industries or regions, often neglecting a holistic global viewpoint and missing out on interdisciplinary syntheses.Given the escalating intricacy and diversifcation of industrial interactions, there is a pressing need for more in-depth exploration and research in this domain.
Te primary objective of this study is to delve deeply into the interactions between industries and their network structures using complex network theory, aiming to elucidate the network's stability and resistance to attacks.Te research unfolds in distinct phases.Initially, leveraging the input-output table data, we employ the maximum entropy method to shape a complex network.Subsequently, we probe into the foundational statistical attributes-degree, node weight, clustering coefcient, shortest path, and network efciency-of the crafted interindustrial complex network to discern the network's architecture.In the concluding phase, we assess the network's four centralities and gauge the stability and resilience of the overarching network through simulations of various attack paradigms.

Literature Review
2.1.Network Construction and Analysis Methods.Te construction and analysis of networks from empirical data have become pivotal in understanding complex systems across various disciplines.Te work of Donner et al. [16] highlights the quantitative assessment of structural properties in systems composed of interacting entities.Tis is particularly relevant for our study as it underscores the importance of understanding dynamic higher-order structures in complex networks, which can be applied to analyzing interindustrial relationships.Christensen et al. [17] provide insights into the universal nature of complex networks.Teir research into the topological similarities among diferent systems can inform our understanding of the interconnected nature of industrial sectors, further enriching our network analysis.Furthermore, Cheng and Scherpen [18] discuss the challenges and solutions to dealing with high-dimensional dynamics and complex interconnections in network systems.Tis perspective is crucial for simplifying and efectively analyzing the intricate network of industrial sectors in our study.Emmert-Streib et al. [19] emphasize the interconnectedness of economic and fnancial entities.Teir approach to network science in economics and fnance ofers valuable parallels to our method of extracting network structures from real-world data, particularly in the context of economic networks.Lastly, Polishchuk's [20] analysis introduces concepts like fow adjacency matrices and dynamic characteristics of system elements.Tese concepts are instrumental in understanding the behavior of complex network systems and can be applied to our study to explore the fow core and dynamic interactions within the industrial network.Similar to our method, these studies often extract network structures from real-world data, underscoring the importance of leveraging empirical data to form directed, weighted complex networks.Te process of forming these networks from input-output tables is akin to the approach taken by economic and ecological network analyses, which extract relational data to understand systemic dependencies and dynamics.Te insights from these papers not only reinforce our methodology but also provide a broader context for understanding the complexities and interdependencies in industrial networks.

Simulation and Attack Modeling in Network Stability
Analysis.Simulation methods, including those for targeted network attacks, are extensively utilized to evaluate network stability and robustness.Researchers simulate the removal of key nodes to determine a network's vulnerability to specifc disruptions.For example, Iyer et al. analyzed complex network centrality to assess network attack robustness, bridging theoretical models with practical applications.Furthering this feld, Dshalalow and White [21] employ stochastic processes to model network attacks, enabling predictions about the timing and scale of network failures.Tis method introduces a probabilistic aspect to network robustness assessment, shedding light on the unpredictability and impact of network attacks.Fabris and Zelazo [22] investigate the resilience of multi-agent consensus networks against attacks that manipulate edge weights.Teir research broadens the scope of network robustness understanding by examining the efects of such attacks on network convergence performance, underscoring the need for structured defense strategies.Sarraute et al. [23] present a prototype for simulating large-scale network attacks, focusing on realism from the attacker's perspective.Tis study highlights the importance of comprehensive simulation environments that accurately refect the complexities of real-world networks.P. Y. Chen and S. M. Chen [24] explore the efectiveness of sequential defense strategies against both random and intentional attacks, emphasizing the importance of adaptive defense mechanisms for maintaining network integrity.Bel et al. [25] introduce a cosimulation framework for generating and monitoring network attacks, especially in power grids with integrated distributed energy resources.Tis study points out the changing nature of network attack surfaces and the necessity for advanced simulation tools for evaluating and improving network resilience.

Complexity
Collectively, these studies enhance the understanding of network stability and robustness, highlighting the critical role of advanced simulation methods in assessing network vulnerabilities and devising defense strategies against various potential attacks.

Assessment of Network Stability.
Evaluating the stability of network structures is crucial for understanding the resilience of complex systems to various perturbations.Our study extends the literature by simulating varied network attacks to assess the stability of the interindustrial network.Tis methodology aligns with the approaches seen in Casali and Heinimann, who evaluated the robustness of the Zurich road network under diferent disruption processes.By employing simulations of targeted attacks, we can identify critical nodes within the network, akin to the analysis of network centrality in determining the pivotal roles certain nodes play in maintaining network integrity and stability.Building upon this foundation, Platig et al. [26] further enhance our understanding of network stability.Teir exploration of the robustness of network measures, including centrality, in the presence of link inaccuracies is particularly relevant.It underscores the importance of considering the reliability of network connections in assessing overall network stability.Ufmtsev et al. [27] reveal that the impact of structural noise on centrality ranks is examined.Tis study complements our approach by highlighting how minor structural changes can signifcantly infuence the stability and centrality of nodes, thereby afecting the network's resilience to disruptions.Saxena et al. [28] argue for the importance of stability in centrality measures under information loss or noise.Tis perspective is crucial for our study as it emphasizes the need for robust centrality measures that can withstand variations in network data, ensuring accurate identifcation of critical nodes.Oldham et al. [29] provide insights into the roles of diferent nodes through various centrality analyses.Understanding the consistency and uniqueness of these measures across diferent network types aids in accurately assessing the roles of nodes in maintaining network stability.Gupta et al. [30] discuss the signifcance of centrality measures in large networks with community structures.Tis research is pertinent to our study as it ofers methods to identify infuential nodes in complex networks, which is key to understanding and enhancing the resilience of the interindustrial network.Together, these studies enrich our methodology by providing a comprehensive view of how centrality measures and network robustness assessments can be efectively utilized to understand and improve the stability of complex networks, such as the interindustrial network in our study.In the course of data processing, input-output data from the 42 industries across each province, region, and city were aggregated.Tis aggregation was undertaken to compute the consolidated input-output fgures for the 42 industries within the eight principal economic regions, yielding a comprehensive input-output table.For analytical clarity, the eight key economic regions are denoted by letters A through H, while the 42 industries are numerically represented from 1 to 42.Refer to Tables 1 and 2 for a detailed breakdown.

Modeling Method.
In this study, the construction of the complex network pivots on the 0-1 adjacency matrix.Within this matrix, an element denoted as "1" signifes that the nodes corresponding to that particular row and column are interconnected, whereas an element marked "0" suggests the absence of such a connection.To achieve this confguration, the matrix undergoes a binarization process to manifest as a 0-1 adjacency matrix.Tis study employs the maximum entropy method for binarization, a statistical approach grounded in information theory tailored to estimate probability distributions contingent on certain predefned constraints.Tis approach is in line with Loaiza-Ganem et al. [32], who highlight the adaptability and efcacy of maximum entropy in statistical models, especially for optimizing network density.In addition, Metzig and Colijn [33] demonstrate the use of Gibbs-Shannon entropy in network size and degree distributions, providing a theoretical foundation for predictive analysis, and Zenil et al. [34] ofer insights into employing maximum entropy for prior probability distributions, essential for understanding the probabilistic aspects of industrial networks.
In information theory, entropy quantifes the uncertainty inherent in random variables and simultaneously refects the informational richness of the data.Te information entropy of an event is, essentially, the expected value of the logarithm of the event's probability.When entropy peaks, it signals the extraction of optimal Complexity where p(x) represents the probability of the event's occurrence.Given a set with n events, the information entropy for this event set is defned as follows: where p(x i ) denotes the probability of the i th event occurring.Te crux of the maximum entropy method lies in its principle: Among all probability distributions that align with given constraints, the distribution with the maximum entropy is the one capturing the optimal amount of information.
In the construction of a complex network, the maximum entropy method plays a pivotal role in determining the network's adjacency matrix.Specifcally, an appropriate threshold is chosen to classify the connections between nodes into two categories: present or absent.When the sum of the average entropies for these two categories is maximized, we attain the greatest amount of information.Tis, in turn, helps ascertain the optimal threshold, which dictates whether a connection between nodes exists.Te primary steps for the maximum entropy algorithm in identifying the best threshold are as follows: (1) Initially, acquire the original matrix indicating the correlation strength between nodes and derive the probability distribution of the correlation strengths.
If the total count of distinct correlation strengths is N, the probability of the i th correlation strength appearing is represented as p i .(2) Set an initial threshold T, which equates to the j th correlation strength, 1 ≤ j < N. Partition the correlation strength matrix into two categories: those less than or equal to T, and those greater than T. (3) Calculate the average relative entropy for both categories: (4) When the value of E 1 + E 2 peaks, the corresponding T is identifed as the optimal threshold.By binarizing the correlation strength matrix using this optimal threshold, we obtain the adjacency matrix.

Building Complex Networks.
To construct an adjacency matrix based on the input-output tables from 42 sectors (industries) within the eight major economic zones, it is frst essential to establish a matrix denoting the linkage intensity between these industries.In this study, the directly calculated consumption coefcient matrix is employed as this linkage intensity matrix.Tis direct consumption coefcient matrix illustrates the coefcient of direct inputs from various sectors into a specifc output, shedding light on the direct dependencies between industries.Let's denote this coefcient matrix as A, where A ij represents the direct input required from industry i to produce 1 unit of the product from industry j. Applying the maximum entropy method described in the previous section, we determined the optimal threshold for the direct consumption coefcient matrix.Tis threshold serves to extract signifcant interindustry relationships from the original direct consumption coefcient matrix.Specifcally, elements in matrix A that are less than or equal to this threshold are assigned a value of 0, while those exceeding the threshold are designated a value of 1.
Consequently, we obtain the adjacency matrix M, which delineates the salient relationships between industries.Tis procedure is referred to as the binarization process.
Based on the adjacency matrix M, we construct a complex network.If M ij � 1 , an oriented edge is drawn from industry i to industry j. Conversely, if M ij � 0 , there is no edge from industry i to industry j .Given the inherent asymmetry in inputs (or consumption) between two industries, typically M ij ≠ M ji .Moreover, some industries have inputs (or consumption) relating to themselves, M ii ≠ 0, indicating the presence of loops within the network.Keeping in mind the practical signifcance of the input-output association in complex networks, it is essential to account for the diferences in the magnitude of inputs (or consumption) between industries.As such, edges within the network carry weights, denoted as W ij , with the set of all weights represented as W. Here, W ij corresponds to the direct consumption coefcient A ij of industry i to industry j. Te resulting complex network is illustrated in Figure 1.In the fgure, blue dots represent network nodes, red labels denote node codes, black lines signify edges, and arrows indicate the direction of the edges.Nodes within the network symbolize industry sectors, with the eight major economic zones contributing 336 nodes in total.Within each economic zone, the 42 industry nodes are distributed in a rectangular uniform layout, while the grouping of these nodes refects the rough geographical placement of the respective economic zones.Edges in the network represent relationships between industries, amounting to 30,864 directed and weighted edges in total.
In summary, starting from the direct consumption coefcient matrix, and using the maximum entropy method to go through a series of calculations and processing, a directed weighted complex network with self-loop is fnally established.Tis network provides the basis for subsequent network statistical properties and stability analysis.

Analysis of Complex Network Statistical Properties
foundational in understanding the organization and dynamics of diverse complex systems, from biological networks to social structures and technological infrastructures.
Te application and signifcance of graph theory in analyzing such systems are extensively supported by a body of research.Christensen and Albert [17] underscore the universal applicability of graph concepts across various felds, highlighting the shared topological features of diferent complex systems.Jalving et al. [35] propose graph-based modeling abstractions that articulate the dependencies and interactions within complex systems.Teir work is particularly pertinent to studies focused on interconnected entities, such as industrial networks, demonstrating the utility of graphbased models in capturing the intricate relationships that defne complex systems.Torres et al. [36] delve into the intricacies of representing complex systems, emphasizing the necessity for efective representation strategies across various domains.Teir discussion on the why, how, and when of complex system representations sheds light on the methodological challenges and considerations in employing graph theory to model complex interactions, providing a critical lens through which to view our research endeavors.Te number of connections or edges linked to a node, or its degree, provides insights into its role and signifcance within the network.In complex networks, the degree of a node is indicative of its connectivity.In directed networks, the concept further bifurcates into in-degree (incoming connections) and out-degree (outgoing connections).From an economic perspective, a higher degree, whether it be indegree or out-degree, generally signifes an industry's infuential role and interconnectedness in the marketplace.A node's degree in economic networks can be seen as a measure of its interdependence with other industries or sectors.
In economic networks, the in-degree can be thought of as the diversity of resources or inputs an industry requires, whereas the out-degree can indicate the variety of outputs or services it provides to other sectors.An industry with a high in-degree, for instance, might be crucial for multiple sectors due to its essential products or services.Conversely, a high out-degree may suggest the industry relies on diverse inputs from various sectors, denoting its intricate integration into the overall economic fabric.
For the established directed weighted network, we computed the degree, in-degree, and out-degree and derived a degree distribution histogram as shown in Figure 2. Statistical analysis reveals that the degree distribution does not follow a power-law distribution, with 85% of industry sectors having a degree ranging between 70 and 220.Tis indicates a tight topological linkage among industries.Te industry with the lowest degree is Petroleum and Natural Gas Extraction Products (East coast) with an in-degree of 35 and an out-degree of 4, suggesting that the East Coast economic zone's petroleum and natural gas extraction heavily relies on input from other industries.Te industries with the highest degrees include Wholesale and Retail (East Coast), Transportation, Warehousing, and Postal Services (East Coast), Nonmetallic Minerals and Other Mining Products (Mid-Yellow River), and Transportation, Warehousing, and Postal Services (North coast) with degrees ranging from 328 to 325.Tese industries maintain connections with almost all other sectors, denoting their signifcant role in the economic network.
Figure 2 also reveals that the in-degree distribution roughly follows a bell-shaped curve, indicative of a Poisson distribution, with 91% of industries having an in-degree ranging between 50 and 130. Te highest in-degree is 163 for construction (Mid-Yellow River), followed by metal products machinery and equipment repair services (Mid-Yellow River) with an in-degree of 156, and then various industries from the Mid-Yellow River and the Northwest.Notably, construction, public facilities, and the service sector are high-consumption industries, particularly those in the Mid-Yellow River and Northwest economic zones, as they require resources and products from nearly half of the industries.
Te out-degree distribution in Figure 2 shows a higher proportion of industries with lower out-degrees.Te industry with the highest out-degree of 326 is Transportation, Warehousing, and Postal Services (Mid-Yellow River), followed closely by Wholesale and Retail (East Coast) with 325.It is evident that transportation, warehousing, and postal services, as well as wholesale and retail, play a central role, as nearly all industries rely on consuming their products and resources, underscoring their critical position in the national economy.
Analyzing our directed weighted network, we found nuances in the connectivity and centrality of various sectors, refected in their degree distributions.A signifcant observation from Figure 2 is that the degree distribution does not align with the typical power-law seen in many real-world networks, implying that our economic network deviates from scale-free characteristics.Instead, the observed Poisson distribution suggests a more homogenous network structure where most nodes have a degree close to the average.Some sectors, as evident from our analysis, act as central hubs.Teir high connectivity, both in terms of inputs and outputs, indicates their pivotal role in the economic Given the directed, weighted nature of our established complex network, our initial degree distribution analysis did not account for the signifcance of edge weights.Consequently, we extended our investigation to ascertain the node strength, in-strength, and out-strength for each industry node within the network.
In weighted network analyses, the node strength represents the sum of the weights of all edges connected to a specifc node.In the context of a directed weighted network, the in-strength embodies the cumulative weight of all incoming edges (edges directed towards the node), while the out-strength conveys the sum of the weights of all outgoing edges (edges originating from the node).
From an economic perspective, these network metrics take on heightened signifcance.Te in-strength can be interpreted as the total volume of inputs an industry receives from other sectors, indicating its dependency or reliance on external factors for operation.Conversely, the out-strength ofers insights into the volume of inputs or resources an industry provides to others, refecting its contribution and potential infuence over other sectors in the economic landscape.
By examining these weighted network measures, we can derive a more nuanced understanding of each industry's role and importance within the broader economic network, leading to deeper insights into intersectoral dependencies and infuences.
Te direct consumption coefcient between industries serves as an edge weight in this analysis.Te higher its value, the more signifcant the input or consumption volume between two industries, indicating a closer interrelation.Terefore, this edge weight operates as a similarity weight.Table 3 presents industries with the highest and lowest node weight, in-weight, and out-weight.
A striking observation is that the industries ranking in the top four for node weight also dominate the top four for out-weight.Tese industries, in sequence, are as follows: chemical products (Eastern Coastal), chemical products (Northern Coastal), leasing and business services (Eastern Coastal), and chemical products (Middle Yangtze River).Predominantly, the node weight of these sectors is dictated by their out-weight.Chemical products, as an export-driven sector, provide a signifcant volume of resources or products to other industries.
Regionally, the Jiangsu-Zhejiang-Shanghai area is underscored by the robust standing of its wholesale and retail sector, which ranks ffth in out-weight, reinforcing the prosperous nature of this sector and its substantial contributions to other industries.
On the fip side, the industries ranking high in in-weight include textiles (Northern Coastal), other manufacturing products (Northwest), other manufacturing products (Northeast), textile, clothing, footwear, leather, down and its products (Middle Yellow River), and textile, clothing, footwear, leather, down and its products (Northern Coastal).

Complexity
Tis suggests some regional dynamics at play: other manufacturing product industries in the Northeast and Northwest regions appear to be underdeveloped, demanding signifcant inputs from other sectors.Te textile sector in the Northern Coastal area stands out in its reliance on other industries for resources.From a regional economic perspective, the spatial distribution and development of industries often refect a region's historical, geographical, and socio-economic conditions.Te thriving chemical and leasing businesses along the coastlines indicate the coastal regions' advantages in trade, port logistics, and market accessibility, leading to economies of scale and agglomeration benefts.
Comparing the industries ranking in the bottom fve for both node weight and out-weight, there's a discernible pattern: sectors such as oil and natural gas extraction products and metallic mineral mining products from the eastern coastal region have relatively low in-weights, signaling minimal direct consumption from other industries.Furthermore, waste material sectors in economic zones A, B, C, D, and H are similarly characterized by their low consumption from other sectors.Tis could hint at either the self-sufcient nature of these industries or perhaps a need for more integration and collaboration for sustainable regional economic development.

Clustering Coefcient.
Te clustering characteristic is a critical feature in complex networks, typically used to describe the aggregation tendency among nodes within the network.Te clustering coefcient is a commonly used metric to quantify this characteristic.Although its computation varies slightly across diferent network types (like undirected and directed networks), its core idea revolves around describing the interconnection situation amongst the neighbors of a node.For a given node i in an undirected network, the local clustering coefcient is defned as follows: where (1) C i represents the local clustering coefcient of node i.
(2) E i denotes the number of links between the neighbors of node.
(3) K i signifes the degree of node i, i.e., the number of links connected to node i.
Te local clustering coefcient describes the ratio between the number of actual connections formed among a node's neighbors and the maximum possible number of such connections.C i � 1 indicates that all neighbors of node i are interconnected, while C i � 0 signifes that there are no connections among the neighbors of node i. Te formula essentially captures the ratio of the number of actual links between the neighbors of a node to the maximum possible number of such links.A higher clustering coefcient for a node indicates that its neighbors are more densely interconnected.Te global clustering coefcient C is the average of the local clustering coefcients for all nodes in the network.Mathematically, it can be expressed as follows: where n is the total number of nodes in the network.Te global clustering coefcient is also commonly referred to as the average clustering coefcient of the network.
In the realm of directed networks, computing the clustering coefcient presents intricate challenges, largely due to the inherent directionality of the edges.A key complication arises from the fact that one triangle formation in an undirected network can manifest in seven possible confgurations in a directed network context.To circumvent this complexity, it is standard practice to consider directed edges in the network as bidirectional, undirected edges, thus allowing the application of clustering coefcient calculation methods originally developed for undirected networks.In weighted networks, the calculus extends beyond mere nodeto-node connection states to incorporate the strength of these connections as well.As a result, this study adopts a specifc defnition of the clustering coefcient that is tailored for weighted networks: where is the normalized weight.In this section, our study lays the groundwork by constructing an undirected weighted network, a subset of the complex network discussed earlier.In a regional economic context, this exercise serves as a crucial step in understanding spatial interdependencies and resource allocations among various industrial nodes.Te weight of the edge between two nodes is determined as the sum of the weights of any existing directed edges between them, which is particularly relevant for capturing the fow of goods, services, or information between industries.Tese weights are calculated using the previously defned direct cost coefcients, which function as similarity weights.
Following this, we apply (4) to compute the clustering coefcient for the unweighted, undirected network and (6) for the weighted clustering coefcient of the undirected weighted network.Table 4 presents the top ten industries 8 Complexity ranked by both their clustering and weighted clustering coefcients.In the realm of regional economics, a higher clustering coefcient signifes strong local synergies and interindustrial cooperation.Tis can often be seen in regional clusters where industries beneft from shared resources, expertise, and markets.Similarly, industries with elevated weighted clustering coefcients are indicative of substantial capital fows, both in terms of investments and consumption, among the industries constituting the vertexassociated triangles.Te weighted measures give a nuanced understanding of the economic robustness and the depth of interindustry relationships.According to (5), the network's average clustering coefcient is ascertained to be 0.63737820, while the average weighted clustering coefcient is 0.00252026.Tese metrics can serve as valuable indicators for policymakers and stakeholders in identifying regional economic strengths and potential areas for fostering industrial collaboration.

Shortest Path and Network Efciency.
Te shortest path in a graph or network refers to the path connecting two nodes that minimizes the sum of lengths or weights along that path.In an unweighted network, the length of the shortest path is typically the number of edges it contains.In a weighted network; however, the length is determined by the sum of the edge weights along the path.When the edge weight is used to represent the distance between nodes, a longer path between two nodes implies a greater distance and thus a more distant or weaker relationship.Consequently, in this context, edge weights should serve as dissimilarity measures.
In this section, the study recalibrates the edge weights in the directed weighted network by taking their reciprocal values, which are then used as dissimilarity measures.Tis is consistent with using the direct cost coefcient as an inverse measure.Specifcally, the greater the direct cost coefcient between two industries, the closer and more tightly-knit their relationship is expected to be.Te concept of the shortest path takes on a critical role.It acts as a proxy for transaction costs between industries, and a shorter average path length could indicate a more efcient, agile, and wellintegrated regional economy.Furthermore, understanding the network efciency and the average shortest path length provides actionable insights for policymakers aiming to optimize resource allocation and improve the economic interconnectivity of industrial clusters.After determining the shortest path and its length between every pair of nodes, the network's overall efciency can be evaluated by computing the average shortest path length among all pairs of nodes using the following equation: where n represents the total number of nodes in the network, while d ij denotes the shortest path length between node i and node j.By taking the reciprocal of the shortest path lengths, one can obtain a measure of efciency between nodes.Te average of these reciprocal values across all pairs of nodes in the network is termed the global network efciency.
In this section, we employ the Dijkstra algorithm to calculate the shortest paths and their corresponding lengths within the directed weighted network under investigation.Te fundamental idea behind the algorithm is to start with a source node and incrementally expand the set of nodes for which the shortest paths are known.Tis expansion occurs by exploring nodes that are adjacent to the current set but have not yet been visited.More specifcally, the algorithm maintains two sets: one consisting of nodes with already-known shortest paths and another set comprising candidate nodes.Te algorithm iteratively selects the next node with the shortest path from the candidate set until the shortest paths to all nodes have been identifed.After calculating the shortest path lengths between all node pairs, the paths corresponding to the minimum and maximum values of these lengths are highlighted in the network, as shown in Figure 3. Te shortest path length from node A1 to A6 is the smallest, at 2.286.Tis suggests that the input from the Agriculture, Forestry, Fishing, and Hunting sector in the Northeast to the Food and Tobacco sector in the same region is exceptionally direct, bypassing intermediary industries, and is also signifcant in volume.Te shortest path length in Figure 3 H8 ⟶ H7 ⟶ H30 ⟶ H5 ⟶ H3 ⟶ C11 ⟶ C35 ⟶ A33 ⟶ A23.In the network diagram, the longest shortest path length is 1525.267,extending from the Textile, Clothing, Footwear, and Leather Goods sector in the Greater Northwest region to the Waste and Scrap sector in the Northeast.Tis path sequentially traverses seven different industrial sectors: textiles (Greater Northwest), transportation, warehousing, and postal services (Greater Northwest), nonmetallic minerals and other mining products (Greater Northwest), crude petroleum and natural gas (Greater Northwest), petroleum, coking products, and nuclear fuel processing goods (Eastern Coast), leasing and business services (Eastern Coast), and fnance (Northeast).Importantly, this path crosses three major economic zones.Te pivotal point is the input from crude petroleum and natural gas (Greater Northwest) to petroleum, coking products, and nuclear fuel processing goods (Eastern Coast), highlighting the transfer of abundant petroleum and natural gas resources from the Greater Northwest to the Eastern Coastal economic zone.Tis path sequentially traverses seven diferent industrial sectors, Complexity each of which could be a manifestation of regional economic specialization.For example, the Greater Northwest's focus on Crude Petroleum and Natural Gas might be a function of its resource endowments.Importantly, this path crosses three major economic zones, illustrating how regional capabilities contribute to forming complex interindustrial relationships.Te pivotal point is the input from crude petroleum and natural gas (Greater Northwest) to petroleum, coking products, and nuclear fuel processing goods (Eastern Coast).Tis could signify a vital input-output linkage in the supply chain, reinforcing the structural interconnectedness of these industries across regions.Such relationships might be indicative of high transaction costs, as evidenced by the involvement of sectors like Transportation, Warehousing, and Postal Services in the longest shortest path.Te calculated average shortest path length for the entire network is 219.7,indicating that, on average, the shortest path lengths between industries are relatively long.Tis relatively long average path length suggests that there may be inefciencies or bottlenecks in the system, either due to regulatory hurdles or inherent complexities in production processes.Te global efciency of the weighted network is 0.00728, signifying that the efciency of resource or product propagation among the industries is low.Tis could refect a high degree of market power or anticompetitive behavior among some sectors, impeding the efcient fow of goods and services.

Network Structure Stability Analysis
Tis section explores the stability of the network structure, a critical aspect related to the network's resilience and antidisturbance capabilities.By simulating attacks on the complex network, we analyze its fault tolerance and resistance to targeted disruptions, thereby shedding light on the overall network stability.

Centrality Measures.
A primary focus is the analysis of network centrality, an essential facet of network stability research.Centrality reveals key nodes and vulnerabilities within the network, contributing to our understanding of the structural characteristics and relative importance of individual nodes.Tis section evaluates four types of centrality in the directed, weighted network: degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality.Tis is echoed in the work of Ufmtsev et al. [27], who emphasize the impact of centrality measures on understanding the stability of networks, especially under conditions of noise and disturbance.Tis section evaluates four types of centrality in the directed, weighted network: degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality.Te relevance of these measures in diferent network contexts, as discussed by Grando et al. [37], underscores their utility in identifying infuential nodes and assessing network resilience.Furthermore, the study by Rajeh et al. [38] highlights the importance of considering community structures in centrality analysis, which can be particularly pertinent in complex industrial networks.Te correlation analysis of centrality measures by Ficara et al. [39] provides a comprehensive understanding of how these diferent centrality types interact and infuence each other, enriching the analysis of network stability in this research.
5.1.1.Degree Centrality.Tis measure refects the number of connections a node has, thereby identifying the most popular and active nodes in the network.Te failure of these nodes could severely impact network stability.

Closeness Centrality. Calculated as the average
shortest path length from one node to all other nodes, this centrality measure gauges the node's importance.A higher closeness centrality implies better accessibility and faster information dissemination, indicating that the node may play a critical role in the network.

Betweenness Centrality.
Tis measure is based on the frequency with which a node appears in all shortest paths across the network.It uncovers nodes that act as "bridges" in the network.Tese nodes appear in the shortest paths between many pairs of nodes; thus, their failure could substantially alter the network structure.

Eigenvector Centrality.
Tis form of centrality is a function of the importance of the neighbors to which a node is connected.It refects a node's social infuence within the network.Nodes connected to infuential nodes often have high eigenvector centrality, helping to identify nodes that may be crucial within the network.
Tis multi-dimensional approach to centrality provides a nuanced understanding of the elements that contribute to or jeopardize network stability.By identifying these critical nodes and potential vulnerabilities, we gain insights that can  Complexity inform strategies for enhancing network resilience and efciency.
In Table 5, the top ten industrial nodes are ranked according to the four diferent types of centrality measures.Industries leading in degree centrality are mainly those in the sectors of transportation, warehousing, and postal services, specifcally located in the middle reaches of the Yellow River, the Eastern Coast, and the Northeast of China.Tis highlights the pivotal role these sectors in China's centraleastern and northeastern regions play in the overall national economy.
For closeness centrality, the values are generally low, and the industries that rank the highest are predominantly in the Greater Northwest, specifcally in sectors such as transportation equipment, metal product machinery, electrical machinery, and timber processing and furniture.Despite their peripheral geographical locations, these sectors may have a strategic position that allows for efcient information dissemination.
High betweenness centrality is mainly observed in industries located in the Eastern Coastal economic zone, specifcally in sectors such as petroleum coking products and nuclear fuel processing, leasing and business services.In addition, the petroleum and natural gas extraction industries in the Greater Northwest and Northeast, along with the production and supply of electric and thermal Power in the Northeast and Eastern Coast, play key "bridge" roles in the network.Industries involved in coal mining in the middle reaches of the Yellow River and chemical products on the Eastern coast also exhibit high betweenness centrality.
Lastly, the top ten industries in terms of eigenvector centrality are all concentrated in the Northern Coastal economic zone, predominantly in light industry, manufacturing, and chemical products sectors.Particularly noteworthy are the textile and apparel sectors in this region, whose eigenvector centrality is far higher than all other industries, indicating the critical importance of their neighboring industries.
Analysis of centrality measures can unveil the most infuential nodes, the best nodes for dissemination, and the key "bridge" nodes within the network.Protecting and closely monitoring these nodes can signifcantly enhance the network's stability and resilience to disturbances.Tis analytical approach is particularly important for policymakers and stakeholders who aim to safeguard critical infrastructure and optimize resource allocation.In the realm of regional economics and industrial organization theory, identifying these central nodes can also guide regional development policies, investment decisions, and crisis management strategies.Terefore, understanding centrality measures is not just a theoretical exercise but also an essential practice for ensuring efective economic management and sustainable development.

Network Connectivity Metrics.
Te resilience of a network to attacks refers to its ability to maintain structural and functional integrity in the face of adversarial actions, such as the failure or removal of nodes or edges.In this study, changes in network connectivity serve as a measure for assessing the network's robustness against attacks.Specifcally, two key metrics are employed: network efciency and the size of the largest connected subgraph.
Network efciency, as elaborated in Section 3.3, is an important gauge of global connectivity within the network.Te mathematical expression for network efciency is provided in the following equation: where is d ij the shortest path length between node i and node j.When there is no edge between node d ij � +∞ i and node j, the local efciency between the pair of nodes is 0. After the previous calculation, the global efciency of the initial directed weighted network before the attack started was 0.00728.Te largest connected subgraph refers to the connected portion of the network containing the maximum number of nodes.In directed networks, one can further distinguish between weakly and strongly connected subgraphs.In a weakly connected subgraph, any pair of nodes is mutually reachable through a sequence of directed edges (ignoring edge direction), whereas in a strongly connected subgraph, directionality must be considered.Te size and characteristics of the largest connected subgraph serve as important indicators of network connectivity and stability.In this study, the strongly connected subgraphs of the directed network are computed using Kosaraju's algorithm.Te ratio of the number of nodes in the largest connected subgraph is defned in (9): where m ′ is the number of nodes in the largest connected subgraph postattack and m is the number of nodes in the largest connected subgraph of the initial network.Te ratio Z is used to refect changes in network connectivity subsequent to an attack.After computational analysis, it was established that the initial directed weighted network constitutes a strongly connected subgraph.Terefore, in this specifc case, m � n � 336.

Random vs. Targeted Attacks on Network Resilience.
Network attacks can broadly be classifed into two categories: random attacks and targeted (or deliberate) attacks.Random attacks refer to nonspecifc assaults on nodes or edges in the network.In such scenarios, the elements attacked are randomly chosen without taking into account their unique roles or signifcance within the network structure.Contrastingly, targeted attacks are orchestrated to impact key nodes or edges within the network selectively.Tese attacks can be executed based on various criteria.In the present study, we employ four types of centrality metrics, previously discussed, as the criteria for simulating targeted attacks.

Complexity
Tis section outlines fve distinct attack strategies: random attacks, attacks targeting nodes with the highest degree centrality, the highest closeness centrality, the highest betweenness centrality, and the highest eigenvector centrality.For instance, under the highest degree centrality attack, the node with the maximum degree centrality is removed along with its associated edges.Subsequently, the connectivity indices and degree centrality are recalculated for the newly formed network.Tis process is iteratively repeated until the network's overall connectivity is reduced to zero.Te remaining three targeted attacks follow a similar methodology.
Figure 4 illustrates the changes in network efciency and the ratio of the number of nodes in the largest connected subgraph under the fve attack strategies.From the efciency change curve, it is evident that attacks based on the highest closeness centrality are markedly less efective than random attacks.Intriguingly, after 300 instances of these two attack types, there is a noticeable uptick in global efciency.Specifcally, at 326 instances of highest closeness centrality attacks, the network efciency peaks at 0.0081, surpassing the initial network's efciency, followed by a linear decrease as attacks continue.
Furthermore, the efectiveness of the attack strategies, as observed from the graph, is sequentially best to worst as follows: highest betweenness centrality, highest degree centrality, and highest eigenvector centrality; all three being more efective than random attacks.Te ratio of the number of nodes in the largest connected subgraph reveals a consistent trend across all fve strategies when the number of attacks is below 200.Beyond this point, the performance of targeted attacks based on the highest closeness centrality and random attacks remains similar, but the other targeted attacks exhibit superior performance.Among them, attacks based on the highest betweenness centrality are the most efective, followed by those based on the highest degree centrality and the highest eigenvector centrality.
Te analysis quantifed the resilience of the network by computing the number of attacks needed to completely disrupt its connectivity.Specifcally, for each of the fve strategies-random attacks, highest degree centrality attacks, highest closeness centrality attacks, highest betweenness centrality attacks, and highest eigenvector centrality attacks-the required number of attacks was 332, 305, 331, 312, and 324, respectively.Tis data suggests that the network in question possesses a commendable level of resilience and structural stability.Among the examined attack strategies, the highest betweenness centrality, and the highest degree centrality attacks warrant special attention.Industries characterized by high values of these centrality metrics emerge as critical nodes in the network and substantially infuence its overall stability.Tus, they are pivotal in safeguarding the economic system.Targeted policy interventions and risk management strategies aimed at these high-centrality industries can further fortify the stability of the intricate input-output network under study.

Further Interpretations and Suggestions.
In China, the central, eastern, and northeastern regions serve as pivotal hubs for the transportation, warehousing, and postal sectors, which wield considerable infuence over the national economy and logistics network.Te elevated centrality of these industries is accentuated by their extensive interconnections with other industrial sectors.Concurrently, peripheral industries such as transportation equipment, metal product machinery, electrical machinery, wood processing, and furniture, primarily situated in the remote Great Northwest, manifest notable metrics of network cohesiveness.Tis suggests their capacity to act as early indicators for network perturbations, efciently disseminating information and adaptations across the network.
It is noteworthy that certain sectors, including petroleum coking products and nuclear fuel processing, exhibit pronounced intermediary centrality, thereby serving as critical nodes or "bridges" in the economic network.Tese industries facilitate essential conduits for the transit of resources and information.Furthermore, industries such as light manufacturing and chemical production, concentrated in the northern coastal economic zone, not only maintain high eigenvector centrality but also indicate analogous levels of centrality in their proximate regions, underlining the network resilience within this geographical area.
In light of these fndings, policy implications emerge.Targeted infrastructure investments should be allocated preferentially to regions characterized by both high industrial density and intermediary centrality, thereby catalyzing more expansive economic activities.Concurrently, risk mitigation strategies should be proactively formulated 12 Complexity for highly network-centric industries, even those located in economically peripheral areas, to preempt potential cascading efects stemming from local network disruptions.Given the indispensable "bridging" role of industries with high intermediary centrality, supply chain risk management warrants particular focus to assure the uninterrupted and stable fow of goods and services.To bolster the vigor of regions with elevated eigenvector centrality, the cultivation of innovation clusters is advised to generate industrial synergies.In addition, strategic tax incentives and subsidies can be deployed to sustain competitiveness and stimulate growth in pivotal industries.Lastly, a real-time monitoring mechanism is recommended for tracking critical performance metrics in these key sectors, enabling timely interventions should severe fuctuations in centrality indicators arise.Tis analysis, rooted in empirical data, ofers actionable insights for policy-making, aiming to enhance network resilience through data-driven strategies.(2) Degree analysis reveals that 85% of industrial sectors have degrees ranging between 70 and 220, indicating tight topological connections between industries.Notably, sectors such as transportation, warehousing, and postal services, along with wholesale and retail, play pivotal roles in the economic network.Te distribution of in-degrees and out-degrees refects distinct regional industrial structures and interdependencies.For instance, construction, public utilities, and service sectors are highly consumed in the Yellow River Basin and the Northwest regions.

Conclusion
(3) In the weighted network, point, in-strength, and outstrength metrics further highlight interindustry connections and dependencies.Coastal Eastern regions display strong export-oriented characteristics in chemical product industries, while the Northern Complexity coastal regions excel in textiles.In contrast, the Northeast and Northwest regions exhibit importdependent traits in various manufacturing sectors.Tese patterns are indicative of regional industrial specialization and varying levels of economic development.
Te observed topological structure and industry roles can be situated within broader industrial and regional theories.For example, the concept of agglomeration economies may explain why certain industries like transportation and warehousing are centrally positioned.Tey potentially serve as clusters that generate additional economic advantages for nearby sectors.
Simulated network attacks reveal that the most efective targeted strategies are highest-betweenness centrality attacks, followed by highest-degree centrality and highesteigenvector centrality attacks.Tese deliberate attack methods outperform random attacks.While the industrial network demonstrates high resilience, targeted policy interventions are warranted for industries with high degree and betweenness centrality.Tese sectors emerge as critical nodes, infuencing the stability of the network structure.
In summary, the network exhibits strong resilience but requires nuanced policy and risk management strategies aimed at industries with high centrality metrics, as these sectors are pivotal in maintaining the network's structural stability.
6.2.Limitations and Future Directions.Our investigation into the intricate relationships among industrial sectors within China's economic zones contributes to the understanding of complex networks in an economic context.Despite the insights gained, our study acknowledges several limitations that pave the way for future research directions.Firstly, the reliance on static data for network modeling in existing literature does not adequately refect the dynamic nature of economic activities, resulting in diminished predictive capabilities.Tis highlights the necessity of incorporating dynamic data and models that can more accurately mirror the fuctuations and trends within economic networks.
Second, the simulation of network disruptions often fails to account for real-world economic shocks, such as fnancial crises or sudden market changes, which limits the practical relevance of these models.Future studies should aim to integrate real economic shock scenarios to enhance the applicability and resilience of network models.
Tird, there is a notable gap in cross-disciplinary research regarding the examination of network analysis methods tailored to specifc economic situations.Tis oversight may overlook the complex realities of economic environments, suggesting a need for more nuanced studies that evaluate the efectiveness of network methodologies within economic frameworks.
Fourth, strategies developed to enhance network robustness often do not take into account the unique characteristics and interdependencies of economic networks, particularly those defned by industry-specifc interactions.Tis oversight underscores the importance of devising robustness strategies that are not only theoretically sound but also practically applicable, taking into consideration the distinct nature of economic networks.
To address these limitations, future research should focus on developing dynamic modeling approaches that accurately refect economic network operations and their responses to various shocks.Additionally, there is a critical need for studies that assess the suitability of network analysis methods for economic contexts, ensuring that these tools can capture the complexity of economic scenarios.Tailoring strategies for robustness to address the unique characteristics of economic networks will enhance our comprehension and management of these systems efciently.By tackling these areas, subsequent research can signifcantly advance the application of complex network theory in economic studies, leading to more robust and applicable insights for policy-making and strategic planning [40][41][42][43][44][45][46][47].

Figure 1 :
Figure 1: Te complex network of 42 industries across the eight major economic regions.

Figure 3 :
Figure 3: Schematic diagram of the shortest and longest paths in the shortest path.

6. 1 .
Further Interpretations and Suggestions.Tis study employs the Maximum Entropy Method to construct a directed, weighted complex network comprising 42 industrial sectors across China's eight major economic regions.Te research ofers an in-depth statistical analysis of the network's attributes and stability.Key fndings include (1) Te maximum entropy method efectively uncovers signifcant intersectoral relationships, maximizing the extraction of information from input-output tables.

Figure 4 :
Figure 4: Changes in network efciency and maximum connected subgraph node ratio under diferent attack strategies.

Table 1 :
Division and codes of the eight major economic regions.

Table 3 :
Te top fve and bottom fve industrial nodes in terms of power, entry, and exit.

Table 4 :
Te top ten industrial nodes in clustering coefcient and weighted clustering coefcient.

Table 5 :
Centrality of complex networks.