Firms’ Structural Positions in Patent Citation Networks and Innovation Performance: Evidence from a Large-Scale Chinese Dataset

Abstract

Using a panel of Chinese A-share listed companies from 2007 to 2022, this study examines how firms’ structural positions in patent citation networks affect innovation efficiency. We construct a firm-level patent citation network and use betweenness centrality to capture firms’ brokerage-oriented positions in knowledge flows. Based on firm- and year-fixed-effects models, instrumental-variable estimation, and robustness checks, we find that stronger brokerage positions significantly improve innovation efficiency. Mechanism analyses show that this effect operates through two channels: cross-domain knowledge recombination and organizational boundary spanning. Firms in stronger brokerage positions are more likely to access technologically heterogeneous external knowledge and interact with a wider range of external knowledge-bearing entities, thereby improving the efficiency with which innovation inputs are transformed into patent-based outputs. We further find that digital transformation negatively moderates the relationship between brokerage centrality and innovation efficiency. This suggests that digital transformation reduces firms’ marginal dependence on external brokerage positions by strengthening internal data-processing, coordination, and knowledge-integration capabilities. Additional analyses show that the positive effect of brokerage centrality is broadly shared across ownership groups. Regional heterogeneity is more evident in the stronger brokerage premium observed in the western region than in the eastern region.

1. Introduction

Inter-firm network positions matter for innovation, including those embedded in supply-chain networks [1] and collaboration networks [2], by shaping the relational environment in which firms search for, access, and utilize external resources. Knowledge spillovers and inter-firm learning constitute central drivers of innovative activity. Patent citation data provides a valuable window into these processes, as the citation of one firm’s patent by another signals the transfer of technological knowledge. Beyond capturing the intensity of knowledge spillovers, patent citation data also reveals structural differences in technological learning capabilities across firms with different ownership types, thereby offering micro-level evidence for the design of differentiated innovation policies [3]. By linking patents to assignees, it is possible to construct a large-scale directed network of inter-firm knowledge flows, which makes explicit who builds upon whose technological ideas. Firms occupying key positions in such networks—for example, acting as bridges or hubs—are more likely to access diverse external ideas and recombine them into novel innovations. Indeed, recent studies document that network centrality and structural holes in patent and knowledge networks significantly enhance firms’ innovation capability [4]. Such positions matter not merely because they increase information exposure, but because they broaden firms’ access to heterogeneous external knowledge and improve their ability to coordinate, filter, and recombine dispersed technological inputs [5]. In particular, core firms embedded in broad and heterogeneous knowledge networks tend to achieve superior innovation performance by leveraging diversified external knowledge sources [6], thus forming a coherent chain of development from relational networks to knowledge acquisition and innovative performance [7].

Despite extensive research on knowledge spillovers, two important gaps remain. First, at the theoretical level, prior studies often treat network position as a general relational advantage, while paying insufficient attention to how brokerage-oriented structural positions are translated into innovation efficiency through specific knowledge-integration mechanisms [5]. Second, at the methodological level, existing research relies heavily on patent counts or simplified spillover proxies and less frequently exploits the topology of firm-level patent citation networks to identify firms’ structural roles and relate them to efficiency-oriented innovation outcomes [8]. To address these gaps, this study constructs a firm-level patent citation network from a large Chinese patent dataset and examines how firms’ brokerage positions affect innovation efficiency through cross-domain knowledge recombination and boundary-spanning integration across heterogeneous knowledge entities. Applying social network analysis, we use betweenness centrality as the baseline measure of brokerage position and an inverse structural-hole measure for robustness checks. To ensure credible identification, we combine firm and year fixed effects with instrumental-variable specifications and a Heckman two-stage procedure. The results show that firms occupying stronger brokerage positions in citation networks achieve significantly higher innovation efficiency and patent output, while digital transformation reduces firms’ marginal dependence on brokerage positions by strengthening internal knowledge integration and coordination capabilities. Overall, the findings highlight the economic importance of brokerage-based network embeddedness in shaping firm-level innovation and provide new evidence from an emerging-economy context.

2. Literature Review and Research Hypotheses

2.1. Knowledge Network Centrality and Innovation Performance

Enterprise innovation does not rely solely on internal research and development capabilities; rather, it increasingly depends on interactions with external partners, research institutions, and peer firms to acquire technological, knowledge, and market information [9]. Social relationships and the networks these relationships constitute are influential in explaining the processes of knowledge creation, diffusion, absorption and use [10]. Through participation in innovation networks and processes of technological learning, enterprises are able to enhance their innovation performance [11]. A foundational stream of the literature demonstrates that knowledge spillovers constitute a key mechanism underlying such network-based innovation. The seminal contribution by Jaffe, Trajtenberg, and Henderson (1993) shows that patent citations can be used to trace knowledge spillovers across inventors and firms [12], and the robustness of this approach has been repeatedly verified in subsequent studies [13]. Consequently, patent citations have become a widely accepted proxy for inter-firm knowledge flows. In empirical research, a citation from patent A to patent B is generally interpreted as evidence that knowledge embodied in patent B has been utilized by the inventor of patent A.

However, knowledge spillovers do not occur in a vacuum. Prior studies emphasize that an individual’s or organization’s position within a network is often more consequential than the mere strength of dyadic relationships [14,15]. Network embeddedness therefore provides a crucial analytical framework for understanding knowledge exchange and innovation processes, and is widely regarded as an important prerequisite for achieving superior innovation performance [16].

From an evolutionary perspective, major inventions often emerge from the cumulative integration of existing technological components and principles, a process that is intrinsically linked to knowledge spillover effects [17]. In this context, the publicly accessible knowledge held by a “source” enterprise can be more easily absorbed by a “recipient” enterprise and subsequently transformed into innovation inputs [18]. Moreover, research adopting a knowledge network perspective demonstrates that knowledge spillovers are embedded in networks characterized by specific structural patterns. Network robustness and structural integrity have been shown to significantly increase the likelihood of generating high-value inventions, thereby serving as critical structural indicators of technological value [19].

Against this background, firms occupying intermediary positions in patent citation networks are situated at the intersection of multiple technological trajectories, rather than being confined to isolated knowledge domains. Such intermediary positions reflect a structural characteristic of the knowledge flow architecture shaped by cumulative and historically embedded citation patterns, as they connect otherwise weakly linked technological domains or innovation actors. As a result, firms with higher betweenness centrality are exposed to a broader and more diverse set of knowledge sources, which expands their innovation search space. Moreover, by spanning distinct knowledge streams, intermediary positions create more opportunities to integrate and recombine heterogeneous knowledge elements. This structural advantage facilitates the effective coordination and synthesis of dispersed technological inputs and constrains the informational environment in which firms conduct innovation activities, which is particularly conducive to improving the efficiency with which innovation inputs are transformed into innovation outputs. Accordingly, this study proposes the following hypothesis:

Hypothesis 1.

A firm’s betweenness centrality in the patent citation network (Betw_cite) is positively associated with its innovation efficiency.

2.2. Knowledge Acquisition and Cross-Border Knowledge Flow

Absorbing external knowledge has become an important means for enterprises to enhance their innovation performance [20]. From the perspective of knowledge networks, the improvement of enterprise innovation performance does not only depend on whether it can enter the innovation network, but more importantly, on the way and structural characteristics of acquiring and integrating external knowledge through the network. The absorptive capacity theory states that external knowledge can only be truly transformed into innovative output after it is effectively identified, absorbed, and internalized by the enterprise [21]. Therefore, knowledge acquisition is not a simple process of “information contact”, but a structured learning process that highly relies on the existing knowledge foundation and network location of the enterprise.

From the perspective of evolutionary economics, technological innovation often stems from the continuous restructuring and expansion of existing knowledge elements, rather than sudden breakthroughs that completely deviate from historical trajectories. Related studies have shown that enterprise innovation activities exhibit obvious path dependence characteristics, and their search space is often constrained by existing technological accumulation [22]. Technological innovation is not a sudden breakthrough but a path-dependent evolution based on existing knowledge elements [23]. In this context, crossing existing technological boundaries and absorbing heterogeneous external knowledge have become important ways for enterprises to break through path locking and achieve innovation efficiency improvement.

2.2.1. Cross-Domain Knowledge Recombination and Innovation Efficiency

Firstly, in the technological dimension, interdisciplinary knowledge flow refers to the reorganization of existing technological elements by enterprises through absorbing external knowledge from different technological fields or knowledge domains. Research indicates that a combination of heterogeneous knowledge from different scientific fields may drive highly innovative outcomes [24]. As a key to obtaining raw materials for knowledge recombination, diversity in knowledge search enables enterprises to overcome resource constraints and accelerate the realization of breakthrough technological innovation [25]. With the increasing complexity of technology, knowledge accumulation within a single technological field has become difficult to support sustained innovation breakthroughs, and innovation activities are increasingly dependent on knowledge integration across disciplines and technological paradigms [26,27]. Meanwhile, the latest evidence on “remote knowledge recombination and invention value” also supports that the combination of scientific knowledge and technical knowledge at different distance levels affects the technical impact and economic value of inventions, further reinforcing the micro-mechanism of “cross-border recombination” [28]. External knowledge with a large technological distance often contains higher potential for recombination. Although it is more difficult to absorb, once successfully integrated, it can often lead to a more significant improvement in innovation performance [29].

Firms occupying brokerage positions are more likely to bridge otherwise disconnected knowledge streams and to access technologically distant external knowledge, thereby expanding the space of potential recombinations. This mechanism is therefore concerned with the content heterogeneity of knowledge inputs and the technological distance among them. When firms successfully combine heterogeneous knowledge elements, they are more likely to overcome path dependence and improve the efficiency with which innovation inputs are transformed into outputs. Therefore, interdisciplinary knowledge flows are an important condition for breaking existing technological path dependence and enhancing innovation efficiency.

2.2.2. Organizational Boundary Spanning and Innovation Efficiency

Secondly, in the organizational dimension, knowledge is embedded not only in technological systems but also in different types of organizational entities [30]. Enterprises, universities, research institutions, and other non-market-oriented innovation organizations exhibit systematic differences in knowledge production objectives, research logic, and forms of knowledge expression [31]. Cognitive distance is defined as the manifestation of technological knowledge differences between enterprises [32]. However, with the continuous advancement of industry–university–research cooperation, cognitive distance is not only reflected in technological knowledge differences between enterprises, but also between enterprises and non-enterprise organizations. The impact of different external sources on innovation performance varies significantly, and exhibits systematically explainable heterogeneity under different organizational types and situational conditions. The more fully a company utilizes the knowledge acquired from the market or scientific entities, the higher its innovation performance will be [33]. Compared to relying on a single knowledge source, the diversity of knowledge holders enables enterprises to flexibly allocate knowledge resources across different stages of innovation, thereby enhancing overall innovation efficiency. This diversity enhances a firm’s absorptive capacity by expanding the breadth of knowledge acquisition and assimilation (potential absorptive capacity) [34], while social integration mechanisms further bridge the gap between potential and realized capacity, thereby boosting the efficiency of knowledge transformation and exploitation in innovation processes [35]. Accordingly, innovation efficiency depends not only on what knowledge is acquired, but also on whether firms can span organizational boundaries and maintain effective interactions with diverse knowledge holders. Firms occupying brokerage positions are structurally advantaged in connecting heterogeneous actors and organizing external knowledge relationships across organizational boundaries. This mechanism is therefore concerned with the diversity and governance of knowledge sources, rather than with the technological distance of knowledge content itself. By broadening and structuring external interactions, organizational boundary spanning improves firms’ absorptive capacity and enhances the efficiency of knowledge transformation and exploitation.

In the knowledge network structure, whether an enterprise can achieve the aforementioned two types of cross-border knowledge acquisition is closely related to its network position [36]. Enterprises occupying higher intermediary centrality are usually located at the intersection of different knowledge flow paths, capable of connecting originally dispersed technological fields and heterogeneous knowledge entities. Compared to enterprises at the edge of the network, such enterprises are not only more likely to access diverse external knowledge sources, but also have obvious information advantages in the process of knowledge screening, comparison, and integration. By embedding multiple knowledge flow channels, enterprises are able to explore in a broader knowledge space and achieve more efficient knowledge recombination based on existing technologies. It can be seen that network brokerage centrality does not enhance innovation efficiency through a single generic knowledge-acquisition channel. Instead, it operates through two analytically distinct mechanisms: cross-domain knowledge recombination in the technological dimension and organizational boundary spanning in the relational–organizational dimension.

Hypothesis 2a.

Cross-domain knowledge recombination plays a mediating role in the relationship between patent citation network betweenness centrality and innovation efficiency.

Hypothesis 2b.

Organizational boundary spanning plays a mediating role in the relationship between patent citation network betweenness centrality and innovation efficiency.

2.3. Moderating Effect of Digital Transformation

Digital technology plays a central role in enhancing innovation capabilities and competitive advantages [37]. Digital transformation reshapes not only firms’ internal innovation processes but also the way they rely on external network positions to acquire and utilize knowledge. From a Schumpeterian perspective, digital transformation should not be understood merely as the adoption of new technologies, but as part of a broader restructuring of the innovation paradigm. In Schumpeter’s sense, innovation consists of “new combinations”, while creative destruction implies that new technological systems may erode existing sources of competitive advantage and reallocate them to new organizational forms [38,39]. In this context, digitalization changes the way firms search for, coordinate, and recombine knowledge, thereby altering the relative importance of external network positions in the innovation process.

Building on the neo-Schumpeterian and evolutionary view, firms’ innovation outcomes are shaped by routines, capabilities, and path-dependent search processes rather than by one-off optimization [22]. At lower levels of digitalization, firms often rely more heavily on brokerage positions in external knowledge networks to access heterogeneous knowledge, connect dispersed actors, and organize cross-boundary interactions. However, as digital transformation deepens, firms gradually strengthen their internal data-processing, coordination, and knowledge-integration capabilities through digital platforms, intelligent systems, and data-driven routines. These improvements enable firms to internalize part of the search, matching, and recombination functions that previously depended more strongly on external brokerage positions.

Therefore, digital transformation does not weaken firms’ innovation capability per se. Rather, it reduces the marginal dependence of innovation efficiency on external brokerage advantages. In other words, digital transformation attenuates the brokerage premium associated with network centrality by shifting part of the basis of competitive advantage from external relational intermediation toward internally enabled digital coordination and knowledge integration.

Hypothesis 3.

Digital transformation negatively moderates the relationship between network brokerage centrality and enterprise innovation efficiency by reducing firms’ marginal dependence on external brokerage positions.

Figure 1 presents the theoretical framework and summarizes the proposed hypotheses.

3. Model Specification and Variable Definitions

3.1. Model Design and Data Source

3.1.1. Benchmark Model, Mediation Analysis, and Moderation Specification

This paper initially employs a panel regression model with two-way fixed effects to examine the relationship between the structural position of enterprises in the innovation network and their innovation performance. The benchmark model is set as follows:

$${\mathit{Ino}\text{\_}\mathit{perform}}_{it}={\alpha }_0+{\alpha }_1{\mathit{Centrality}}_{it}+\gamma X_{it}+{\phi }_t+{\lambda }_i+{\epsilon }_{it}$$

(1)

A key empirical concern is that network position may be jointly determined with innovation outcomes, giving rise to simultaneity and reverse causality. To mitigate this concern, the main regressions use lagged centrality measures (detailed below), so that network position is measured prior to the realization of contemporaneous innovation performance. Under TWFE, identification relies on within-firm changes in network position over time and the associated changes in innovation performance, net of observed time-varying controls and economy-wide shocks captured by year fixed effects.

$${\mathit{Ino}\text{\_}\mathit{perform}}_{it}={\alpha }_0+{\alpha }_1{\mathit{Centrality}}_{it}+{{\alpha }_2\mathit{Med}}_{it}+\gamma X_{it}+{\phi }_t+{\lambda }_i+{\epsilon }_{it}$$

(2)

$${\mathit{Med}}_{it}={\beta }_0+{\beta }_1{\mathit{Centrality}}_{it}+\gamma X_{it}+{\phi }_t+{\lambda }_i+{\epsilon }_{it}$$

(3)

To examine mechanisms, we estimate a mediation framework in which ${\mathit{Med}}_{it}$ denotes a hypothesized channel (e.g., cross-domain knowledge recombination or organizational boundary-spanning diversity). The mediation analysis proceeds in three steps: (i) estimating the total effect of centrality on innovation performance; (ii) testing whether centrality predicts the mediator; and (iii) assessing whether the mediator is associated with innovation performance while conditioning on centrality:

Finally, we analyze heterogeneity by firms’ digital transformation (DT) using an interaction specification:

$${\mathit{Ino}\text{\_}\mathit{perform}}_{it}={\alpha }_0+{\alpha }_1{\mathit{Centrality}}_{it}+\delta {DT}_{it}+\xi {\left(\mathit{DT}\text{×}\mathit{Centrality}\right)}_{it}+\gamma X_{it}+{\phi }_t+{\lambda }_i+{\epsilon }_{it}$$

(4)

The coefficient $\xi$ captures whether digital transformation amplifies or attenuates the marginal effect of network centrality. In the empirical section, we further report and visualize the conditional marginal effect of ${\mathit{Centrality}}_{\mathit{it}}$ over the support of ${DT}_{it}$, providing an interpretable assessment of the moderating mechanism.

3.1.2. Data Source and Sample Construction

Patent citation data are obtained from the Patent Office of the China National Intellectual Property Administration (CNIPA), from which we identify patent–patent citations and construct firm-level patent citation networks. Firm-level financial and accounting variables are collected from the China Stock Market and Accounting Research (CSMAR) database, which also provides the key identifiers used to merge patent-based measures with listed-firm information. Detailed definitions and construction procedures for the main variables are provided in Section 3.2.

The benchmark regression sample covers Chinese A-share listed companies over the period 2007–2022. This time span corresponds to the effective estimation sample used in the baseline regressions after applying the variable-construction requirements, lag structure, and missing-value screening. Accordingly, the reported sample period refers to the actual estimation window of the baseline specification rather than the full raw-data availability period.

3.2. Variable Definitions

3.2.1. Independent Variable

The core explanatory variable, ${\mathit{Centrality}}_{it}$, captures firm i’s structural position in the knowledge citation network in year t. In the baseline specification, this study employs betweenness centrality, denoted by ${betw\text{\_}cite}_{it}$ [40], to measure firms’ intermediary roles in knowledge flows. The measure is computed using standard network algorithms implemented in the NetworkX (3.5) package of Python (3.12.2), based on firm-level patent citation networks constructed from Chinese A-share listed companies’ patent citation data.

The firm-year patent citation network used to compute firms’ centrality measures is constructed using a three-year rolling window. Specifically, for each year t, we identify patents applied for by sample listed companies during the period [t − 2, t] and collect the backward citations made by these patents. A directed citation linkage from firm i to firm j is recorded when at least one patent applied for by firm i within the rolling window cites a patent assigned to firm j. These patent-level citation relations are then aggregated to the firm level to form a time-varying directed inter-firm citation network.

The cited patents are not restricted to the same three-year window. That is, patents applied for during [t − 2, t] may cite patents assigned in earlier years, so the resulting network captures recently realized inter-firm knowledge flows while still reflecting historically accumulated knowledge stocks embodied in cited patents. The three-year rolling-window design also helps reduce excessive sensitivity to single-year fluctuations in patenting and citation activity. The boundary of the centrality network is defined at the level of listed companies in the sample. Accordingly, the betweenness centrality measures used in the benchmark regressions characterize firms’ structural positions within the listed-firm citation network.

This network boundary should be distinguished from the construction of certain mechanism variables. In particular, the variables used to capture organizational boundary spanning incorporate citations involving heterogeneous knowledge-bearing entities such as universities, research institutes, and other organizational types. Thus, non-firm organizations are introduced in the mechanism analysis, but they are not included as nodes in the benchmark inter-firm centrality network.

Betweenness centrality reflects the extent to which a firm lies on the shortest paths connecting other firms in the citation network. A higher level of betweenness centrality indicates that a firm occupies a brokerage position connecting otherwise sparsely linked and non-redundant knowledge sources, enabling it to act as an intermediary in the transmission and recombination of heterogeneous technological knowledge.

$${\mathit{Betw}\text{\_}\mathit{cite}}_{it}=\sum _{s\text{≠}i\text{≠}j}{\displaystyle \frac{{\sigma }_{sj}\mathit{\left(}\mathit{i}\mathit{\right)}}{{\sigma }_{sj}}}$$

(5)

• ${\sigma }_{sj}$: The number of all shortest paths between node s and node j.
• ${\sigma }_{sj}\left(\mathit{i}\right)$: The number of shortest paths passing through enterprise i.

In addition to betweenness centrality, this study employs a structural-hole-based measure as an alternative indicator of firms’ network position to examine the robustness of the baseline results across different dimensions of network structure. Specifically, we use Burt’s network constraint, denoted by structure_hole, which captures the extent to which a firm’s network ties are redundant and embedded within closed clusters. A higher level of constraint indicates stronger network closure and fewer brokerage opportunities, whereas a lower level of constraint reflects greater access to structural holes.

To maintain consistency in interpretation with betweenness centrality—where larger values represent stronger brokerage advantages—we follow the common practice in the literature and use the negative of the constraint index, denoted as ${Constraint}_{it}$, as the robustness measure. Formally, the inverse structural-hole measure is defined as

$${Constraint}_{it}=-{\mathit{Stucture}\text{\_}\mathit{hole}}_{it}=-\sum _{j\text{∈}{N}_{it}}{(p_{ij,t}+\sum _{q\text{∈}{N}_{it}}{p}_{iq,t}{p}_{qj,t})}^2$$

(6)

• $j,q$: Network neighbors of enterprise i in year t.
• $N_{it}$: The set of directly connected enterprises of enterprise i in year t.
• $p_{iq,t}$: The relative relationship strength between enterprise i and enterprise j.

This transformation ensures that higher values of ${Constraint}_{it}$ correspond to weaker network constraints and greater structural-hole advantages. From an innovation perspective, firms with higher betweenness centrality benefit from bridging otherwise disconnected knowledge domains and coordinating dispersed knowledge flows, while firms with higher values of ${Constraint}_{it}$ are less constrained by redundant ties and can more flexibly recombine heterogeneous external knowledge. Together, betweenness centrality and the inverse structural-hole measure capture complementary aspects of firms’ structural advantages in knowledge networks—namely, brokerage positions that facilitate cross-domain knowledge integration and reduced network closure that lowers informational redundancy.

3.2.2. Dependent Variable

Among them, ${\mathit{Ino}\text{\_}\mathit{perform}}_{it}$ represents the innovation performance of enterprise i in year t. This article measures innovation performance from two dimensions: innovation efficiency and innovation output, specifically including innovation efficiency indicators (${\mathit{Innoeff}1}_{it}$, ${\mathit{Innoeff}2}_{it}$).

The dependent variables are two measures of innovation efficiency, denoted by ${\mathit{Innoeff}1}_{it}$ and ${\mathit{Innoeff}2}_{it}$. In both cases, innovation efficiency is defined as the ratio of innovation output to innovation input. Innovation input is measured by firms’ R&D expenditure, while innovation output is measured using patent applications.

First, ${\mathit{Ino}\text{\_}\mathit{eff}1}_{it}$ is an unweighted innovation-efficiency measure. Its numerator is the total number of patent applications filed by firm i in year t, including invention patents, utility model patents, and design patents. Its denominator is the logarithm of one plus R&D expenditure. Formally,

$${\mathit{Innoeff}1}_{it}={\displaystyle \frac{{Patent}_{it}}{\mathrm{l}\mathrm{n}(1+{R\&D}_{it})}}$$

(7)

where ${Patent}_{it}$ denotes the total number of patent applications filed by firm i in year t. This indicator mainly captures the efficiency with which firms transform R&D inputs into the overall quantity of patent output.

Second, ${\mathit{Ino}\text{\_}\mathit{eff}2}_{it}$ is a quality-adjusted innovation-efficiency measure. Its numerator is a weighted patent index, where invention patents, utility model patents, and design patents are assigned weights of 3, 2, and 1, respectively. The denominator is again the logarithm of one plus R&D expenditure. Formally,

$${\mathit{Innoeff}2}_{it}={\displaystyle \frac{3\times {Invention}_{it}+2\times {Utility}_{it}+1\times {Design}_{it}}{\mathrm{l}\mathrm{n}(1+{R\&D}_{it})}}$$

(8)

where ${Invention}_{it}$, ${Utility}_{it}$, ${Design}_{it}$ denote the numbers of invention patents, utility model patents, and design patents applied for by firm i in year t, respectively.

Compared with ${\mathit{Innoeff}1}_{it}$, this specification assigns greater weight to invention patents and therefore places more emphasis on the quality structure of innovation output rather than on patent quantity alone. In this sense, ${\mathit{Innoeff}2}_{it}$ is better suited to capturing firms’ efficiency in transforming R&D inputs into relatively higher-value and more substantive innovative output.

Taken together, ${\mathit{Innoeff}1}_{it}$ reflects quantity-based innovation efficiency, whereas ${\mathit{Innoeff}2}_{it}$ reflects quality-adjusted innovation efficiency with greater emphasis on invention patents. Using both indicators allows us to assess whether the empirical findings are robust across different dimensions of innovation efficiency.

3.2.3. Mediating Variables: Cross-Domain Knowledge Recombination

To capture cross-domain knowledge recombination, we use two complementary indicators. The first, ${\mathit{KF}\text{\_}\mathit{cross}\text{\_}\mathit{crossdomain}}_{it}$, measures the cross-domain external citation ratio, i.e., the share of a firm’s external patent citations that go to patents belonging to IPC4 classes outside the firm’s existing technological portfolio.

Formally, for firm i in year t:

$${\mathit{KF}\text{\_}\mathit{cross}\text{\_}\mathit{crossdomain}}_{it}={\displaystyle \frac{{\mathit{Cross-domain}externalcitations}_{it}}{{Totalexternalcitations}_{it}}}$$

(9)

where cross-domain external citations refer to citations to patents classified in IPC4 classes different from those of the citing firm’s existing technological portfolio. A higher value indicates a greater reliance on technologically heterogeneous external knowledge.

The second indicator,${KF\text{\_}cross\text{\_}techdist}_{it}$, measures the technological distance of external knowledge sourcing. It is constructed as the average Jaccard distance between the IPC4 technological class set of a focal firm’s patent and that of each cited external patent. A higher value indicates that the firm draws upon technologically more distant external knowledge, reflecting a stronger tendency toward cross-domain knowledge recombination.

Formally, the Jaccard distance is defined as

$$JaccardDistance(A,B)=1-{\displaystyle \frac{|A\cap B|}{|A\cup B|}}$$

(10)

It is constructed as the average Jaccard distance between the IPC4 technological class set of a focal firm’s patent (A_i) and that of each cited external patent (B_i), where the distance equals 0 when the two patents share identical technological domains and approaches 1 when they belong to completely different domains. The firm-year level indicator ${KF\text{\_}cross\text{\_}techdist}_{it}$ is calculated as the average Jaccard distance across all external citations in year t.

$${KF\text{\_}cross\text{\_}techdist}_{it}={\displaystyle \frac{1}{N}}\sum _{i=1}^N[1-{\displaystyle \frac{|A_i\cap B_i|}{|A_i\cup B_i|}}]$$

(11)

A higher value of ${KF\text{\_}cross\text{\_}techdist}_{it}$ indicates that a firm draws upon technologically more distant external knowledge, reflecting a stronger tendency toward cross-domain knowledge recombination. Such exposure to heterogeneous technological domains is expected to facilitate recombinant innovation by expanding firms’ search space and enabling novel combinations of existing knowledge elements.

Taken together, ${\mathit{KF}\text{\_}\mathit{cross}\text{\_}\mathit{crossdomain}}_{it}$ captures the breadth of cross-domain sourcing, whereas ${KF\text{\_}cross\text{\_}techdist}_{it}$ captures the distance of that sourcing.

3.2.4. Mediating Variables: Organizational Boundary-Spanning Diversity

To capture organizational boundary spanning, we construct two complementary indicators based on firms’ external patent citations. In this context, “external” means that self-citations are excluded. Specifically, a citation is treated as self-citation if the cited patent belongs to the same listed firm (identified by the parent-company securities code) or if the citing and cited patents share the same applicant. After removing self-citations, we classify cited external organizations into heterogeneous organizational types, including firms, universities, research institutes, medical institutions, government-related entities, and others. Based on this classification, we construct two indicators to capture different dimensions of organizational boundary spanning.

The first indicator,${Diversity\text{\_}ext}_{it}$, measures the diversity of external knowledge source types and reflects the breadth of a firm’s interactions with heterogeneous organizational entities. We first calculate ${Open\text{\_}diversity\text{\_}type}_{it}$, defined as one minus the Herfindahl–Hirschman Index (HHI) of the distribution of external citations across organizational types (1—HHI). A larger value of ${Open\text{\_}diversity\text{\_}type}_{it}$ indicates that the firm’s external knowledge sourcing is spread across a wider range of organizational categories rather than concentrated in a single type of external actor. We then combine this type diversity with the total number of external citations, ${N\text{\_}open\text{\_}cit\text{\_}ext}_{it}$, and take logarithms to obtain the final firm-year indicator:

$${Diversity\text{\_}ext}_{it}=ln({Open\text{\_}diversity\text{\_}type}_{it}\ast {N\text{\_}open\text{\_}cit\text{\_}ext}_{it})$$

(12)

A higher value of ${Diversity\text{\_}ext}_{it}$ therefore indicates that the firm sustains boundary-spanning engagement with a broader and more diversified set of external knowledge holders.

The second indicator, ${Hhi\text{\_}ext}_{it}$, measures the concentration of external knowledge sourcing across organizational entities and reflects the structural depth of firms’ reliance on particular external partners. We first calculate ${open\text{\_}depth\text{\_}hhi}_{it}$, defined as the HHI of the distribution of external citations across distinct partner organizations. A higher value of ${open\text{\_}depth\text{\_}hhi}_{it}$ indicates that the firm’s external knowledge interactions are concentrated among a smaller number of dominant knowledge providers, rather than being broadly distributed across many partners. We then combine this concentration measure with the total number of external citations, ${N\text{\_}open\text{\_}cit\text{\_}ext}_{it}$, and take logarithms to obtain

$${Hhi\text{\_}ext}_{it}=ln({open\text{\_}depth\text{\_}hhi}_{it}\ast {N\text{\_}open\text{\_}cit\text{\_}ext}_{it})$$

(13)

A higher value of ${Hhi\text{\_}ext}_{it}$ indicates greater concentration and a narrower reliance structure in a firm’s external knowledge sourcing.

Taken together, ${Diversity\text{\_}ext}_{it}$ captures the breadth of organizational boundary spanning, whereas ${Hhi\text{\_}ext}_{it}$ captures the concentration structure of external knowledge sourcing. The former emphasizes whether a firm connects with a wider range of heterogeneous organizational types, while the latter emphasizes whether its external knowledge dependence is dispersed or concentrated among a limited number of dominant partners.

3.2.5. Moderating Variable

Digital transformation (DT): Following the prior literature [41], we measure firms’ digital transformation using a text-based annual-report approach. Specifically, we collect firms’ annual reports and extract the full text, and then construct a firm-year digital transformation indicator through keyword identification and frequency counting. The keyword dictionary is built with reference to both the academic literature and major policy documents or reports on digital transformation. It covers two broad dimensions: underlying digital technologies and technology-enabled practical applications. The underlying-technology dimension includes representative terms related to artificial intelligence, blockchain, cloud computing, and big data, while the application dimension includes keywords associated with digital business scenarios and digital technology adoption in practice.

Based on the extracted annual-report text, we search, match, and count the frequencies of these digital-transformation-related keywords at the firm-year level, and then aggregate them into an overall indicator of digital transformation intensity. In constructing the index, expressions containing explicit negation words and references to digital transformation that do not pertain to the focal firm itself are excluded. Because the resulting frequency data are typically right-skewed, the aggregated keyword count is log-transformed to obtain the final DT measure. A larger value of DT indicates that the firm places greater emphasis on digital transformation in its disclosed strategic and operational activities.

3.2.6. Control Variables

Control Variables: The vector Xit includes a set of firm-level control variables commonly used in the innovation economics and corporate finance literature. Specifically, we control for firm size (Size), measured as the logarithm of total assets, to account for scale effects and differences in resource endowments; leverage (Lev) [42,43], defined as the ratio of total liabilities to total assets, to capture financial constraints and risk-taking behavior; return on assets (ROA) and return on equity (ROE), reflecting firms’ profitability; asset turnover (ATO), measuring operational efficiency; Tobin’s Q (TobinQ) [42], capturing market valuation and growth expectations; and R&D investment (Rdspendsum) [44], measured by R&D expenditures, to control for firms’ innovation input intensity.

Together, these variables provide a comprehensive control for firms’ financial conditions, operational performance, and innovation-related investment characteristics.

3.3. Model Settings

The model incorporates both firm fixed effects λi and year fixed effects. Firm fixed effects control for time-invariant unobserved heterogeneity at the firm level, such as persistent technological capabilities, managerial quality, organizational routines, and historically formed innovation trajectories [45]. Year fixed effects absorb macroeconomic conditions, institutional changes, and technology cycles that affect all firms simultaneously. Under this specification, identification primarily relies on within-firm variations in network centrality over time and the associated changes in innovation performance. By exploiting firms’ intertemporal variation, the model substantially mitigates omitted-variable bias arising from cross-sectional heterogeneity. This identification strategy is standard in firm-level innovation research, particularly in contexts where innovation outcomes are strongly shaped by persistent firm-specific characteristics.

To ensure robust statistical inference, we employ heteroskedasticity-robust standard errors with two-way clustering at the firm and industry levels. Clustering at the firm level allows for serial correlation in error terms within firms over time, while clustering at the industry level accounts for correlated shocks shared by firms within the same industry, such as technological paradigm shifts, regulatory changes, or competitive dynamics. This two-way clustering approach follows standard practice in applied microeconometric studies and provides reliable inference in the presence of firm–industry correlation structures.

In sum, by estimating a panel model with firm and year fixed effects, employing two-way clustered robust standard errors, and considering multiple measures of network centrality, this study systematically identifies the impact of firms’ positions in innovation networks on innovation performance from multiple structural dimensions.

4. Results

4.1. Baseline Analysis

4.1.1. Sample Clarification

Table 1. Reports descriptive statistics for the key variables based on the baseline regression sample.

	Count	Mean	sd	Min	Max
Betw_cite	28,066	0.0006	0.0036	0.000	0.171
Constraint	28,066	−0.1703	0.3837	−4.000	−0.003
KF_cross_crossdomain	26,005	0.4973	0.3080	0.000	1.000
KF_cross_techdist	26,005	0.3626	0.2413	0.000	1.000
Diversity_ext	24,504	3.5207	1.5129	0.000	9.852
Hhi_ext	25,740	0.9440	0.6564	−0.000	7.214
DT	28,188	1.5668	1.4654	0.000	6.306
Size	28,215	21.5753	1.4699	14.116	28.806
Lev	28,215	0.4172	0.2014	0.008	1.545
Roa	28,215	0.0390	0.0816	−1.859	1.285
Roe	28,215	0.0413	0.8650	−85.647	8.715
Ato	28,215	0.6529	0.4749	0.000	12.373
TobinQ	28,215	2.0928	1.5051	0.625	29.167
Rdspendsum	28,215	2.6155 × 108	1.2068 × 109	1800.000	7.384 × 1010

Note: The “count” column reports the number of non-missing observations for each variable. Differences in sample size arise because patent-based network measures and mechanism variables require additional information for construction. All descriptive statistics are reported using the baseline regression sample, while subsequent analyses employ the largest available sample conditional on the variables included. Accordingly, the effective sample size varies across empirical specifications.

reports descriptive statistics for the key variables in the broad baseline dataset, and the count column reports the number of non-missing observations for each variable. Effective sample sizes differ across subsequent tables because different equations require different variables. In particular, some auxiliary equations—such as certain IV first-stage regressions and mediation first-stage regressions—do not require non-missing innovation-efficiency measures and may therefore use a larger sample than the baseline regressions. By contrast, some mechanism specifications require additional patent-based variables, such as cross-domain recombination or organizational boundary-spanning indicators, whose construction depends on more detailed citation information and therefore reduces the available sample size. Accordingly, differences in N across tables reflect specification-specific variable availability rather than arbitrary changes in the underlying population.

Table 1 reports descriptive statistics for the key variables. The core regression sample contains 28,215 firm-year observations for financial controls, while patent-based network and mechanism variables have slightly smaller counts (24,504–28,188), reflecting missing values when constructing citation-network measures. Network-position indicators show strong dispersion and skewness: betweenness centrality (Betw_cite) is small on average (mean = 0.0006, sd = 0.0036) but has a pronounced right tail (max = 0.171), implying that most firms have limited brokerage roles, whereas a small subset occupies highly central intermediary positions. The inverse-constraint proxy (Constraint) also varies widely (mean = −0.1703, sd = 0.3837; min = −4.000), indicating substantial heterogeneity in structural-hole advantages across firms. The two cross-domain recombination measures span the full [0,1] interval—KF_cross_crossdomain averages 0.4973 and KF_cross_techdist averages 0.3626—suggesting meaningful variation in both the share of cross-domain external citations and the technological distance of absorbed external knowledge. Boundary-spanning measures likewise exhibit wide ranges (Diversity_ext mean = 3.5207, max = 9.852; Hhi_ext mean = 0.9440, max = 7.214), consistent with sizable differences in the depth and concentration of external knowledge sourcing. Digital transformation (DT) shows substantial dispersion (mean = 1.5668, max = 6.306). Control variables take plausible values but are heterogeneous—firm size (Size) varies markedly, leverage averages 0.4172 with values exceeding one, profitability ratios include extreme observations (roe min = −85.647), and both Tobin’s Q (max = 29.167) and R&D expenditure (Rdspendsum max = 7.384 × 10¹⁰) exhibit strong right-skewness—highlighting large cross-firm differences in scale, financial conditions, market valuation, and innovation input intensity.

4.1.2. Baseline Estimation Results

Table 2. Baseline results and robustness checks.

	(1)	(2)	(3)	(4)
	Innoeff1	Innoeff2	Innoeff1	Innoeff2
L.Betw_cite	1.265768 ***	1.345968 ***
	(0.224)	(0.235)
L2.Betw_cite			0.895587 ***	0.938078 ***
			(0.162)	(0.173)
N	28,215	28,215	24,828	24,828
R-sq	0.703	0.671	0.709	0.678
Controls	Yes	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes	Yes

Note: Robust standard errors two-way clustered by firm and industry are reported in parentheses. *** denote significance at the 1% level. All specifications include firm and year fixed effects. Control variables are included in all regressions.

reports the baseline regression results on the relationship between firms’ positions in the innovation network and innovation performance. Given that firms’ network positions are themselves outcomes of past innovation activities and knowledge interactions, we focus on lagged measures of network centrality to mitigate concerns related to simultaneity and reverse causality. Across specifications, lagged betweenness centrality exhibits a positive and statistically significant association with both innovation efficiency and innovation output. Specifically, firms occupying more central intermediary positions in the patent citation network achieve significantly higher innovation efficiency, measured by two alternative indicators, even after controlling for firm fixed effects, year fixed effects, and a comprehensive set of firm-level covariates. Figure 2 provides a binned comparison of innovation efficiency across lagged betweenness centrality under the L1 and L2 specifications.

In the regression notation, L. denotes a lag operator, c. indicates a continuous variable in interaction notation, and # denotes an interaction term.

Importantly, the positive effects of network centrality remain statistically significant when further lags are introduced. The persistence of the estimated coefficients on both first- and second-order lagged centrality measures suggests that the influence of network position on innovation performance is not transitory, but unfolds over time. This dynamic pattern is consistent with the innovation process, in which changes in firms’ positions within knowledge networks require time to translate into observable innovation outcomes, such as patent applications.

To assess the robustness of these findings,

Table 3. Substitution variables and robustness tests.

	(1)	(2)	(3)	(4)
	Innoeff1	Innoeff2	Innoeff1	Innoeff2
L.Constraint	0.005942 ***	0.006990 ***
	(0.001)	(0.002)
L2.Constraint			0.003733 ***	0.004135 ***
			(0.001)	(0.001)
N	28,215	28,215	24,828	24,828
R-sq	0.703	0.672	0.709	0.678
Controls	Yes	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes	Yes

Note: Robust standard errors two-way clustered by firm and industry are reported in parentheses. *** denote significance at the 1% level. All specifications include firm and year fixed effects. Control variables are included in all regressions.

replaces betweenness centrality with the inverse of structural constraint (i.e., the negative of Burt’s constraint), which captures firms’ access to structural holes from an alternative network perspective. Figure 3 presents a binned comparison of innovation efficiency across lagged structural-hole advantage under the L1 and L2 specifications. The estimated coefficients on the lagged alternative centrality measures remain positive and highly significant across all specifications, with magnitudes and significance levels comparable to the baseline results. This consistency indicates that the observed relationship is not driven by a particular operationalization of network centrality but rather reflects a robust effect of firms’ structural positions in the innovation network.

Taken together, the baseline and robustness results suggest that firms’ advantages in innovation performance stem from their structural embeddedness in the knowledge network, rather than from any single metric of network centrality. Firms occupying key intermediary or brokerage positions are better able to access, filter, and recombine heterogeneous external knowledge, thereby improving both the efficiency with which R&D inputs are transformed into innovative outputs and the overall scale of innovation outcomes. By relying on lagged centrality measures, the analysis strengthens the causal interpretation of the results and enhances the credibility of the conclusion that innovation network structure exerts a substantive and systematic influence on firm-level innovation performance. These findings provide a solid foundation for the subsequent analyses addressing endogeneity concerns and underlying mechanisms.

4.1.3. Addressing Endogeneity: Instrumental Variable Evidence

To further address potential endogeneity arising from reverse causality and time-varying unobservables, we implement two alternative instrumental-variable specifications for lagged betweenness centrality. In the first specification, L.Betw_cite is instrumented by its third-order lag, L3.Betw_cite. This instrument exploits the persistence of firms’ brokerage positions in the patent citation network: firms’ intermediary positions are path-dependent over time, so earlier network positions remain strongly associated with subsequent centrality. At the same time, conditional on firm fixed effects, year fixed effects, and time-varying firm characteristics, the third-order lag is less likely to affect current innovation efficiency except through its effect on the firm’s current brokerage position.

In the second specification, L.Betw_cite is instrumented by Ind_con_mean, defined as the industry-year mean of ${Constraint}_{it}$, the inverse structural-hole measure introduced earlier in the paper. Since larger values of ${Constraint}_{it}$ indicate stronger brokerage advantages, this reflects an industry-year network environment in which brokerage opportunities are, on average, more abundant. The relevance of this instrument follows from the fact that firms’ betweenness centrality is shaped not only by their own network behavior, but also by the broader opportunity structure of the industry-level citation network in which they are embedded. When the average inverse structural-hole level in a given industry-year is higher, the surrounding knowledge network is, on average, less redundant and more conducive to intermediary positioning, making it more likely that an individual firm can occupy a brokerage role in the citation network. The exclusion restriction relies on the distinction between the external industry-year network environment and the focal firm’s own innovation outcome. After controlling for firm fixed effects, year fixed effects, and time-varying firm characteristics, Ind_con_mean is not intended to proxy for the focal firm’s internal innovation capability, R&D effort, or patenting quality per se. Instead, it captures an external structural condition that affects firm-level innovation efficiency primarily by altering the firm’s opportunity to occupy an intermediary network position. For this reason, its influence on current innovation efficiency is expected to operate mainly through L.Betw_cite rather than through an independent direct channel.

Table 4. Instrumental Variable Evidence.

	(1)	(2)	(3)	(4)	(5)	(6)
	First Stage: L.Betw_cite	Second Stage: Innoeff1	Second Stage: Innoeff2	First Stage: L.Betw_cite	Second Stage: Innoeff1	Second Stage: Innoeff2
L3.Betw_cite	0.490492 ***
	(0.098)
L.Betw_cite		1.205501 **	1.322901 **		20.515418 *	27.003633 **
		(0.569)	(0.593)		(11.026)	(12.548)
Ind_con_mean				0.000545 ***
				(0.000)
N	25,753	21,790	21,790	33,403	28,215	28,215
R-sq	0.866	0.039	0.033	0.791	−0.508	−0.702
Cragg–Donald Wald F statistic	6621.634			25.246
Kleibergen–Paap rk Wald F statistic	21.651			8.209
Kleibergen–Paap rk LM statistic	4.488			8.052
p-value of K-P LM statistic	(0.0341)			(0.0045)
Controls	Yes	Yes	Yes	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes	Yes	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. For IV specifications with clustered standard errors, instrument relevance is assessed primarily using the Kleibergen–Paap statistics. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. All specifications include firm and year fixed effects. Control variables are included in all regressions. Negative R-sq values in the second stage may arise in IV estimation and do not imply model misspecification.

reports the corresponding identification diagnostics. For the specification using L3.Betw_cite, the Kleibergen–Paap rk LM statistic is 4.488 (p = 0.0341), rejecting the null of underidentification, and the Kleibergen–Paap rk Wald F statistic is 21.651, indicating strong first-stage relevance. For the specification using Ind_con_mean, the Kleibergen–Paap rk LM statistic is 8.052 (p = 0.0045), also rejecting underidentification, while the Kleibergen–Paap rk Wald F statistic is 8.209, suggesting that the instrument retains meaningful first-stage relevance, although its identifying strength is weaker than that of L3.Betw_cite. Because these are alternative just-identified IV specifications with clustered standard errors, we focus primarily on the Kleibergen–Paap statistics when assessing instrument relevance.

Across both IV specifications, the second-stage coefficients on L.Betw_cite remain positive and statistically significant for both innovation-efficiency measures. This pattern is consistent with the baseline fixed-effects results and suggests that the positive association between firms’ brokerage positions in the knowledge citation network and innovation efficiency is unlikely to be driven solely by reverse causality or omitted-variable bias. Overall, the IV evidence provides additional support for our baseline interpretation that intermediary network positions contribute to firm-level innovation efficiency.

4.1.4. Addressing Endogeneity: Heckman Two-Stage Test (Auxiliary Evidence)

As an auxiliary check for potential sample-selection bias, we implement a Heckman two-stage model. In the first-stage Probit equation, the selection variable select_ext equals one if a firm has any external patent citations in a given year and zero otherwise. The exclusion restriction is Ind_ext_supply, constructed as the leave-one-out industry-year mean of external patent citations (n_cit_external). Formally, for firm i in industry k and year t,

$${Ind\text{\_}ext\text{\_}supply}_{ikt}={\displaystyle \frac{{\sum }_{j\in (k,t)}{n\text{\_}cit\text{\_}external}_{jkt}-{n\text{\_}cit\text{\_}external}_{jkt}}{N_{kt}-1}}$$

(14)

where $N_{kt}$ is the number of firms in industry k and year t. This variable captures the intensity of external knowledge sourcing in the focal firm’s industry-year environment, excluding the firm’s own contribution. The underlying logic is that a richer external knowledge-search environment increases the likelihood that a firm engages in external patent citation behavior, while it is not intended to directly determine the focal firm’s innovation efficiency conditional on firm characteristics and fixed effects. The Heckman specification is therefore used only as a supplementary robustness check rather than as the main identification strategy.

Table 5. Heckman Two-Stage Test.

	(1)	(2)	(3)
	Selection (Probit)	Outcome: Inno_eff1	Outcome: Inno_eff2
main
Ind_ext_supply	0.000307 *
	(0.000)
imr		0.002060	−0.000367
		(0.017)	(0.020)
N	33,548	25,864	25,864
R-sq		0.696	0.660
Controls	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. * denote significance at the 10% level. All specifications include firm and year fixed effects. Control variables are included in all regressions.

reports the results of the Heckman two-stage model used as an auxiliary check for potential sample-selection bias.

4.2. Further Analysis

4.2.1. Mechanism Analysis: Cross-Domain Knowledge Recombination

To further elucidate how firms’ structural positions in innovation networks translate into superior innovation performance, this study investigates two complementary mediating mechanisms: cross-domain knowledge recombination and organizational boundary spanning. These two mediating mechanisms reflect complementary dimensions of knowledge integration: cross-domain recombination captures what knowledge is combined, while organizational boundary spanning reflects how firms structure and govern external knowledge interactions.

On the one hand, innovation increasingly depends on the recombination of heterogeneous knowledge elements across technological domains. Firms occupying brokerage positions in knowledge networks are expected to face lower cognitive and informational barriers in accessing technologically distant knowledge, thereby facilitating cross-domain recombination. On the other hand, effective knowledge recombination often requires sustained interactions across organizational boundaries. Network centrality may therefore enhance innovation efficiency not only by expanding the scope of accessible knowledge, but also by deepening firms’ engagement with external knowledge sources.

The mechanism analysis follows a stepwise mediation framework. In the first step, where mediating variables are regressed on lagged network centrality, the estimation relies on the maximum available sample for which the mediator and explanatory variables are observed, as innovation efficiency measures are not required at this stage. In subsequent steps, which examine the association between mediators and innovation efficiency, the sample is restricted to firm-year observations with non-missing innovation performance indicators. As a result, the effective sample size varies across columns in

Table 6. Mechanism Analysis: Cross-domain Knowledge.

	(1)	(2)	(3)	(4)	(5)	(6)
	KF_cross_crossdomain	Innoeff1	Innoeff2	KF_cross_techdist	Innoeff1	Innoeff2
L.Betw_cite	2.593941 ***	1.184295 ***	1.237706 ***	1.872149 ***	1.187933 ***	1.243096 ***
	(0.760)	(0.206)	(0.207)	(0.506)	(0.207)	(0.209)
KF_cross_crossdomain		0.012727 ***	0.015891 ***
		(0.001)	(0.002)
kf_cross_techdist					0.015633 ***	0.019061 ***
					(0.002)	(0.002)
N	29,536	25,879	25,879	29,536	25,879	25,879
R-sq	0.306	0.700	0.667	0.303	0.700	0.667
Controls	Yes	Yes	Yes	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes	Yes	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. *** denote significance at the 1% level. All specifications include firm and year fixed effects. Control variables are included in all regressions.

and

Table 7. Organizational boundary-spanning diversity.

	(1)	(2)	(3)	(4)	(5)	(6)
	Diversity_ext	Innoeff1	Innoeff2	Hhi_ext	Innoeff1	Innoeff2
L.Betw_cite	35.507577 ***	0.510343 ***	0.470116 ***	22.768821 ***	0.778018 ***	0.766630 **
	(4.781)	(0.139)	(0.134)	(8.128)	(0.280)	(0.300)
Diversity_ext		0.020311 ***	0.023161 ***
		(0.001)	(0.001)
Hhi_ext					0.020874 ***	0.024193 ***
					(0.001)	(0.002)
N	28,215	28,215	24,828	29,175	25,604	25,604
R-sq	0.703	0.672	0.709	0.628	0.711	0.679
Controls	Yes	Yes	Yes	Yes	Yes	Yes
Individual fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
Time fixed effect	Yes	Yes	Yes	Yes	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. ***, and ** denote significance at the 1%, and 5% levels, respectively. All specifications include firm and year fixed effects. Control variables are included in all regressions.

.

Cross-domain knowledge recombination. Panels A and B of Table 6 examine cross-domain knowledge recombination as a mediating channel, using two alternative indicators: the cross-domain citation ratio and technological distance.

Columns (1)–(3) show that lagged betweenness centrality is positively and significantly associated with both measures of cross-domain knowledge sourcing, indicating that firms in brokerage positions are more likely to draw upon technologically distant knowledge domains. This finding suggests that intermediary firms are structurally advantaged in bridging otherwise disconnected technological areas.

When the cross-domain recombination measures are included in the innovation efficiency regressions, they enter with positive and highly significant coefficients for both innovation efficiency indicators. At the same time, the coefficient on betweenness centrality remains positive and statistically significant, albeit reduced in magnitude. This pattern is consistent with a partial mediation effect, implying that network centrality enhances innovation efficiency partly by facilitating cross-domain knowledge recombination. This interpretation aligns with the recombinant innovation literature, which emphasizes that innovation breakthroughs often arise from combining heterogeneous knowledge elements across technological domains, and that access to technologically distant knowledge expands the space of potential recombinations and increases innovation productivity [26,46].

4.2.2. Mechanism Analysis: Organizational Boundary-Spanning Diversity

Organizational boundary-spanning diversity: Table 7 examines organizational boundary spanning as a complementary mechanism through which firms’ network centrality influences innovation efficiency. Beyond technological recombination, innovation performance also depends on how firms organize and sustain interactions with external knowledge providers across organizational boundaries. Firms occupying brokerage positions in knowledge networks are expected not only to access external knowledge more frequently, but also to shape the depth and structure of their boundary-spanning engagements in ways that facilitate effective knowledge absorption and recombination.

The results in Table 7 provide direct support for Hypothesis H2b, which posits that cross-organizational knowledge flows mediate the relationship between network centrality and innovation efficiency. Firms with higher betweenness centrality exhibit significantly greater organizational boundary-spanning diversity, indicating that brokerage positions facilitate sustained interactions with heterogeneous external knowledge holders. In turn, both the depth of boundary-spanning engagement (Diversity_ext) and the structure of external knowledge sourcing (Hhi_ext) are positively associated with innovation efficiency. Importantly, the inclusion of these boundary-spanning indicators attenuates the estimated effect of betweenness centrality, consistent with a partial mediation effect.

Taken together, these findings suggest that firms’ advantages derived from central network positions operate in part through their ability to organize and manage cross-organizational knowledge relationships more effectively. This mechanism is consistent with the boundary-spanning and open innovation literature, which argues that sustained engagement with diverse external knowledge sources enhances absorptive capacity, facilitates knowledge integration, and improves innovation efficiency, particularly when firms are able to structure and govern boundary-spanning interactions effectively [21,47,48].

4.2.3. Moderating Role of Digital Transformation

The results reported in

Table 8. Moderating effect: digital transformation.

	(1)	(2)
	Innoeff1	Innoeff2
L.Betw_cite	1.673392 ***	1.806709 ***
	(0.256)	(0.259)
DT	0.000428	0.000514
	(0.001)	(0.001)
c.DT#cL.Betw_cite	−0.281258 ***	−0.318037 ***
	(0.070)	(0.070)
N	28187	28187
R-sq	0.703	0.672
Controls	Yes	Yes
Individual fixed effects	Yes	Yes
Time fixed effect	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. *** denote significance at the 1% level. All specifications include firm and year fixed effects. Control variables are included in all regressions.

indicate a significant moderating role of digital transformation in the relationship between network centrality and innovation efficiency. While betweenness centrality exhibits a positive and significant main effect on innovation efficiency, the interaction term between digital transformation and betweenness centrality is negative and statistically significant across both innovation efficiency measures. This suggests that the marginal effect of network centrality on innovation efficiency decreases as firms’ level of digital transformation increases. Figure 4 presents a heatmap of the joint association of lagged betweenness centrality and digital transformation with innovation efficiency. Figure 5 further illustrates how the marginal effect of lagged betweenness centrality on innovation efficiency varies across different levels of digital transformation.

Substantively, these findings point to a substitution effect between digital transformation and network-based brokerage advantages. As firms deepen their digital transformation, they enhance internal capabilities for information processing, knowledge integration, and cross-domain recombination, thereby becoming less dependent on occupying structurally central positions in inter-firm knowledge networks to access and exploit external knowledge. This interpretation is consistent with the digital innovation literature, which emphasizes that digital technologies reduce search and coordination costs and enable firms to integrate dispersed knowledge beyond traditional relational embeddedness [49,50]. Consequently, the incremental innovation benefits associated with brokerage positions are weakened as digital capabilities improve.

Economic interpretation. Taken together, the analyses show that the innovation advantages of network centrality arise from qualitative changes in firms’ knowledge integration processes, rather than from a simple expansion in the scale of accessible knowledge. Brokerage positions enable firms to recombine technologically distant knowledge and to organize more effective interactions across organizational boundaries, thereby reducing search and coordination costs and improving the efficiency of R&D-to-innovation transformation. These mechanisms indicate that network centrality shapes not only how much knowledge firms access, but more fundamentally how knowledge is sourced, structured, and recombined. At the same time, the moderating effect of digital transformation suggests that technology-enabled knowledge integration can partially substitute for network-based advantages, attenuating the marginal role of centrality as digital capabilities deepen. Overall, the findings underscore the context-dependent role of innovation network structure in driving firm-level innovation efficiency.

To further address the reviewer’s concern that the moderating effect of digital transformation may vary across different stages of digitalisation, we estimate an additional nonlinear specification by introducing a quadratic interaction term between lagged betweenness centrality and digital transformation. The results show that the linear interaction between lagged betweenness centrality and digital transformation remains negative and highly significant for both innovation-efficiency measures, while the quadratic interaction term is positive and statistically significant in both models. This pattern indicates that the moderating effect of digital transformation is nonlinear rather than purely linear. Specifically, as digital transformation increases, the positive effect of brokerage centrality on innovation efficiency is initially weakened, but this attenuating effect diminishes at higher levels of digitalisation. In other words, digital transformation predominantly reduces the brokerage premium, yet this substitution effect does not intensify indefinitely and appears to moderate once firms reach more advanced stages of digitalisation. Overall, the results support a stage-dependent interpretation of the moderation effect while confirming that the dominant pattern remains a negative marginal moderation over most of the observed range. Detailed results are reported in Appendix A,

Table A1. Nonlinear Moderation Test of Digital Transformation.

	(1)	(2)
	Innoeff1	Innoeff2
L.Betw_cite	1.261661 ***	1.924813 ***
	(0.215)	(0.262)
DT	0.001267 *
	(0.001)
DT2	−0.000760
	(0.000)
cL.Betw_cite#c.DT	−0.405686 ***
	(0.087)
cL.Betw_cite#c.DT2	0.087939 **
	(0.038)
cL.Betw_cite#c.DT		−0.522548 ***
		(0.142)
cL.Betw_cite#c.DT2		0.056781 *
		(0.034)
N	28,187	28,187
R-sq	0.704	0.672
Controls	Yes	Yes
Individual fixed effects	Yes	Yes
Time fixed effect	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. All specifications include firm and year fixed effects. Control variables are included in all regressions.

.

4.3. Heterogeneity

4.3.1. State-Ownership

Table 9. Heterogeneity: state-ownership.

	(1)	(2)
	Innoeff1	Innoeff2
L.Betw_cite	1.259244 ***	1.338094 ***
	(0.403)	(0.440)
soe#cL.Betw_cite	0.168740	0.191849
	(0.424)	(0.468)
p-value_soe_vs_nonsoe	0.691539	0.682720
N	27,555	27,555
R-sq	0.702915	0.671626
Controls	Yes	Yes
Individual fixed effects	Yes	Yes
Time fixed effect	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. *** denote significance at the 1% level. All specifications include firm and year fixed effects. Control variables are included in all regressions. p-values report formal coefficient-difference tests based on linear combinations of interaction terms.

reports the formal ownership-heterogeneity tests based on full-sample interaction models. The coefficient on lagged betweenness centrality is positive and statistically significant in both columns, indicating that occupying brokerage positions in innovation networks improves subsequent innovation efficiency across ownership groups. However, the interaction term between the SOE indicator and lagged betweenness centrality is not statistically significant, and the coefficient-difference tests do not reject the null hypothesis that the effects are equal for SOEs and non-SOEs. Therefore, the revised evidence does not support a statistically significant ownership difference in the effect of brokerage centrality on innovation efficiency.

Although the estimated coefficients are numerically larger for SOEs, this pattern should be interpreted cautiously. One possible explanation is that listed SOEs, having undergone market-oriented reforms, may be better positioned to convert network-based brokerage advantages into innovation outcomes through stronger coordination capacity, more stable access to strategic resources, and deeper embeddedness in formal innovation platforms. However, because the formal interaction tests do not show statistically significant coefficient differences, this explanation remains suggestive rather than conclusive.

Overall, Table 9 is better interpreted as showing a broadly shared positive network effect across SOEs and non-SOEs, rather than as evidence of clear ownership-based heterogeneity.

4.3.2. Regional Heterogeneity

Table 10. Regional heterogeneity: formal coefficient-difference tests.

	(1)	(2)
	Innoeff1	Innoeff2
L.Betw_cite	1.176305 ***	1.255134 ***
	(0.203)	(0.218)
Mid	−0.017598 *	−0.018608 *
	(0.009)	(0.010)
West	−0.036198 *	−0.035746 *
	(0.020)	(0.020)
Mid#cL.Betw_cite	0.842790	0.902997
	(0.899)	(0.892)
West#cL.Betw_cite	2.043714 **	2.061111 *
	(0.947)	(1.054)
p-value_Mid_vs_East	0.351459	0.314496
p-value_West_vs_East	0.033925	0.054047
p-value_Mid_vs_West	0.440360	0.453243
N	28,215	28,215
R-sq	0.703309	0.671596
Controls	Yes	Yes
Individual fixed effects	Yes	Yes
Time fixed effect	Yes	Yes

Note: Robust standard errors clustered at the firm and industry levels are reported in parentheses. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. All specifications include firm and year fixed effects. Control variables are included in all regressions. p-values report formal coefficient-difference tests based on linear combinations of interaction terms.

reports the formal regional-heterogeneity tests based on full-sample interaction models. The coefficient on lagged betweenness centrality is positive and statistically significant in both columns, indicating that for the baseline group of eastern firms, occupying brokerage positions in innovation networks is associated with higher subsequent innovation efficiency. The interaction term between the middle-region indicator and lagged betweenness centrality is positive but statistically insignificant for both innovation-efficiency measures, and the corresponding coefficient-difference tests do not reject the null of equal coefficients between the central and eastern regions. By contrast, the interaction term between the western-region indicator and lagged betweenness centrality is positive in both columns and statistically distinguishable from the eastern-region baseline, indicating that the effect of brokerage centrality is stronger in the western region than in the eastern region. However, the difference between the central and western regions is not statistically significant.

This pattern suggests that the positive effect of brokerage centrality on innovation efficiency is broadly present across regions, but its strength is not uniform. In particular, the innovation return to brokerage position appears to be stronger in the western region than in the eastern region. A possible explanation is that firms in western regions often face relatively greater constraints in direct access to high-quality innovation resources, formal research platforms, and dense local innovation networks. Under such conditions, occupying brokerage positions may generate larger marginal returns by helping firms bridge otherwise disconnected knowledge sources and external collaborators. By contrast, although the coefficients in the central region differ numerically from those in the eastern region, the formal tests do not support a statistically distinguishable regional effect.

Overall, Table 10 is better interpreted as showing a common positive network effect across regions, with evidence of a stronger brokerage premium in the western region relative to the eastern region, but limited evidence of a distinct middle-region effect.

5. Conclusions and Discussion

5.1. Conclusions

This paper investigates how firms’ structural positions in inter-firm patent citation networks shape innovation performance, using a large panel of Chinese listed companies and detailed patent citation data. By constructing a firm-level knowledge-flow network and measuring network centrality, the study provides systematic evidence that occupying brokerage positions in innovation networks significantly enhances firms’ innovation efficiency. These effects are robust across alternative measures of innovation performance and network centrality, persist when lagged centrality is used, and remain valid after addressing endogeneity through instrumental variable and selection-correction approaches.

Beyond establishing a robust centrality–innovation link, the analysis uncovers the underlying mechanisms through which network position translates into innovation advantages. Firms with higher betweenness centrality are more likely to engage in cross-domain knowledge recombination and to organize deeper and more structured interactions across organizational boundaries. These channels partially mediate the effect of network centrality on innovation efficiency, indicating that central positions matter not merely because they increase access to external knowledge, but because they reshape how knowledge is sourced, integrated, and recombined within firms.

The paper further shows that the innovation benefits of network centrality are contingent on technological and institutional contexts. Digital transformation weakens the marginal effect of network centrality, suggesting that technology-enabled knowledge integration can partially substitute for relational advantages embedded in innovation networks. Moreover, heterogeneity analyses indicate that the positive effect of brokerage centrality is broadly shared across ownership groups, while regional heterogeneity is more clearly reflected in a stronger brokerage premium in the western region relative to the eastern region.

5.2. Discussion

The findings of this study deepen the theoretical dialogue on network embeddedness, knowledge recombination, and digital transformation in three main ways. First, rather than treating network position as a broad relational advantage, this paper shows that the economically meaningful structural feature is firms’ brokerage-oriented position in the patent citation network. The positive effect of betweenness centrality on innovation efficiency suggests that what matters is not simply being embedded in a network, but occupying intermediary positions that connect otherwise weakly linked knowledge domains and innovation actors. In this sense, the study refines the network embeddedness literature by linking brokerage structure more explicitly to the efficiency dimension of innovation performance.

Second, the study clarifies the mechanisms through which brokerage position is translated into innovation efficiency. The results show that the effect of network centrality is not a mechanical direct effect of “more connectivity”, but operates through two analytically distinct channels: cross-domain knowledge recombination and organizational boundary spanning. The first mechanism highlights the role of technologically heterogeneous knowledge inputs and distant recombination, whereas the second emphasizes the diversity and governance of external knowledge-bearing entities. This distinction extends prior research on knowledge spillovers by specifying how network-based advantages are converted into innovation outcomes.

Third, the study shows that the value of brokerage position is context-dependent. In particular, digital transformation weakens the marginal effect of brokerage centrality, indicating that internal digital coordination and knowledge-integration capabilities can partially substitute for the brokerage premium associated with external network positions. This finding suggests that network advantage should not be understood as fixed, but as contingent on firms’ organizational and technological capabilities. In this sense, digital transformation does not reduce firms’ innovation capability per se; rather, it reduces the extent to which innovation efficiency depends on occupying brokerage positions in external knowledge networks.

These findings also provide a more nuanced interpretation of the heterogeneous results. The revised coefficient-difference tests indicate that the positive effect of brokerage centrality is broadly shared across ownership groups, while providing limited evidence of a statistically significant ownership gap. This suggests that the innovation value of brokerage position is not confined to a particular ownership type, but reflects a more general structural advantage in knowledge networks. By contrast, the regional analysis indicates that the brokerage premium is more pronounced in the western region than in the eastern region, whereas the central region does not differ significantly from the eastern benchmark. A plausible interpretation is that in relatively resource-constrained innovation environments, brokerage positions generate larger marginal returns by helping firms bridge otherwise disconnected knowledge sources and external collaborators.

The practical implications of these findings should therefore be closely aligned with the identified mechanisms. From a policy perspective, innovation policy should not focus solely on increasing the number of network ties or expanding the volume of knowledge exchange. Instead, policies should encourage cross-domain knowledge linkage and boundary-spanning collaboration so that firms can occupy more valuable brokerage positions in the innovation system. This is particularly important in regions with relatively limited local innovation resources, where structural brokerage can compensate for gaps in the surrounding knowledge environment. From a managerial perspective, firms should recognize that the value of network centrality lies not only in accessing more knowledge, but in organizing how heterogeneous knowledge is selected, integrated, and recombined. Managers should therefore invest in both boundary-spanning governance capabilities and internal digital integration capabilities. The former helps firms benefit from brokerage positions, while the latter reduces excessive dependence on external relational advantages and enables more flexible innovation coordination in the digital era.

Limitations: This study has several limitations. First, the sample covers Chinese A-share listed companies from 2007 to 2022, so the findings should be interpreted within this temporal scope and may not fully reflect more recent post-2022 developments in innovation networks and digital transformation. Second, although the patent citation network provides an observable proxy for knowledge flows, it cannot fully capture informal, tacit, or non-patented knowledge exchange. Third, while the empirical strategy combines fixed effects, lag structures, instrumental-variable tests, and supplementary selection corrections, no single empirical design can fully eliminate all endogeneity concerns. Future research may extend the sample period, compare alternative innovation-efficiency measures, and examine broader knowledge-network settings.

Another limitation concerns the sample end year. Because patent authorization and citation realization are subject to grant lag, patent-based indicators for the most recent years are not yet fully observed. The raw annual data show a sharp decline in usable patent citation records after 2022, suggesting that extending the sample further would introduce instability and truncation bias into the key variables. Future research may revisit the analysis when newer patent cohorts become more fully observed.