Too much of a good thing? Network brokerage within and between regions and innovation performance

ABSTRACT The paper offers a multilevel understanding of the effect of brokerage on regional innovation. We develop a typology of regional networks based on the extent to which the inventors of a region connect the otherwise disconnected group of local inventors (internal brokerage) or connect the local network with other regions (external boundary-spanning). Using data on regional innovation performance and co-inventing networks within and between US metropolitan statistical areas (MSAs) between 2000 and 2014, we show that configurations balancing high (low) internal brokerage and low (high) external boundary-spanning lead to higher innovation performance than those where brokerage occurs at both levels of analysis.

Collaboration networks are not confined within regional boundaries. Rather, actors often collaborate simultaneously with other actors within and outside their regions (Bathelt et al., 2004). These multilevel nature of collaboration networks represent a key feature that may advance our understanding of innovation processes. Yet, empirical evidence on multilevel innovation networks remains scant (Paruchuri et al., 2019), with most work focusing either on intra-regional networks (e.g., Fleming et al., 2007;Saxenian, 1996) or on connections between regions (e.g., Breschi & Lenzi, 2015;Miguelez & Moreno, 2018). Our study makes a step forward in this direction by conducting a multilevel analysis to understand the effects of brokerage on regional innovation performance. Brokerage enhances regional innovation performance by both increasing knowledge flows between distinct groups within the same region (determined by the structure of the internal collaboration network) and knowledge flows that occur with actors located in other regions (through collaboration ties that cut across regional boundaries). 1 Accordingly, we develop a theoretical framework explaining how regional innovation performance varies depending on whether actors span structural holes in the collaboration network within the region or connect the local network with actors located outside the region. Elaborating on foundational work on brokerage (Allen, 1977;Gould & Fernandez, 1989;Marsden, 1982), we propose that brokerage in internal and external networks implies trade-offs between different roles (coordinator, gatekeeper and representative). On these premises, we contend that the network configurations with low internal brokerage and high external boundary-spanning, or high internal brokerage and low external boundary-spanning, lead to higher innovation performance than the configurations with low internal brokerage and low external boundary-spanning, or high internal closure and high external boundary-spanning.
We test our hypothesis in a longitudinal study of collaboration networks within and between US metropolitan statistical areas (MSAs) between 2000 and 2014. In line with previous studies (Fleming et al., 2007), we capture collaboration networks using inventors' co-patenting relationships. We measure brokerage within regions by looking at the additive inverse of the average inventor constraint in the network within each MSA (Burt, 1992(Burt, , 2004, and brokerage between regions by looking at the connections maintained by inventors within the MSAs with those in other metropolitan areas in the United States (Breschi & Lenzi, 2015;Zhao & Anand, 2013).
Trade-offs between intra-and interregional knowledge spillovers and their effect on innovation has been analysed by previous research (e.g., Autant-Bernard, 2001;Kang & Dall'Erba, 2016). Our proposed framework differs from and extends these findings in three ways. First, we focus on collaborative spillovers, that is, those channelled by formal collaborations between actors, rather than inferring spillovers from the co-location of actors in a bounded geographical space. Second, while existing works studying collaboration networks looked primarily at the amount/ intensity of collaboration, captured by degree-centrality or tie strength (e.g., Coffano et al., 2017), our work looks at the structure of collaboration networks, focusing on brokerage within and between region. Finally, the use of longitudinal data on a large sample of MSAs allows us to capture the theorized effects controlling for alternative mechanisms proposed by past research (e.g., Anselin et al., 1997). By doing so, we offer a robust, more nuanced understanding of the trade-offs between different types of brokerage in spatially bounded collaboration networks.

SOURCES OF REGIONAL INNOVATION: FROM KNOWLEDGE SPILLOVERS TO COLLABORATION NETWORKS
Innovative activities tend to be concentrated in geographical space. Knowledge-based perspectives attribute this tendency to the presence of localized knowledge spillovers among co-located individuals and organizations (Feldman & Kogler, 2010;Malmberg & Maskell, 2002). Colocation enables the transmission of tacit and contextual knowledge through face-to-face contacts (Storper & Venables, 2004). Co-located actors are also more likely to share a common language and cultural context that favours communication, knowledge recombination and integration (Cohen & Levinthal, 1990;Uzzi, 1996). These processes are crucial to explaining heterogeneity in innovation performance between regions (Fleming, 2001;Weitzman, 1998).
Past research identified three factors affecting the magnitude of localized knowledge spillovers: the nature of the underlying knowledge, the type of actors participating in knowledge creation, and the physical distance between actors. For example, some studies (e.g., Audretsch & Feldman, 2004;Beaudry & Schiffauerova, 2009) looked at the relative benefits of specialized (diversified) knowledge basesthe concentration (dispersion) of co-located research activities in similar (different) industries. 2 Other scholars explored the different contribution of universities and private firms to the local knowledge production function, with studies documenting positive spillovers from research institutions to commercial ones (e.g., Anselin et al., 1997;Jaffe, 1989) while others identifying a weaker, less straightforward relationship (Co, 2002;Ó hUallacháin & Leslie, 2007;Riddel & Schwer, 2003). A third line of research focused on spatial distance as a possible moderator reconciling these contradictory findings. Kang and Dall'Erba (2016) found that industry spillovers benefit primarily metropolitan counties. In contrast, spillovers from universities are smaller but spatially homogeneous over long distances, affecting both metropolitan and rural counties. Autant-Bernard and LeSage (2011) found that industry research and development (R&D) creates positive spillovers both locally and in distant locations, especially in diversified regions.
While these studies advance our understanding of differences in regional innovation performance, they remain short of identifying the mechanism underlying knowledge spillovers. Their assumption that the simple co-location of agents is sufficient to guarantee intense knowledge spillovers has been criticized and empirically challenged (Breschi & Lissoni, 2001. Addressing this limitation, scholars embraced a structural approach, arguing that knowledge spillovers are channelled by formal collaboration networks within and between regions. Actors' embeddedness in the local (regional) and the distant (interregional) collaboration networks has a sizeable positive impact on a region's innovation performance (e.g., Bathelt et al., 2004;Coffano et al., 2017;Fleming et al., 2007).
Our study extends this line of research by focusing on the effects of network brokerage. By brokerage, we mean creating bridges between actors that have not been connected before (Burt, 1992;Gould & Fernandez, 1989;Marsden, 1982). Brokerage may occur between co-located actors or can span geographical boundaries. Drawing on multilevel network research (Paruchuri et al., 2019), our framework clarifies the role of brokers within and across regions, and sheds light on the trade-offs entailed in different brokerage configurations.

BROKERAGE WITHIN AND BETWEEN LOCATIONS
Local brokers 3 foster innovation by creating horizontal ties between otherwise disconnected actors belonging to separate closed groups or working in different organizations in the same location. They connect members of the regional community who never collaborated before, acting as 'coordinators' within their group (Gould & Fernandez, 1989;Marsden, 1982). In doing so, they foster the integration and recombination of local knowledge that underlie innovation.
Too much of a good thing? Network brokerage within and between regions and innovation performance 301 A substantial body of empirical research has validated the benefits of brokerage within regions. Fleming et al. (2007), in a study of co-inventing networks in the United States, found that brokers increased innovation within cities by shortening path lengths in the community, favouring the exchange of distant and diverse information. Cowan and Jonard's (2004) agent-based model demonstrated that decreased path lengths due to brokerage increase innovation-diffusion in a spatially bounded network. Evidence establishing the innovation benefits of local brokerage has prompted policy measures incentivizing actors to bridge groups within their local community (e.g., De Silva et al., 2018;Ponds et al., 2007).
Brokerage may also occur between different geographical locations (Bathelt et al., 2004;Breschi & Lenzi, 2015;Miguelez & Moreno, 2018;Sigler et al., 2019). External boundary-spanners fulfil the functions of mediation and diffusion (Allen, 1977;Morrison, 2008), connecting regional actors with individuals or organizations located in other regions. They act as 'gatekeepers'who access external knowledge and translate it to the members of their local communityor as 'representatives'who contact outsiders and promote local ideas to spur external impact (Gould & Fernandez, 1989;Martinus et al., 2021).
Prior research indicates that both mediation and diffusion processes increase regional innovation. Exposure to knowledge generated outside the region increases actors' absorptive capacity, defined as their ability to process and apply extra-regional knowledge to the problems they encounter, reducing the risk of technology redundancy and lock-in (Lazaric et al., 2008). The gatekeeper is also able to evaluate which technological trajectories are promising and should be pursued and which ones should be avoided (Reagans & Zuckerman, 2001), thus prompting effective idea selection and external championing. Empirical evidence supporting this view is abundant and consistent. Using US patent data, Breschi and Lenzi (2015) showed that the intensity of external linkages is positively associated with the renewal and expansion of the local knowledge base in a city. Using patent data and co-inventor networks in genomics between 1990 and 2006, Le Gallo and Plunket (2020) demonstrated that gatekeepers with direct access to non-local knowledge leads to higher team inventive performance, with benefits for the entire local community.
A multilevel typology Taken together, these studies underscore the importance of internal and external brokerage for regional innovation. Although research streams on intra-and interregional collaboration ties have developed in parallel, there may be benefits in integrating the two. Internal and external networks are concatenated to form a tightly interdependent structure. Benefits at each level are not necessarily additive (Ibarra et al., 2005, p. 367;Paruchuri et al., 2019). For this reason, network scholars called for studies exploring 'the degree to which constituent components of a phenomenon and the relationships among the components are similar across levels of analysis' (House et al., 1995, p. 88). Operti et al. (2020, p. 100) suggested that 'interactions between levels of analysis afford researchers greater theoretical insights into the system as a whole'. Zaheer et al. (2010, p. 74) pointed out that 'an important challenge for future research is to tease out the mechanisms that cross levels of analysis and those that do not, and when and under what conditions they do'.
Consistent with these recommendations, in this study we join research in innovation and geography (e.g., Coffano et al., 2017;Galaso & Kovářík, 2021;Le Gallo & Plunket, 2020) that exploits the possibility of cross-level theorizing stemming from joint consideration of internal and external networks (Rousseau, 1985, p. 14). Combining internal brokerage and external boundary-spanning, we obtain four different configurations, presented in Figure 1.
In the first configuration, which we label 'fortress', actors operate in regions whose internal network lacks structural holes and also have limited opportunities to form collaboration ties with actors located in other regions. The ensuing structure is a self-standing cohesive local network, where most individuals work for the same few local firms or universities with limited exposure to the external national and international context. All actors are directly or indirectly connected and share overlapping knowledge. Charlotte in North Carolina is an exemplary fortress region that developed closed collaboration networks in sectors like energy, steel and transportation, with most inventors working for the local universities and a large steel corporation, Nucor. Another example is the province of Friesland in the Netherlands, where Philips conducts R&D to develop shavers and hair clippers. The fortress's cohesive social networks and isolated location facilitate knowledge protection and secrecy (Lells, 2005).
In the second configuration, which we label 'playing fields,' the local (intra-regional) network is rich in structural holes, with several actors bridging disconnected small groups, often mapping onto different similarly sized, organizations. Still, the region has limited collaborative ties with other regions within the nation. According to Saxenian (1996), this configuration characterized San Jose-Sunnyvale-Santa Clara, one of the metropolitan areas in Silicon Valley, in its early days (until 2010, according to our data). Seattle-Tacoma-Bellevue in the United States is another exemplar of 'playing field'. According to Gray et al. (1996), this district was highly innovative in biotechnology, aerospace and software, thanks to a hub-and-spoke network of isolated suppliers and entrepreneurs, coordinated by a few hub firms (Microsoft and Boeing) and one research centre (the Fred Hutchinson Cancer Center).
In the third configuration, which we label 'absorber', actors in the region confront cohesive internal networks, with limited local opportunities to bridge structural holes. Yet, they collaborate extensively with actors located in other regions. For example, the metropolitan area of Helsinki has historically been characterized by high demographic homogeneity and cohesive collaboration networks within one large, geographically diversified firm, Nokia. However, thanks to its location, Helsinki intermediated between West Europe and Russia in the 1980s, or between the different Finnish areas (e.g., North Ostrobothnia and Pirkanmaa) where Nokia operated in the 1990s (Van den Berg & Braun, 2017). A similar example characterized by internal cohesion and external boundary spanning between the East and the West is Hong Kong (Martinus et al., 2021).
In the last configuration, which we label 'multilevel brokerage,' actors span structural holes within their region and are also exposed to several opportunities to act as boundary spanners in the network outside their region. This may be the case of vibrant international metropolitan areas, that present opportunities to bridge structural holes between local actors while also preserving ties to national and global players. Examples include San Francisco-Oakland-Hayward (since 2000) and San Jose-Sunnyvale-Santa Clara (after 2012) in Silicon Valley (Breznitz, 2014).
We expect the role of brokerage across these two levels to be non-additive primarily because of two mechanisms: 'information overload' and 'mobilization failure'. Consider the case of multilevel brokerage. Suppose that an actor occupies the role of 'coordinator' within a region but also serves as a 'gatekeeper' or as 'representative' outside the region. S/he runs the risk of being flooded by external and internal information without the necessary time to process, translate and recombine knowledge. Information overload, in turn, triggers energy depletion and information loss (Oldroyd & Morris, 2012). This can be highly problematic, especially when knowledge is complex and the actor is bridging distinct contexts, characterized by different norms and routines (Ponds et al., 2007). Even if different actors perform internal brokerage and external boundary-spanning, the sparseness of internal network and the coordination efforts required between these actors will in part deplete the informational advantage associated with brokerage across levels. By contrast, 'playing field' and 'absorber' configurations enable recombination of diverse knowledge but also feature cohesive groups characterized by trust and shared language, where assimilation and learning unfold (Burt, 2008). The resulting in-depth assimilation and recombination of distant knowledge serve as a precondition for innovation (Castaldi et al., 2015).
Moreover, in 'multilevel brokerage' configurations, a broker in the intra-regional network lacks the support of a cohesive community. If such an actor is also an external boundary-spanner and receives knowledge from outside, s/ he will lack the mobilization power necessary to champion and implement ideas developed outside the region. The broker will be questioned as a coordinator and treated as traitors for diffusing local knowledge outside. Indeed, network research shows that brokers are often located in the best position to pursue their own advantage at the community's expense (Burt, 2004(Burt, , 2008. For either reason, one can expect that cutting across both internal and external boundaries entails less effective network mobilization than more balanced configurations. Thus, the innovation performance of regions with high internal brokerage and low external boundary-spanning ('playing field'), and low internal brokerage and high external boundary-spanning ('absorber') will thus be higher than the performance of regions with low internal brokerage and low external boundary-spanning ('fortress'), and high internal brokerage and high external boundary-spanning ('multilevel brokerage'). Our main hypothesis reads as follows: Hypothesis 1: Ceteris paribus, the positive effect of internal brokerage (within a region) on regional innovation performance decreases with the degree of external (between regions) boundary-spanning.

METHODS
We tested our proposition using a longitudinal database capturing networks and innovation activities in US MSAs between 2000 and 2014. According to the Bureau of Labor Statistics, 4 an MSA is a geographical unit comprising an urban nucleus with a minimum population of 50,000, and the adjacent communities economically and socially integrated with the urban nucleus.
We used co-patenting relationships between inventors within and between MSAs to construct collaboration networks (Breschi & Lenzi, 2015;Fleming et al., 2007;Lobo & Strumsky, 2008), using time windows of three years (t -1, t -2 and t -3). This approach assumes that two inventors have a collaboration tie if they collaborated at least once in a time window. We used inventors' location from Patents View to determine which patents were filed from an MSA.

Dependent variable
Based on prior research, we measured a region's innovation performance using the total number of patents produced in the region in a given year t, weighted by the forward citations received (Fleming et al., 2007;Hagedoorn & Cloodt, 2003). To address the truncation problem, we computed the number of forward citations patents received in the first five years after patent grant (Nagaoka et al., 2010(Nagaoka et al., , p. 1113). Number of forward citations received by a patent positively correlates with Too much of a good thing? Network brokerage within and between regions and innovation performance 303 innovation's technological impact (Jee et al., 2019) and social and economic value Trajtenberg, 1990). 5 Given that the variable is highly skewed, we used the natural log of this variable.

Independent variables
External boundary-spanning External connections help regional actors learn from external knowledge sources (Coffano et al., 2017). Collaboration between a source and a recipient region can be conceptualized as an incidence of external boundary-spanning (Zhao & Anand, 2013). Accordingly, we measured external boundary-spanning as the average number of collaboration ties maintained by its inventors with the alters located in other MSAs within the United States.

Internal brokerage
Brokerage within a local network can be conceptualized in terms of connecting actors who have not been connected before, holding constant the number of collaborations (Burt, 1992). We measured the incidence of internal brokerage in an MSA using the average brokerage score of all its inventors in the intra-regional co-invention network. The brokerage measure is computed as 1 -Burt's constraint (Burt, 1992) using the igraph R-package.
Burt's constraint captures the extent to which a focal inventor i's collaboration investment is directly (p ij ) or indirectly (Σp iq p qj ) spent on inventor j. The ego network constraint C i of inventor i is given by the sum of c ij over all the contacts in the inventor's network: where individual constraints are obtained as c ij ¼ (p ij + Σp iq p qj ) 2 .

Control variables
We controlled for several factors that previous research had shown to impact innovation performance at the regional level.
Our goal is to capture the effect of brokerage configurations controlling for the amount of regional collaboration. Thus, we controlled for collaboration network connectedness, measured as the share of local inventors belonging to region's network largest connected component (Fleming et al., 2007). This measure varies between 0 (all inventors are isolates) and 1 (all inventors are directly or indirectly connected). 6 Also, while our study focuses on collaborations within the United States (within and between MSAs), we acknowledge the effects of global connections on regional innovation (Bathelt et al., 2004;Fitjar & Huber, 2014). We thus controlled for the average number of collaboration ties maintained by the inventors of the focal MSA with those outside the United States.
Second, innovation performance partly reflects the existing stock of regional knowledge. Thus, we controlled for the cumulated patent stock in a given MSAs (Bettencourt et al., 2007;Lobo & Strumsky, 2008). To account for the degree of region's knowledge localness, we controlled for the ratio of local self-citations (backward citations referring to patents previously filed from the region) (Gambardella & Giarratana, 2010). The presence of Marshall-Arrow-Romer (stemming from regional specialization) or Jacob (stemming from variety) externalities is considered an important driver of regional innovation performance (for an empirical test, see Audretsch & Feldman, 2004or Beaudry & Schiffauerova, 2009). We accounted for regional specialization using the Herfindahl-Hirschman index of dispersion of patenting activity in the US Patent Classification (USPC) two-digit technology classes (Fritsch & Slavtchev, 2010). We used Hall's (2005) correction to remove the downward bias in Herfindahl-Hirschman index. In addition, novel and original technological areas are relatively fertile both in terms of innovation (patenting) and impact (citations). We used Hall et al.'s (2001) originality index to proxy knowledge novelty.
Regional innovation performance also depends on the type of actors performing research activities. For example, scholars explored the contribution of universities and private firm investments to regional innovation, with studies documenting positive spillovers from local universities (e.g., Anselin et al., 1997;Jaffe, 1989) and others identifying a weaker, less straightforward relationship (Co, 2002;Ó hUallacháin & Leslie, 2007;Riddel & Schwer, 2003). Through the HERD survey, the National Science Foundation (NSF) collects data on R&D expenditures carried out by US colleges and universities. We used these data to track university R&D investments, mapping each university to its MSA, using the corresponding ZIP code. We also controlled for private and federal agencies R&D investments. These data are at the state level, given that data on industry R&D are not available from the NSF at the MSA level for confidentiality reasons. We used the number of university patents (ln) in each MSA to gauge the relative weight of university knowledge in a region. To track whether MSAs dominated by a few organizations are better than MSAs where inventive activity is distributed between many organizations, we controlled for the corrected Herfindahl-Hirschman index of concentration of inventions by assignees (Hall, 2005).
Structural factors also positively affect regional innovation performance (Ó hUallacháin & Leslie, 2007;Sleuwaegen & Boiardi, 2014). We measured the quality of human capital in the region using the per capita payroll of employees in the high-tech sectors in each MSA (source: US BEA). Finally, we used FRED economic data to control the share of bank branches controlled by international banks at the MSA level. Local banks decrease the extent to which firms engage in innovation and take risk, while international banks promote high risk innovation projects (Chava et al., 2013). We also included MSA and year fixed effects to account for stable, unobserved regional and year differences. We used NBER two-digit industry fixed effects to account for stable sectoral heterogeneity in collaboration network structure and research intensity. A focal dummy is set to 1 if at 304 Elisa Operti and Amit Kumar least 10% of the patents filed by the region belong to this category (Plum & Hassink, 2011).

Estimation
Given that some MSAs cut across state boundaries, the unit of analysis used in our study is the MSA-state-year.
In line with past research (Breschi & Lenzi, 2015), we dropped fully disconnected networks (networks of isolates). Further, we considered only MSAs with at least 10 active inventors in a given three-year window to limit the impact of outliers. 7 We used a fixed effect panel regression, taking the natural log of our dependent variable, which is highly skewed. This approach aligns with previous knowledge production function research and allows to account for region-fixed effects 8 and a more straightforward interpretation of the interaction effects. We collected data concerning innovation activities between 2000 and 2014; however, as constructing networks requires a three-year window, our analyses start in 2003. We use a one-year lag between the dependent and the independent/control variables to avoid simultaneity biases. We use robust standard errors (clustered by MSA) to account for heteroskedasticity (White, 1980).

Descriptive statistics
To illustrate our typology using the data we collected, in Table 1 we provide a few examples of MSAs classified in each quadrant, based on the distribution of the variables external boundary-spanning and internal brokerage between 2003 and 2014. 9 We also computed mean descriptive statistics of selected variables of interest by each quadrant to better understand the distinctive traits of each configuration. 'Multilevel brokerage' and 'playing field' configurations tend to be larger regions, which also feature dispersion of these inventive activities across multiple organizations, with a few hubs coordinating collaboration activities, with a significant weight of universities in brokering internal knowledge flows. Collaboration networks characterized as 'absorbers' and 'fortress' are smaller in size, and the inventive activities therein tend to be concentrated in a few cohesive large organizations, typically private firms. Similar levels of specialization characterize all four configurations: each quadrant features both examples of highly specialized regions, akin to what studied by past research on industrial clusters, and diversified regions. Tables 2 and 3 summarize our key variables and controls, their operationalization, data sources and the descriptive statistics. Table 4 reports pairwise correlations. The mean variance inflation factor (VIF) of the full model is 2.35, below the recommended thresholds (Neter et al., 1989).
Based on our theory, we expected positive effects of external boundary-spanning and internal brokerage structure, and a negative interaction term of the two. Table 5 reports the results of the models. Model 1 is the baseline model, in which we estimated the effects on our control variables. The direction of effects is in line with our expectations. Network connectedness (β ¼ 0.027; p ¼ 0.909) and global connections (β ¼ 0.062; p ¼ 0.425) have a positive non-significant effect on innovation. The latter finding is not surprising, given that global pipelines may convey spillover advantages, but also expose to increased coordination costs, risk of imitation and limited appropriability of inventions, depending on country-level institutional factors. The Stock of regional knowledge has a positive and significant effect (β ¼ 0.323; p < 0.001) on regional innovation performance. Local knowledge sourcing also has a negative but non-significant effect on regional innovation performance (β ¼ −0.541; p ¼ Too much of a good thing? Network brokerage within and between regions and innovation performance 305 0.242). Supporting the presence of Jacobian externalities, regional specialization has a significant, negative effect on innovation performance (β ¼ −0.760; p < 0.01). Knowledge novelty has positive, non-significant effects on innovation performance (β ¼ 0.310; p ¼ 0.341). In terms of the relative role of universities and firms in regional innovation, we find that state-level investment in R&D by private firms and agencies has a positive and significant effect (β ¼ 0.120; p < 0.05) on regional innovation performance, while university R&D investments in the MSA 10 has a positive, but non-significant effect (β ¼ 0.004; p ¼ 0.523). The number of university patents is also not associated with regional innovation performance (β ¼ 0.002; p ¼ 0.994). While these results may at first be surprising, they are in line with some studies challenging the size of and mechanisms behind knowledge spillovers from universities (Co, 2002;Riddel & Schwer, 2003), especially when controlling for structural endowments (Ó hUallacháin & Leslie, 2007). We also do not find significant differences between regions where inventive activities are concentrated in a few assignees and regions wherein activity is dispersed in multiple organizations (β ¼ −0.007; p ¼ 0.967). In structural terms, we find that international risk-taking banking institutions have positive, but non-significant effects on innovation.
In model 2, we introduce the independent variables. External boundary-spanning has a positive but nonsignificant effect (β ¼ 0.018; p ¼ 0.501), whereas internal brokerage has a positive and significant effect (β ¼ 0.514; p ¼ 0.050) on regional innovation performance. In model Table 2. Variables, their operationalization and data sources.

Region's innovation performance
Natural log of the number of patents from the MSA weighted by forward citations received within five years (Fleming et al., 2007;Hagedoorn & Cloodt, 2003;Hall et al., 2005) USPTO

Independent variables
External boundary-spanning Average number of ties of MSA inventors with those located in other MSAs within the United States (Breschi & Lenzi, 2015) USPTO Internal brokerage Mean brokerage of all inventors in the MSA. Brokerage measured as 1 -Burt's constraint (Burt, 1992) USPTO

Control variables
Network connectedness Percentage of inventors forming the largest connected component in an MSA's co-patenting network (Fleming et al., 2007) USPTO Global connections Average number of ties of MSA inventors with those located in foreign countries (Fitjar & Huber, 2014;Zeller, 2004) USPTO Stock of regional knowledge Natural log of the number of patents from the MSA (Bettencourt et al., 2007;Lobo & Strumsky, 2008) USPTO Knowledge localness Share of backward citations from the focal MSA (Gambardella & Giarratana, 2010) USPTO Regional specialization HHI of patenting in MSA by USPCs (Fritsch & Slavtchev, 2010), corrected by portfolio size (Hall, 2005) USPTO Knowledge novelty Originality index (Hall et al., 2001) USPTO University R&D investments Total R&D expenditure by universities and colleges located in the MSA (Anselin et al., 1997) NSF HERD surveys Private state-level R&D investments Natural log of the R&D investments by firms and federal and non-federal agencies at the state level (Feldman, 1994;Feldman & Florida, 1994)  3, when we introduce the interaction term of the two independent variables, the effects of external boundary-spanning (β ¼ 0.134; p < 0.01) and internal brokerage (β ¼ 1.352; p < 0.001) became positive and strongly significant. The coefficient of the interaction term is negative and significant (β ¼ −0.641; p < 0.01). The results support our main hypothesis, postulating the non-additive nature of brokerage across levels of analysis. To illustrate the magnitude of the interaction effect and substantiate our claims intuitively, we plot the interaction effects in Figure 2.
We consider the effect of internal brokerage, at low and high (mean ± 2 SD) levels of external boundary spanning. The results highlight increasing returns to internal brokerage at low levels of external boundary-spanning, with maximum innovation performance associated with the configuration we labelled 'playing fields'. We also show that regions characterized by high external boundary-spanning are highly innovative at low levels of internal brokerage ('absorbers'). Yet, configurations characterized by 'multilevel brokerage' are less innovative than those configurations characterized by cross-level balance.

Robustness checks
We conducted several robustness analyses (Tables 6 and  7). First, we used alternative operations of our key variables, internal brokerage, external boundary-spanning and innovation performance. Concerning internal brokerage (1) we recoded all isolates, considering them as missing values, rather than assigning them a constraint value of 1 (Burt, 1992); (2) we replaced Burt's measure with the average normalized betweenness centrality of inventors in the internal network. With respect to external boundary-spanning: (3) we weighted the number of external collaborations by the geodesic distance between the collaborating inventors, 11 given that more than 75% of external ties entail distances greater than 75 miles; and (4) we used the median number of external collaboration ties, rather than the average. Finally, given that there may be temporal autocorrelation between sets of patents filed in different years, we checked whether our results are robust to the removal of the patent stock variable from the control list, given that it is the control variable with the highest VIF score, to the absence of citation weights on the dependent variable, or to both choices. The results, presented in models 4-10 support our hypothesis. 12 We tested whether the effect of each focal variable (internal brokerage, external boundary spanning) is curvilinear. The results did not support this conjecture (see models 11 and 12): the main effect and the squared term of both variables are not significant in both models. The R 2 is also lower than in Table 5, with an increased mean VIF compared with the full model. These results support the choice of a monotonic relationship, rather than an inverted 'U'-shaped one.
In unreported additional analyses (omitted here due to space constraints, but see Appendix A in the supplemental data online), we ran our analyses using an employmentbased index of regional specialization and fixed effects based on MSA employment by sector, rather than relying on industry measures derived from patent/technology classifications (e.g., Kang & Dall'Erba, 2016). We also ran our analyses using different regional size proxies, including a weighted cumulated patent stock measure and a population measure. Our results remained consistent.

CONCLUSIONS
This paper contributes to research on collaboration networks and regional innovation (e.g., Breschi   Too much of a good thing? Network brokerage within and between regions and innovation performance studying the effect of brokerage within and across regions. Addressing a long-standing call for research exploring network advantage across levels of analyses (Paruchuri et al., 2019;Zaheer et al., 2010), we propose a typology of network configurations that better accounts for interdependencies in network structures across levels (Ibarra et al., 2005). Our findings show that the most innovative regional configurations are those that combine both knowledge recombination through brokerage and knowledge assimilation through cohesive communities, highlighting the trade-offs of brokerage within or across levels in spatially bounded collaboration networks. Trade-offs between local and non-local knowledge and their effect on innovation have been analysed by previous Too much of a good thing? Network brokerage within and between regions and innovation performance 309 research (e.g., Autant-Bernard, 2001;Kang & Dall'Erba, 2016;Neal, 2013). Our framework differs from and extends these findings in several ways. First, we focus on collaborative spillovers, that is, spillovers channelled by formal collaboration networks, rather than inferring spillovers from the co-location of agents in a bounded geographical space, as done in previous studies. Second, while existing works studying collaboration networks looked primarily at the amount/intensity of collaboration, captured by degree centrality or tie strength (e.g., Coffano et al., 2017), our work focuses on brokeragea structural property of collaboration networks. Third, the use of longitudinal data on a large sample of MSAs allows us to capture the effects of structural properties of the collaboration network controlling for alternative mechanisms that prior research linked to knowledge spillovers (e.g., Anselin et al., 1997). The findings presented in this paper also have some limitations. First, we used co-patenting data to capture collaboration networks, in line with prior work (e.g., Fleming et al., 2007), based on the assumption that 'the network of collaborators is the most immediate and influential environment from which inventors draw ideas and information' (Breschi & Lenzi, 2015, p. 792). We could also have used patent citations to construct intra and interregional networks. Although both approaches have been used in past research, we used the first given that citations can be a less accurate indicator of knowledge spillovers due to actors' strategic motives and examiner added citations (Alcacer & Gittelman, 2006;Breschi & Lissoni, 2009). Second, in our empirical analysis, we focused on local vs. external collaboration within the United States, controlling for connections outside the United States. Global connections can be an excellent source of innovation, but they bring about significant heterogeneity in endowments and institutional contexts, which we cannot tackle within this paper. Further studies can apply our framework to within and between state collaborations, identifying the trade-offs entailed in cross-national collaboration. Finally, our framework and findings shed light on the effect of brokerage configurations irrespective of the type of organization occupying a brokerage role. However, our descriptive statistics show that inventors working for universities and private firms display different tendencies towards internal vs. external brokerage, with regions characterized by strong university investments displaying higher values of internal brokerage. Future qualitative work can provide additional insights on the division of brokerage labour, and how different types of organizations may alter the trade-offs presented in each of the four configurations. For example, researchers can explore the challenges that organizations face when they operate both as local 'coordinators' within territorial boundaries, and as 'gatekeepers' or 'representatives' in the national network. It would be also interesting to understand whether individuals working for universities or private firms face similar challenges when occupying these positions.
The proposed framework also presents opportunities for additional future research and theory development. Our framework is static; further work can add a dynamic spin on it. For example, a promising line of action would be to explore these configurations over time. For instance, some accounts suggest that some metropolitan areas in California have moved over time from 'playing field' to 'multilevel brokerage', bringing some inventive productivity challenges (Breznitz, 2014), while others remained relatively stable in their quadrant (San Francisco-Oakland-Hayward as multilevel brokerage). Why did this occur and how does it relate with regional specialization or technology life cycles? Examining how regions move in our frameworkwhich policies, norms or exogenous events explain a region's transition from one configuration to anothercan enhance our dynamic understanding of brokerage. In addition, future studies can overlay regions and sectors and explore how the proposed relationships vary within and across industries. Sectors matter because knowledge-networking in a regional context differs significantly by industry, with Too much of a good thing? Network brokerage within and between regions and innovation performance Robust standard errors are shown in parentheses. ***p < 0.001; **p < 0.01; *p < 0.05; + p < 0.1.
remarkable variations between analytical (science-based) and synthetic (engineering-based) sectors (Plum & Hassink, 2011). An aggregated approach, like the one proposed in this paper, cannot fully capture the variation in the effects of collaboration within and across sectors. Further research is thus required to unpack this dimension. The sectoral approach and methodology used in sectoral research based on patent data (e.g., Kekezi et al., Robust standard errors are shown in parentheses. ***p < 0.001; **p < 0.01; *p < 0.05; + p < 0.1. Too much of a good thing? Network brokerage within and between regions and innovation performance 313 2021) can help scholars understand the interplay of intra and interregional collaboration and intra and inter-sectoral knowledge flows.