Knowledge stickiness and technological concentration in the AI industry: an empirical study of Chinese cities.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 11;16:16905. doi: 10.1038/s41598-026-48244-5 Knowledge stickiness and technological concentration in the AI industry: an empirical study of Chinese cities Lin Zhang 1 , Quanjun Zhang 1 , Houyuan Meng 2, ✉ Author information Article notes Copyright and License information 1 China-ASEAN School of Economics, Guangxi University, Nanning, Guangxi China 2 College of Ethnology and Sociology, Guangxi Minzu University, Nanning, Guangxi China ✉ Corresponding author. Received 2025 Dec 25; Accepted 2026 Apr 7; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13230755  PMID: 41965423 Abstract The rapid diffusion of AI in China has proceeded to an uneven pace across cities. In this study we develop a motivation-resistance framework to help study AI knowledge stickiness: motivation in capturing within-city diffusion potential, while resistance captures frictions preventing knowledge transfer across cities and inducing local lock-in. Combining AI patent applications with urban statistics in a city-year panel for the years 2014–2023, and using a two-way fixed-effects model, we find an inverted U-shaped association between AI knowledge stickiness and technological concentration, where higher stickiness up to a limit leads to more concentration and thereafter the opposite. Technological complexity moderates this nonlinear association by altering its strength and curvature, rather than indicating a simple and uniform shift in the turning point. In heterogeneity analyses, the nonlinear pattern is more clearly detected in eastern cities and in small and medium-sized cities, while the evidence for large cities is weaker because the quadratic term is not statistically significant. Collectively, these results show that local embedding conditions shape the internal allocation of AI activity along mapped sub-technology branches, with implications for place-based AI innovation policy. Supplementary Information The online version contains supplementary material available at 10.1038/s41598-026-48244-5. Keywords: Artificial intelligence, Knowledge stickiness, Technological concentration, Technological complexity Subject terms: Geography; Geography; Mathematics and computing; Science, technology and society Introduction Artificial intelligence (AI) is an important and rising general-purpose technology which is remaking, in varying ways, innovation and urbanization 1 . AI innovation is concentrated and networked: large cities attract greater R&D capacity and industrial foundations and have data endowments that enhance AI innovation, while many other cities are relatively reliant on collaboration and inflows of knowledge. In China, this has produced stark core-periphery patterns, with leading cities upgrading rapidly while many small and medium cities are constrained by a lack of strong knowledge networks 2 , 3 . This uneven diffusion points to path dependence and frictions of knowledge. Mapping research additionally show that AI is a continually evolving knowledge base, with core inventions embedded in adjacent enablers and application-facing technologies with different recombination opportunities 4 . Accordingly, we analyze AI patents by sub-technology, and conceptualize knowledge stickiness as a portfolio-level balance between potential for diffusion, and resistance to lock-in in the landscape of AI knowledge. The accumulation and diffusion of locally embodied knowledge in the digital economy interacts with knowledge stickiness, with consequences for urban competitiveness and transformation 1 , 5 . Knowledge stickiness is the difficulty of transferring, absorbing, and reproducing knowledge, arising from the attributes of knowledge, actors’ incentives, and the contextual circumstances where knowledge is embedded 6 , 7 . Stickiness can be advantageous in moderation, allowing for cumulative innovation, but in excess it brings lock-in risks and constraints diffusion and collaboration 8 , 9 . Although previous studies have analyzed the spatial agglomeration characteristics of the artificial intelligence industry from quite a few perspectives 2 , 10 , 11 , two shortcomings in the present literature. First, mainstream studies have mainly focused on the national, firm, or inter industry level which insufficient attention has been paid to the role of cities as the core spatial units of innovation activities 1 – 3 , 11 . This makes it hard to reveal the intrinsic heterogeneity of cities in terms of knowledge flow and technological concentration. Second, the coupling relationship among knowledge stickiness, technological complexity and spatial concentration is yet to be verified empirically. We still lack quantitative research exploring the moderating role of knowledge relatedness, reducing the risk of technological lock-in and the ability to achieve collaborative innovation at the urban scale. The inherent complexity and interconnectedness of AI technologies make traditional linear diffusion models inadequate for explaining the high concentration, interrelatedness, and path dependence observed among cities 12 , 13 . Recent AI knowledge-based mapping studies reveal that mapped AI knowledge structures have economic consequences. Taking a cue from this methodology to the city level, we look at how knowledge stickiness associates with the concentration of AI patenting across mapped sub-technology branches, and how local AI complexity then moderates the nonlinear relationships 4 , 14 . Finding the interacts mechanism between knowledge stickiness and technological concentration at the urban scale has become a key scientific issue in understanding the spatial evolution of AI technologies. This study contributes to the literature in three ways. First, it advances the emerging AI knowledge-based mapping perspective by conceptualizing knowledge stickiness as a two-dimensional balance between diffusion potential and resistance to lock-in, thereby treating AI as a structured and evolving knowledge landscape. Second, it provides fine-grained city-level evidence on AI technological concentration in China, bridging a key gap between firm-level and national-level analyses. Third, it documents an inverted U-shaped association between knowledge stickiness and city-level technological concentration and shows how this relationship varies with technological complexity and urban/regional context, clarifying when localization supports clustering and when it aligns with lock-in, and complementing evidence that AI knowledge availability is economically consequential. Theoretical review and hypothesis analysis Knowledge stickiness and technological concentration AI knowledge-based mapping and AI measurement studies show how patent-based AI taxonomies/relatedness measures can be used to map and quantify AI technological landscapes 14 , 15 . Building on this framing, we treat cities as selective environments where AI knowledge locally sticks, and relate knowledge stickiness to how concentrated city-level AI patenting is across our mapped sub-technology branches, testing a non-linear relationship moderated by local AI complexity. Knowledge stickiness refers to frictions and barriers to transfer and diffusion of knowledge 16 . AI innovation is highly interdisciplinary, drawing on computer science, mathematics, statistics, and linguistics, increasing recombination needs and raising coordination costs and thereby strengthening stickiness 17 , 18 . A key source of stickiness is tacit knowledge, e.g. algorithm tuning and data validation know-how, that cannot be so readily codified and hence tends to stay locally embedded 6 , 19 . Studies suggest that such stickiness from tacit knowledge matters not only within organizations but also across urban space, shaping local capability accumulation and limiting wider diffusion 20 , 21 . At the firm level, evidence indicates that interdisciplinary integration difficulties raise AI adoption costs, while tacit stickiness can slow diffusion and reinforce uneven innovation outcomes for AI across places 22 , 23 . Technological concentration is a motif widely discussed in the literature on agglomeration and the spatial evolution of innovation, reflecting how innovation output and trajectories become unevenly distributed across places through knowledge networks and similarly 24 . Prior work links concentration patterns to industrial structure, economic scale, knowledge embeddedness and inter-organizational access to knowledge 10 . In this case, knowledge stickiness can support stake, e.g. through cumulative local learning and retention of specialized capabilities, while also creating lock in (when search robustness in the local is excessive). Stronger knowledge relatedness between cities can increase technologically related linkages and potentially reinforce cumulative concentration dynamics 25 . For AI as a general-purpose technology, these dynamics are likely most significant: leading cities with strong R&D capacity, talent pools, and dense local networks tend to concentrate AI activity within them, while small cities experience significant and persistent gaps in AI patents, publications and products 26 . Cities are the sites where AI knowledge is created, recombined, and commercialized, via dense innovation networks and localized complementarities in the form of specialized labor markets, data resources and infrastructure. Urban conditions suggest a nonlinear relationship between knowledge stickiness and technological concentration: where stickiness is low knowledge circulates widely but local actors are unable to retain and build specialized capabilities, and the cumulative learning and sustained specialization that come with it, is weakened. Where stickiness is moderate, repeated local interaction and recombination can enable capability building and “embedded openness” such that cities can accumulate trajectories while continuing to benefit from selective knowledge externalities, strengthening technological concentration 27 , 28 . Where stickiness is too high however, local search weakens and the ability to learn from elsewhere is constrained, and associated with rising risks of lock-in and path dependence that diminish the scope for renewal and even the gains from inter-urban collaboration 8 , 9 , 29 . Accumulation-lock-in trade-offs are acute for complex general-purpose technologies like AI in which complementary assets and cumulative capabilities tend to agglomerate in leading cities 26 , 30 . Because of these issues, the following hypothesis is being proposed: H1: At the urban level, AI knowledge stickiness is associated with technological concentration in an inverted U-shaped manner. Moderating effect Technological complexity is theoretically one of the factors shaping urban knowledge networks and competitive advantage. Complex technologies incorporate many substantive elements, increasing uncertainty and coordination requirements and reinforcing path dependency 31 . The higher the level of complexity, the harder the knowledge is to transfer across organizational and spatial boundaries 27 , and complex outputs are less replicable, as widely distributed learning is more time intensive 32 . The combined effect of these arguments is to imply that complexity makes knowledge stickiness more pertinent for explaining technological concentration. This moderation is important given that, at the urban level, complex innovation often depends on city-based complementarities, including specialized labor markets, dense inter-firm and cross-field collaboration, supporting infrastructure and data resources, through which knowledge is recombined, tested, and tried. The interaction of stickiness and complexity is heightened because AI consists of many sub-field with differing component structures and data requirements, and differing trajectories 33 . Some application domains (e.g. image recognition, voice recognition) have relatively standardized methodological routines, while others are more demanding of large scale data and cross-industry, collaborative approaches. This increases the frictions of coordination, and is likely to generate concentrated “pockets” of invention in particular cities 11 . Technological complexity can therefore enhance technological concentration in several ways simultaneously; it heightens the necessity of embedded local learning, while supporting dense agglomerations of expertise. Empirical work finds a positive correlation between the complexity of a knowledge base and technological concentration 34 , and shows that the diversity and specialization of a knowledge base shapes its complexity, and are linked to the yield of competitive advantage across locations, or the sustainability and geographical concentration of advantages 21 . Where complexity is relatively low, knowledge can be relatively easily codified and transferred, so concentration may emerge even if stickiness is comparatively low; further stickiness yields diminishing returns, and too much stickiness can entrench lock-in, cutting vitality in the knowledge pool substantially 23 . Where complexity is high, transfer and imitation become more difficult in practice, and sustained advantage more dependent upon locally embedded accumulation and coordination. Related studies of high-complexity clusters, and those of very large technology firms cite maintenance of lids of stickiness for support of long-run innovation 35 , 36 . Other work demonstrates that knowledge complexity helps account for productivity outputs and explains the sustainability of advantage across locations 19 . On this basis, this study proposes Hypothesis 2. H2: Technological complexity moderates the inverted U-shaped relationship between AI knowledge stickiness and technological concentration, strengthening the positive association at low-to-moderate levels of stickiness while intensifying the negative association beyond the turning point. Heterogeneity of the effect Large cities and eastern regions possess higher research capabilities and industrial diversity, granting them stronger knowledge absorption and integration capacities 37 . Cohen and Levinthal’s 38 theory of absorptive capacity posits that organizations or regions with stronger absorptive capacities are better equipped to maintain innovation advantages in environments characterized by high knowledge stickiness. Zheng et al. 33 further elaborated on the “local buzz—global pipelines” theory, which indicates that large cities can mitigate the adverse effects of excessive lock-in through cross-regional and cross-national networks. Recent studies have shown that large cities, with their diversified networks and international connections, can maintain high technological concentration even in high-knowledge-stickiness environments 26 , 39 . In addition, small and medium-sized cities, as well as central and western regions, are relatively weaker in terms of innovation resources and network structures. Castellacci 40 pointed out that these regions are more likely to fall into path dependence when knowledge is excessively locked in, lacking external knowledge supplementation, and innovation vitality quickly declines. Recent research has similarly shown that these regions may enter the declining phase of the inverted U-shaped curve with only slightly higher levels of knowledge stickiness, leading to a rapid decline in technological concentration 41 , 42 . Furthermore, the differences between types of cities are not only reflected in the threshold level of stickiness but also in the shape of the curve. This suggests that urban heterogeneity is a crucial boundary condition that is related to the mechanism of knowledge stickiness. Therefore, it can be inferred that large cities and eastern regions can withstand higher levels of knowledge stickiness without losing technological concentration. In contrast, small and medium-sized cities and central and western regions more quickly enter the declining segment and lose their innovation advantages. This spatial disparity reveals the imbalance in regional innovation development and provides a crucial basis for formulating differentiated policies. H3: Compared with cities with weaker developmental foundations, cities with stronger developmental foundations and more advantageous spatial positions (e.g., large and eastern cities) exhibit a more pronounced inverted U-shaped relationship, with a stronger positive association at low-to-moderate stickiness and a weaker tendency toward dispersion before the turning point. Methodology Model specification To test the inverted U-shaped relationship between knowledge stickiness and concentration, this study specifies the following baseline model: Here, if >0, <0, it indicates an inverted U-shaped relationship between the two variables. This model incorporates the quadratic term of knowledge stickiness to capture the nonlinear characteristics of its impact on agglomeration degree. Similar quadratic term regression methods have been widely applied in studies of technology agglomeration and knowledge spillovers 3 , 43 , 44 . At the theoretical level, knowledge stickiness has a dual effect on regional technology concentration. On the one hand, moderate knowledge stickiness can reduce the cost of intra-regional knowledge transfer, promote collaborative innovation among technology entities, and accelerate the agglomeration and diffusion of core technologies 45 . On the other hand, excessively high knowledge stickiness will trigger an innovation lock-in effect, restrict the inflow of external heterogeneous knowledge, hinder the iterative upgrading of the technology system, and thus inhibit the continuous improvement of technology concentration 46 . The trade-off between these dual effects constitutes the core logic of the inverted U-shaped relationship between the two. To further examine the role of technological complexity as a boundary condition, this study constructs the following moderating effect model: If the interaction terms are significant, it indicates that technological complexity can alter the peak or slope of the inverted U-shaped curve, reflecting its moderating effect. If β 3 and β 4 are significantly positive, it suggests that in high-complexity domains, moderate stickiness has a more substantial promoting impact on concentration. This reflects that complex knowledge is more dependent on the capabilities and network positions of specific cities 1 , 47 , 48 . The empirical results are interpreted as conditional associations rather than causal effects. Variables and measurements The dependent variable is city-level technological concentration in AI, measured using the Herfindahl–Hirschman Index (HHI) following standard concentration metrics in industrial organization research 49 . For each city-year, we calculate the share of AI patents in each AI sub-technology branch and compute HHI as the sum of squared shares, and the formula is shown as (1). This approach treats AI as a portfolio of sub-technology branches and captures how a city’s AI activity is distributed across major segments of the mapped AI knowledge base. The branches follow the 2023 Classification System for Patents of Key Digital Technologies and cover both core algorithmic domains and enabling/interface technologies. 1 In the formula, denotes the degree of artificial intelligence technology concentration of city i in year t . represents the number of patents in the k-th AI sub-technical branch of the city in year t , and stands for the total number of all AI-related patents of the city in year t . K refers to the total number of tertiary technical branches in the field of artificial intelligence. A larger HHI indicates that a city’s AI patenting is concentrated in a smaller number of AI sub-technology branches, whereas a smaller HHI implies a more diversified distribution across branches, reflecting a broader local AI knowledge base. Conversely, a smaller HHI value implies that the city’s innovation activities are distributed in a more dispersed manner, and the AI technological ecosystem is more diverse and open. This study adopts knowledge stickiness as its independent variable. At the city level, knowledge stickiness represents the tensions between the forces of forces that promote diffusion across organizations and domains and the forces that entrench lock-in through concentration, path dependence, and knowledge closure 50 . In the AI sector, this balance is pertinent because innovation draws on a structured and evolving knowledge base comprising differentiated sub-domains and adjacent enabling technologies, rather than a single homogeneous field 4 . Thus, knowledge stickiness is not considered as a single proxy for diffusion or lock-in, but as a system-level attribute constituting how urban AI knowledge can circulate while remaining sufficiently at least suspended to sustain accumulation 51 . To operationalize knowledge stickiness, we develop a two-dimensional indicator system that captures diffusion potential and lock-in resistance (Table 1 ). The diffusion dimension captures how easily a city’s AI knowledge circulates and recombines across organizations and technological domains, including recombination across AI sub-technology branches, using three sub-indicators: average citation frequency, entropy of IPC classifications, and the share of collaborative patents, corresponding to the intensity of knowledge spillovers, technological variety, and inter-organizational knowledge mobility 50 , 52 . The resistance dimension captures structural conditions that may favor lock-in to a narrow set of trajectories, including the share of sole applicants, the concentration of hot technologies, and within-city patent concentration, reflecting knowledge closure, path dependence, and structural concentration 53 , 54 . Table 1. Knowledge stickiness indicator system. Primary indicator Secondary indicator Data source Indicator definition Knowledge diffusion Average citation count Patent citation data Average citation count of city patents, reflecting the intensity of external knowledge diffusion IPC classification entropy Patent IPC classification code Entropy value of the city’s IPC distribution, reflecting technical field diversity and cross-domain diffusion potential Share of collaborative patents Patent applicant information Proportion of patents with multiple applicants, reflecting inter-organizational knowledge mobility Knowledge lock-in Share of single applicants Patent applicant information Proportion of single-applicant entities, reflecting knowledge closure Concentration of popular technologies Patent IPC classification code Proportion of the city’s most popular IPC categories, reflecting the intensity of technological path dependence Internal patent concentration within the city Applicant distribution HHI index at the applicant level, measuring the structural concentration of innovation entities Open in a new tab In terms of indicator processing, first, min–max normalization is applied to all indicators to eliminate deviations caused by different dimensions. Subsequently, in accordance with the principle of theoretical symmetry and considerations of robustness, the equal-weighted average method is adopted to synthesize the indicators within each of the two dimensions. Antonelli (2008) 55 pointed out that technological diffusion and knowledge lock-in are bidirectional dynamic mechanisms of the regional innovation system, and weights should be adjusted over time to reflect changes in institutional and market environments. Based on this, to dynamically reflect the relative changes in the forces of diffusion and lock-in in different periods, this paper introduces the annual weighting coefficient , and correlated with the weights through the annual standard deviation ratio method. The formula is shown as (2) and (3), and the knowledge stickiness index is finally obtained. We conducted additional validation checks for the knowledge stickiness index. The results from correlation analyses suggest that the index is systematically related to patent-based proxies for innovation and local innovation conditions, and sensitivity tests using alternative transformations (standardization and winsorization) yield similar signs and inferences in the baseline regression. 2 3 Herein, a higher knowledge stickiness index signifies that the regional knowledge system has achieved a benign balance between open diffusion and stable accumulation, with strong knowledge liquidity and robust innovation mechanisms. In contrast, low index values imply that system occupancy is in danger of excessive looseness or extreme aggregation of knowledge, which will result in a restricted flow of knowledge between people and a complete lack of innovative vitality. Thus, this research has created an integrated knowledge stickiness approach, which is theoretical and includes both the dualistic nature of knowledge diffusion and knowledge lock-in as well as indicators with symmetrical and methodologically comparable, time-related elements. The validation and sensitivity checks are reported in the Supplementary Information. Technological complexity is the core variable in this study. The analysis utilizes patent data of Chinese prefecture-level cities from 2014 to 2023. Based on the International Patent Classification (IPC) codes of patent applications by city and applicant, a city-technology matrix is constructed to map the knowledge distribution of cities across different technological domains. To mitigate bias arising from the heterogeneity in total patent counts across cities, this study employs Revealed Technological Advantage (RTA) to measure each city’s relative technological specialization in a given patent-based technological domain. Although the indicator is mathematically identical to the Balassa-type revealed advantage index widely used in economic complexity research 32 , we use the term RTA rather than RCA because the context here is patents and technological fields, not trade flows 56 , 57 . Formally, the RTA for city c in technological field i at time t is defined as formula ( 4 ). 4 where denotes the number of patents of city c in technological field i at time t; represents the total number of patents of city c; refers to the number of national patents in field i; and represents the total number of national patents. When ≥1, the proportion of patents in a specific field within city ccc exceeds the national average, indicating that the city has a comparative advantage in that technological domain. To construct the city–technology incidence matrix, a binary indicator variable is further defined based on the RTA value, shown as formula ( 5 ) 5 This threshold-based discretization removes scale effects from patent counts and allows the complexity indicator to capture the structural diversity and specialization of a city’s knowledge base 1 . Higher values indicate a broader and more diversified technological structure, whereas lower values suggest stronger specialization and path dependence. To estimate the association between AI knowledge stickiness and technological concentration, we control for key confounders related to innovation inputs, openness, institutional conditions, and economic structure. Innovation investment is proxied by science and technology expenditure and science expenditure (STE, SE), which are particularly relevant for capital- and R&D-intensive AI development 58 , 59 . Openness and access to external knowledge are captured by foreign-invested enterprises (FIE) and fixed asset investment (FNA), the latter also reflecting infrastructure capacity for data- and computing-intensive activities 58 , 60 . Institutional stability and capability endowments are measured by social security expenditure (SCR) and human capital (HCL) 61 , 62 . Finally, market size and structural conditions are controlled using registered population (LIE), industrial revenue (RHP), and primary-industry value added (PIV), which capture industrialization, demand conditions, and the scope for AI application and diffusion 63 – 65 . Data and descriptive statistics The main source of data for our study is the Patent Database of the State Intellectual Property Office (SIPO) in China and city-year statistical yearbooks. The study identify AI patents using the 2023 Classification System for Patents of Key Digital Technologies, which includes categories like machine learning, natural language processing, computer vision, human–computer interaction, and intelligent chips. These branches are chosen to represent major segments of the AI knowledge base, spanning core algorithmic areas and enabling/interface technologies. Our final panel consists of 236 prefecture-level cities for the period 2014–2023, totaling 1,073 city-year observations. The panel is unbalanced because we retain only city-year observations for which there is at least one AI patent; city-years with zero AI patenting are excluded by construction, so not all cities are observed in every year. Descriptive statistics for all variables are reported in Table 2 . Table 2. Descriptive statistics. Variable Obs Mean Std. Dev Min Max HHI 1073 0.369 0.362 0.143 1 Stickiness 1073 0.276 0.044 0.196 0.519 Complexity 1073 − 0.037 0.964 − 4.487 1.673 STE 1073 0.029 0.024 0.001 0.178 FIE 1073 0.02 0.041 0 0.322 SE 1073 35.605 72.45 0.182 554.982 FLL 1073 1.037 1.456 0.036 10.86 FNA 1073 0.416 0.698 0.001 6.82 SCR 1073 0.24 0.256 0.005 1.852 HCL 1073 0.036 0.034 0 0.147 RHP 1073 0.585 0.402 0 3.416 LIE 1073 0.241 0.24 0.002 1.498 PIV 1073 0.028 0.021 0 0.207 Open in a new tab Result Basic regression analysis Table 3 shows the more complete baseline regressions, in which technological concentration (HHI) is the dependent variable and knowledge stickiness enters (both in linear and quadratic form) to explore for nonlinearity. For robustness we estimate four specifications: no controls or fixed effects (Model (1)), plus the full set of controls (Model (2)), plus fixed effects but no controls (Model (3)), and Model (4) plus controls and fixed effects. In all four specifications the linear stickiness term is positively significant and the quadratic term negative, suggesting an inverted U-shaped association between knowledge stickiness and technological concentration. Using the estimates in Model (2), the implied turning point is approximately 0.340. Below this threshold, higher knowledge stickiness is associated with higher technological concentration; beyond it, the marginal association becomes negative. It is consistent with an interpretation in which intermediate levels of localization frictions support cumulative learning and specialization, but excessive stickiness is associated with increased lock-in risks and poorer renewal dynamics. Table 3. Basic regression analysis. (1) (2) (3) (4) HHI HHI HHI HHI Stickiness 7.455*** 8.908*** 6.723*** 6.638*** (1.649) (1.503) (1.620) (1.641) Stickiness 2 − 10.508*** − 13.089*** − 9.094*** − 9.069*** (2.632) (2.387) (2.575) (2.600) STE − 1.114* 3.337** (0.646) (1.445) FIE − 0.891** − 5.123* (0.395) (2.855) SE 0.000 − 0.001 (0.000) (0.001) FLL − 0.025 0.094** (0.027) (0.043) FNA 0.016 − 0.056 (0.017) (0.038) SCR − 0.103 − 0.347 (0.143) (0.226) HCL − 2.825*** − 1.382 (0.439) (1.829) RHP 0.130** − 0.672 (0.053) (0.534) LIE − 0.155** 0.348* (0.077) (0.178) PIV − 3.847*** 3.923 (0.840) (2.447) Cons − 0.867*** − 0.807*** − 0.771*** − 0.374 (0.254) (0.235) (0.256) (0.375) Time fixed effects NO NO Yes Yes R 2 0.026 0.221 − 0.192 − 0.185 N 1073 1073 1073 1073 Open in a new tab Standard errors in parentheses * p < 0.1, ** p < 0.05, *** p < 0.01. Moderating effect To ascertain whether technological complexity moderates the nonlinear effect of knowledge stickiness on technological concentration, we then fit a moderation model that tests the linear and quadratic terms of knowledge stickiness as well as their interaction with technological complexity. The results are presented in Table 4 . Models 1–2 are identical to the regressions in the full sample while models 3–4 distinguish the comparison between subsamples whose city technological complexity is above vs below the sample mean. All models contain the aforementioned fixed effects in our baseline regressions. The results of the full-sample regression on Model 2 indicate that while the linear term for Knowledge Stickiness was again statistically significantly positive at 7.021, the quadratic term (Knowledge Stickiness 2 ) was statistically significantly negative (− 9.622). Thus, the benchmark inverted-U shaped relationship remains stable once we have included the moderator variable. A further examination of the interaction terms produced an interaction term between demeaned knowledge stickiness and the linear term of knowledge stickiness with a coefficient of 2.960 and statistically significant at the 10% level. The interaction term for the quadratic term is -4.611 and also statistically significant at the 10% level. Full-sample interaction results tend to suggest that technological complexity moderates the nonlinear knowledge stickiness-technological concentration relationship, with the positive interaction on the linear term and the negative on the quadratic term suggesting complexity accentuates both the steepness and the curvature of the relationship. Such estimates are not easily interpreted as implying simple leftward or rightward shifts of the turning point. In the full-sample interaction model, more technological complexity is associated with a stronger positive association at low-to-moderate levels of stickiness and a steeper drop once stickiness becomes excessive. Put differently, this should be viewed as a change in slope and curvature, not evidence of a turning-point shift. The subsample comparison should be interpreted cautiously. In the low-complexity group, analyzed via Model 3, the linear term of knowledge stickiness (8.713) and quadratic term (− 12.292) remain significant, and the inverted U-shaped relationship persists. In the high-complexity group, examined through Model 4, however, the linear term (− 9.648) and quadratic term (20.007) of knowledge stickiness are both insignificant and show reversed signs. When technological complexity exceeds the average, the inverted U-shaped relationship between knowledge stickiness and the concentration of technology in different regions becomes dramatically different. This does not imply a reversal of the mechanism as such, but rather diminished statistical power and greater heterogeneity in high-complexity city environments, in which we conjecture the innovation activity is concentrated and which may be more sensitive to choice of specification. We therefore weight our conclusions principal on the full-sample interaction pattern and the implied marginal effects, and do not overreach into interpreting coefficient sign changes in the high-complexity subsample. Figure 1 depicts the marginal-effect implications of moderation by technological complexity. We intend it merely as an interpretive aid as to how complexity has the potential to reshape the nonlinear pattern. Table 4. Analysis of moderating effects. (1) (2) (3) (4) HHI HHI HHI HHI Stickiness 6.555*** 7.021*** 8.713** -9.648 (1.636) (1.656) (3.805) (7.603) Stickiness 2 − 8.906*** − 9.622*** − 12.292** 20.007 (2.591) (2.628) (6.160) (13.148) Complexity 0.030*** − 0.424* − 0.270 − 4.408** (0.012) (0.250) (0.457) (2.047) Stickiness*complexity 2.960* 1.821 32.973** (1.669) (3.074) (14.537) Stickiness 2 *complexity − 4.611* − 2.490 − 58.775** (2.739) (5.058) (25.817) Cons − 0.318 − 0.357 0.284 1.412 (0.374) (0.375) (0.850) (1.181) Control variables Yes Yes Yes Yes Time fixed effects Yes Yes Yes Yes R 2 − 0.177 − 0.175 − 0.518 − 0.226 N 1073 1073 536 537 Open in a new tab Standard errors in parentheses * p < 0.1, ** p < 0.05, *** p < 0.01. Fig. 1. Open in a new tab Moderation effect plot. Heterogeneity analysis To provide further clarity on such contextual variation in the non-monotonic relationship between knowledge stickiness and city-level technological concentration (HHI), we conduct heterogeneity analyses both by region (eastern, central, and western China) and by city size. All grouped regressions include the same full set of controls and fixed effects. The results are reported in Table 5 . For the city size split, we order cities by permanent population and split at the median into small and medium-sized cities versus large cities; both types are large enough to be distinct categories. The regional split yields more distinctly geographical heterogeneity relative to the full sample. In eastern China, the linear stickiness term is positive (10.084) and the quadratic term is negative (− 14.666), both significant at the 1% level, consistent with an inverted U-shaped pattern. The turning point implied by these estimates is around 0.347, close to the full sample benchmark. In the central and western regions, the stickiness terms are not statistically significant, and there is no evidence of an inverted U-shaped association in these subsamples. A modest takeaway from this is that the stickiness–concentration relationship is more readily detected where there is a cadre of AI-related activity taking place and complementary assets to which a city can attach learning. The results by city size suggest a similar, albeit superficially parallel pattern. For small and medium-sized cities, the linear term is positive (9.750) and the quadratic term is negative (− 14.618), both significant at 1%, implying an inverted U-shaped association with a turning point around 0.334. For large cities, the linear term remains positive (3.579, significant at 5%) while the quadratic term (− 3.604) is not statistically significant. This suggests a weaker or less clearly nonlinear pattern; one likely reason for this is that large cities may have available to them more diversified knowledge sources and have thicker innovation networks, implying less dependence on lock-in within a single trajectory, while smaller cities depend more on narrower local linkages and as such the balance between cumulative learning and lock-in becomes more evident. Overall, Table 5 suggests the inverted U-shaped pattern is most evident in eastern China and among small and medium-sized cities, indicating that the stickiness-concentration relationship is predicated on urban and regional context rather than being a universally applicable rule. The heterogeneity tests refine, and do not refute, our baseline mechanism. The clearer inverted-U shape in eastern cities is plausible due to cross-firm learning and market integration being stronger, increasing both the benefits of coordination of moderate stickiness and the lock-in costs of excessive stickiness. In contrast, the weaker curvature in larger cities suggests that the scale of these places provides a cushion from the negative marginal effect past the turning point, leading to a slope that is flatter and therefore less likely to be detected. Overall, the evidence points to the stickiness-concentration mechanism being highly contextual, with regional integration strengthening the non-linearity, and urban scale moderating its steepness. Table 5. Analysis of heterogeneity. Eastern Central Western Small cities Large cities HHI HHI HHI HHI HHI Stickiness 10.084*** 2.999 7.542 9.750*** 3.579** (2.202) (3.284) (5.044) (2.963) (1.661) Stickiness 2 − 14.666*** − 3.378 − 11.553 − 14.618*** − 3.604 (3.565) (5.029) (8.252) (4.732) (2.617) Cons − 1.168** 1.220 1.294 0.345 − 0.419 (0.541) (1.286) (1.295) (1.253) (0.379) Control Variables Yes Yes Yes Yes Yes Time fixed effects Yes Yes Yes Yes Yes R 2 − 0.046 − 0.362 − 0.441 − 0.321 − 0.054 N 562 314 197 535 538 Open in a new tab Standard errors in parentheses, * p < 0.1, ** p < 0.05, *** p < 0.01. Robustness tests To assess the extent to which our baseline findings depend on our choices regarding timing and inference, we conduct three additional tests on requested robustness checks, results shown in Table S3 of the Supplementary Material. First, we re-estimate the full model while replacing knowledge stickiness and its square with a one-year lag. While the signs of the coefficients remain as expected, neither are statistically significant. These coefficients being insignificant may reflect the limited number of observations remaining in the unbalanced panel since lagged specifications need consecutive city or year observations 66 , 67 . Second, we estimate a two-year-lag specification, which makes the inverted U-shaped pattern clearer, yielding a positive and significant coefficient on knowledge stickiness (β = 4.767, p < 0.01)and a negative and significant coefficient on the squared term of knowledge stickiness (β = − 7.059, p < 0.01). This again yields a turning point of approximately 0.338, close to the baseline estimate, further suggesting that the nonlinear association is not limited to the contemporaneous specification 26 , 68 . Third, we re-estimate the contemporaneous baseline using two-way clustered standard errors by city and year. The core pattern remains the same : knowledge stickiness is positive and significant (β = 8.915, p < 0.01) and the squared term of knowledge stickiness is negative and significant (β = − 12.773, p < 0.01). Though inference based on a relatively small number of year clusters should be treated with caution anyway, this again suggests a degree of stability in the estimated association 69 . Collectively, these results indicate that the inverted U-relationship between knowledge stickiness and technological concentration is not particular to a single specification, and although the one-year lag model is less well-defined, both the two-year lag model and alternative clustering reproduce the same pattern. Discussion and implications Discussion Utilizing city-level panel data for China from 2014 to 2023, this study examines how knowledge stickiness is associated with patterns of AI technological concentration. China provides a theoretically useful context for studying AI knowledge dynamics because its rapid AI diffusion practices come paired with acute urban hierarchy and intense spatial policy efforts. In its city system, China contains globally connected innovation hubs and many smaller cities who rely more deeply on external flows, creating great variance in local absorptive capacity, network openness, and innovation infrastructure that let us watch how diffusion potential and lock in behaviors intermingle in different urban contexts, with implications for debates on knowledge localization and clustering that reach beyond China. The results are consistent with an understanding that there is a nonlinear (more specifically inverted U shaped) relationship between knowledge stickiness and technological concentration: when stickiness is low, knowledge circulates more easily but local systems may struggle to retain and accumulate specialized capabilities; as stickiness rises to more moderate levels, repeated local interaction and recombination support cumulative learning and specialization, which is associated with higher concentration; when stickiness becomes excessive, local search loses its focus and local actors can no longer draw on external learning, associated with increased risks of lock-in and an overall weakening of renewal. This pattern aligns with existing arguments that moderate localization frictions may enable cumulative innovation while excessive lock-in constrains knowledge renewal 26 and resonates with evidence that spatial frictions suggest a selective filter rather than a pure barrier, one that shapes how collaboration and recombination function in complex knowledge systems 21 . Beyond providing evidence of a nonlinear association, our measure of knowledge stickiness operationalizes the tradeoff between diffusion potential and lock-in resistance using a transparent indicator structure. This provides a practical way to characterize how city-level AI knowledge systems combine circulation and embedded accumulation, and it can be applied to other complex technology fields where tacit knowledge and recombination matter. The empirical results suggest that technological complexity moderates the association between knowledge stickiness and technological concentration. The interaction estimates suggest that complexity alters the strength and curvature of the nonlinear relationship. Instead of suggesting a common shift in the turning point from context to context, our findings suggest that in high-complexity city environments the positive association is strong at low-to-moderate levels of stickiness while the decline beyond the turning point becomes steeper under lock-in pressure. This is consistent with the premise that dimensionality high-value parts depend on the ability to combine multiple, interrelated dimensions of knowledge, which in turn tend to accumulate over the long-term, conferring a more cumulative view of both advantages and risk of lock-in 27 , reports positive links between urban knowledge complexity and economic fitness 48 , accumulation and complexity are viewed as two aspects of innovation policy debates 47 . Altogether our results indicate that complexity conditions when knowledge embedding favors concentration and when it becomes constraining, but we do not suggest a causal assumption framing. City size and geography also shape the stickiness-concentration relationship. The heterogeneity results imply that the inverted U-shaped relationship is easier to detect in eastern cities and in smaller to medium sized cities. Evidence for large cities, by contrast, is weaker since the quadratic term is not statistically significant for this subsample. By contrast, small and medium-sized cities in central and western regions appear more vulnerable to lock-in when knowledge becomes highly embedded, consistent with thinner networks and more limited external knowledge inflows. This divergence plausibly reflects structural differences in local innovation ecosystems: large cities host multi-tier knowledge actors and broader inter-city and global linkages, sustaining external inputs and internal recombination under higher stickiness. SMSCs often face narrower actor bases and shorter knowledge chains, reducing recombination opportunities and increasing path dependence. Lehmann et al. 41 show that high knowledge density can delay lock-in in innovation-rich regions, while resource-constrained regions fall into path dependence more quickly. Xie et al. 42 similarly report lower recombination efficiency in SMSCs than in major innovation hubs. Policy implications Starting from these findings, policy should seek to create conditions under which circulation and embedded accumulation are balanced such that cities accrue benefits from the external, without entrenching lock-in. This might take the form of improving cross-city mechanisms for collaboration: data-sharing arrangements, patent and knowledge-access platforms, R&D–industry exchange to facilitate learning from foundational research while improving IP protection and local ability to mobilize AI research into core technologies and algorithms. Innovation governance should more deeply connect policy tools to technological complexity: the high-complexity domains of AI like foundation models or intelligent chips, benefit from stable knowledge networks, long-term collaboration, and sustained support for original research; in subfields of lower complexity such as applications, diffusion channels and open innovation, as well as mechanisms to reduce premature path dependence are more useful; in funding allocation and project evaluation, the frontiers of AI research can more explicitly incorporate this distinction. Finally, small and medium cities need stronger absorptive and recombination capacity, which can be boosted by enclave-style R&D centers and innovation corridors to bring in talent and research resources, joint laboratories between universities and local leading firms, linking them to local industrial parks, as well as more effective technology-transfer intermediaries and achievement transformation platforms, to widen access to external knowledge and improve local recombination and stop cities falling behind too badly in AI. Research limitations and future directions There are also caveats to keep in mind when interpreting the findings. First, we rely on patent-based indicators, which capture codified technological activity but do not fully reflect tacit knowledge exchange, informal collaboration, or non-patented AI innovation. Second, our panel is unbalanced by construction since the city year observations with zero AI patenting are dropped, so these results should be interpreted as characteristic of cities with visible AI patent activity rather than all Chinese cities. Third, the knowledge stickiness measure is a composite index constructed from multiple patent-based proxies and weighting decisions; we carry out validation and sensitivity checks, but alternative choices of indicators will produce at least some variation in estimated magnitudes. Fourth, the dependent variable concentrates on the concentration across AI sub-technology branches within a city and is not meant to reflect, much less serve as a measure of, overall, AI competitiveness or innovation quality. Finally, fixed effects, lagged specifications, and alternative ways of clustering standard errors help provide robustness but do not eliminate possible endogeneity from reverse causality and omitted variables entirely. For these reasons we treat the empirics as presenting robust evidence of conditional associations—not proof of causal impacts. Future research could combine patent data with publication, investment, and collaboration-network data, and employ stronger identification strategies or spatial dynamic models to further examine inter-city knowledge interactions and policy spillovers. Supplementary Information Below is the link to the electronic supplementary material. Supplementary Material 1 (20.3KB, docx) Author contributions Lin Zhang Conceptualization, Data curation, Investigation, Resources, Writing—review & editing. Quanjun Zhang Conceptualization, Supervision, Writing—original draft, Writing—review & editing. Houyuan Meng (Corresponding author) Formal analysis, Methodology, Software, Writing—review & editing. Funding The author(s) received no specific funding for this work. Data availability The data used in this study can be available from the author upon reasonable request. 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