Synergy in the Knowledge Base of U.S. Innovation Systems at National, State, and Regional Levels: The Contributions of High-Tech Manufacturing and Knowledge-Intensive Services

Using information theory, we measure innovation systemness as synergy among size-classes, zip-codes, and technological classes (NACE-codes) for 8.5 million American companies. The synergy at the national level is decomposed at the level of states, Core-Based Statistical Areas (CBSA), and Combined Statistical Areas (CSA). We zoom in to the state of California and in more detail to Silicon Valley. Our results do not support the assumption of a national system of innovations in the U.S.A. Innovation systems appear to operate at the level of the states; the CBSA are too small, so that systemness spills across their borders. Decomposition of the sample in terms of high-tech manufacturing (HTM), medium-high-tech manufacturing (MHTM), knowledge-intensive services (KIS), and high-tech services (HTKIS) does not change this pattern, but refines it. The East Coast -- New Jersey, Boston, and New York -- and California are the major players, with Texas a third one in the case of HTKIS. Chicago and industrial centers in the Midwest also contribute synergy. Within California, Los Angeles contributes synergy in the sectors of manufacturing, the San Francisco area in KIS. Knowledge-intensive services in Silicon Valley and the Bay area -- a CSA composed of seven CBSA -- spill over to other regions and even globally.


Introduction
The metaphor of "national innovation systems" (NIS) induces the question of whether innovation systems are nationally organized? (Carlsson, 2006). Innovation dynamics do not honor national borders, nor are innovation opportunities limited to cities (Florida, 2002;Jacobs, 1961;Storper et al., 2015) or regions (Cooke, 2002). As a model of innovation systems, however, NIS combines the ideas that innovation is systemic (Lundvall, 1988) and that innovation systems are evolving (Nelson, 1993), organized institutionally, and therefore susceptible to government policies at national levels (Freeman, 1987). Thus, the perspectives of policy analysis, institutional analysis, and (neo)evolutionary theorizing are combined.
The delineation of innovation systems in institutional terms offers the advantage of compatibility with (e.g., national) statistics (Griliches, 1994). However, an institutional perspective on innovation leads to a theory of entrepreneurship (Casson, 1997) rather than accounting for the relational dynamics of communication and innovation, which is at the core of what one seeks to measure (Carter, 1996;Godin, 2006). Using a relational perspective on innovation, the emphasis has increasingly been on co-evolutions between regional economics, economic geography, and technological options (Audretsch & Feldman, 1996;Boschma, Balland, & Kogler, 2014;Feldman & Storper, 2016). This literature suggests a mutual shaping among the various factors of knowledge production inducing trajectories and niches (Geels & Schot, 2007).
In this study, we propose a methodology that combines a relational with a positional (e.g., geographical) perspective to test the assumption of systemness at national, state, and regional levels by using interactions among the geographical, technological, and organizational distributions of companies at different levels or sectors. Storper (1997, at pp. 26 ff.) considered the mutually reflexive relations among these three dimensions as a "Holy Trinity" in regional development. The distributions of these relations, however, can be systemic to varying degrees.
Our "Triple-Helix" methodology is based on entropy statistics and thus rooted in evolutionary systems theory. Synergy can be measured as negative entropy. Leydesdorff & Ivanova (2014) showed that negative information in a Triple-Helix configuration finds its origin in redundancies that are generated when uncertainty is selected from different perspectives (Leydesdorff & Ahrweiler, 2014). New options can be generated in interactions among selection mechanisms.
The total number of options-the maximum entropy-is thus increased. The increase in the redundancy may outweigh the increase of uncertainty generated in ongoing processes of variation. Additional redundancy reduces relative uncertainty by adding options to the system.
Increasing the number of options may be more important for the viability of an innovation system than the options realized hitherto (Fritsch, 2004;Petersen et al., 2016). Furthermore, reduction of uncertainty can be expected to improve the climate for investments (e.g., Freeman & Soete, 1997, pp. 242 ff.).
We assume that three different dynamics-industrial, R&D, and political-are operating selectively upon one another. While two selection mechanisms can be shaped mutually along a trajectory, a complex dynamics is generated when three or more subdynamics interact. A third variable, for example, may make a correlation between the other two spurious. A triangle of relations can rotate clockwise in terms of feedforwards or counter-clockwise in terms of feedbacks (Ulanowicz, 2009). Feedforwards can make a system prosperous, while with the opposite sign, hyper-selectivity may lead to lock-ins and historical stagnation (Bruckner, Ebeling, Montaño, & Scharnhorst, 1996).
From this perspective, the national, regional, or sectorial levels can be considered as specific integrations among the (sub)dynamics (Carlsson, 2006). Both integration in local instantiations and differentiation among the next-order (global) selection environments operate continuously in systems of innovation. The local combinations instantiate historical trajectories, while the interactions among the selection environments (markets, governance, R&D) develop at a nextorder regime level. Interactions among selection mechanisms generate redundancy when the selection mechanisms overlap. Since the two processes-the historical generation of variation and the evolutionary interactions among selection environments-are operating concurrently, the trade-off between uncertainty generation and reduction can be expected to vary among regions, sectors, etc. This trade-off can be measured in bits using the TH indicator (Leydesdorff, Park, & Lengyel, 2014).
The research question thus becomes: to what extent can a given configuration such as a national or regional portfolio be expected to operate not only as a system, but also as an innovation system? A measure for systemness can easily be developed, for example, on the basis of the Markov property: the current state of a system provides a better prediction of its next state than what can be derived from the history of its elements. Using publication data, for example, Leydesdorff (2000) showed that the European Union (at that time) was evolving as a set of national research systems more than at the European level. German unification, however, led to the shaping of a single publication system in Germany during the 1990s.
An innovation system would not only evolve as a system, but also generate new options.
Redundancy generation increases the maximum entropy. Biological systems increase uncertainty following the entropy law (Brooks & Wiley, 1986). Technological innovation, however, extends the number of options. For example, the capacity of transport across the Alps could be considered as constrained by the capacity of roads and railways such as at the Brenner Pass. As one invents new channels, however, other options became available, such as for example air transport across the Alps or tunnels underneath, which are not constrained by the geological or weather conditions on the ground.
Both redundancy and information are generated in TH-type innovation systems. The feedback and feedforward loops precondition each other: the phenotypical variation can be organized historically (for example, by governments and in enterprises). The selection mechanisms have the status of hypotheses; they can be considered as self-organizing "genotypes" (Hodgson & Knudsen, 2011;Langton, 1989). Unlike biological code (DNA), selection is not hard-wired but operates as a code in the communication. The selection criteria can be expected to adapt evolutionarily to the opportunities provided in the historical layer. Using the TH-indicator for the measurement of the trade-off, positive mutual information among the three helices indicates that the generation of (Shannon-type) information prevails; when this measure is negative, the nonlinear generation of redundancy (in loops) prevails, and uncertainty is reduced (cf. Krippendorff, 2009).
In this study, we apply this methodology to studying the knowledge base of the American economy. We have applied the approach in a number of (mainly European) country studies. 6 However, the application to data about the U.S.A. is expected to provide new insights concerning both the effectiveness of the measurement model and the knowledge base of the U.S. economy.
Our methodology enables us to test whether or specify the extent to which synergy among distributions is generated and systems can be considered as innovation systems. We focus on geographical scales, but will distinguish also in terms of sectors such as high-and medium-tech manufacturing and knowledge-intensive services (Carlsson, 2013). We thus endogenize the technological dimension into the model (Nelson & Winter, 1977).

The American innovation system
In a review of the U.S. innovation system and innovation policy, Shapira & Youtie (2010) argue that the U.S. system is marked by diversity and multiple layers and levels to the extent that one may question whether a national system of innovations is even a useful concept. The authors emphasize the role of the States, which they formulate (at pp. 4-5) as follows: State governments tend to be much more active in the innovation area than the federal government has been, primarily because there has traditionally been reluctance at the federal level to intervene in industrial policy, while state governments are closer to the needs of the particular industries that make up their regional economies. Many recent federal programs have had historic roots in long standing state and local innovation initiatives.  (Storper et al., 2015).
According to Audretsch & Feldman (1996) and many other authors, Silicon Valley has been the region with the largest number of innovations, followed by New York, New Jersey, and Massachusetts. However, LA is more important in terms of high-and medium-tech manufacturing than the Bay area (Feldman & Florida, 1994), while San Francisco dominates in terms of knowledge-intensive services (Whittington et al., 2009). Silicon Valley provides a mixture of high-tech manufacturing and knowledge-intensive services (Bresnahan & Gambardella, 2004), but the economic activity of this region is less rooted geographically than in 7 https://ssti.org/blog/useful-stats-share-us-venture-capital-investment-state-2009-2014 the other two areas (Saxenian, 1996). The more detailed analysis of California and Silicon Valley will enable us to discuss some of the limitations of the methodology.
We use companies as the units of analysis and specify three codes as most relevant for innovation systems: (1) ZIP codes indicating company addresses in the geographical dimension, (OMB) as geographical zones of one or more counties (or equivalents) anchored by an urban center of at least 10,000 people and including adjacent counties that are socioeconomically tied to the urban center via commuting. CBSAs can be metropolitan or micropolitan (e.g., rural; Brown et al., 2004;Hall, 2009). CSAs can be defined (by the OMB) when multiple metropolitan or micropolitan areas have an employment interchange of at least 15%; 9 CSAs often represent regions with overlapping labor and media markets. 8 The District of Columbia is included as a state. 9 OMB Bulletin No. 17-01: Revised Delineations of Metropolitan Statistical Areas, Micropolitan Statistical Areas, and Combined Statistical Areas, and Guidance on Uses of the Delineations of These Areas, at https://www.whitehouse.gov/sites/whitehouse.gov/files/omb/bulletins/2017/b-17-01.pdf

Methods and data
Data Data were retrieved from the ORBIS database of Bureau van Dijk on May 4-6, 2017, 10 using the search string "United States of America" for all active companies with data covering a known value and a last available year, including estimates for the number of employees where necessary. Companies with no recent financial data were excluded, as were public authorities, states, and governments. We follow the definition and delineation of companies as provided in ORBIS. This constraint on the data is a major limitation. ZIP codes, for example, vary over geographical regions; however, in reference to the other two dimensions, the distribution of ZIP codes indicates local constraints (such as infrastructure) operating as a (non-market) selection environment.
In addition to the assignment of NACE and ZIP-codes, companies are scaled in terms of the number of their employees as a third dimension. SMEs are commonly defined in these terms.
Financial turn-over is available in the ORBIS data as an alternative indicator of economic structure. However, we chose the number of employees as one can expect this number to exhibit less volatility than turn-over, which may vary with stock value and economic conjecture more readily than numbers of employees. Numbers of employees are sensitive to other activities, such as outsourcing. 10 When we entered the ORBIS database again on September 27, 2018 (after the review process), the coverage had grown from approximately180,000 to 230,000 companies, of which 53,624,319 with an address in the USA.   Small, medium-sized, and large enterprises As noted, we use the number of employees as a proxy for size of the company (Table 4). Small and medium-sized companies (etc.) are commonly defined in terms of numbers of employees.
However, the definitions of small and medium-sized businesses versus large enterprises vary among world regions. Most classifications use six or so categories for summary statistics. We use the eleven classes provided in Table 3 because this finer-grained scheme produces richer results (Blau & Schoenherr, 1971;Pugh, Hickson, & Hinings, 1969a and b;Leydesdorff, Dolfsma, & Van der Panne, 2006;Leydesdorff & Porto-Gomez, 2017;Rocha, 1999). Note that micro-enterprises (with fewer than five employees) constitute 66.3% of the companies under study.

Statistics
Using Shannon's (1948) information theory, uncertainty in the distribution of a random variable x can be defined as = − ∑ log 2 . The values of px are the relative frequencies of x: = ∑ ⁄ . When base two is used for the logarithm, uncertainty is expressed in bits of information.
The uncertainty in the case of a system with two variables can be formulated analogously as In this case of two variables with interaction, the uncertainty of the system is reduced by mutual information as follows: One can derive (e.g., McGill, 1954;Yeung, 2008, pp. 59f.) that in the case of three dimensions, mutual information corresponds to: Eq. 3 can yield negative values and is therefore not a Shannon-type information (Krippendorff, 2009). Shannon-type information measures variation, but this negative entropy is generated by next-order loops in the communication; for example, when different codes interact as selection environments.
Note that uncertainty is implicated by the variation in historical relations. From an evolutionary perspective, the historical networks of relations function as retention mechanisms. Our measure, in other words, does not measure action (e.g., academic entrepreneurship) or output, but the investment climate as a structural consequence of correlations among distributions of relations; the correlations can be spurious. However, the distinction between the structural dynamics and the historical dynamics of relations is analytical. The two layers reflect each other in the events.

Eq. 3 models this trade-off between variation and selection as positive and negative contributions
to the prevailing uncertainty. The question of systemness can thus be made empirical and amenable to measurement.
In the case of groups (or subsamples), one can decompose the information as follows: = 0 + ∑ (Theil (1972, pp. 20f.). The right-hand term (∑ ) provides the average uncertainty in the groups and H0 the additional uncertainty in-between groups. Since T values are decomposable in terms of H values (Eq. 3), one can analogously derive (Leydesdorff & Strand, 2013, at p. 1895): In this formula, TG provides a measure of uncertainty at the geographical scale G; nG is the number of companies at this scale, and N is the total number of companies under study. One can also decompose across regions, in terms of company sizes, or in terms of combinations of dimensions.
Because the scales are sample-dependent, one may wish to normalize for comparisons across samples, for example as percentages. After normalization, the geographical contributions of regions or states can be compared in bits (or other measures) of information. In this design, the between-group term T0 provides us with a measure of what the next-order system (e.g., the nation) adds in terms of synergy to the sum of the regional systems or states. The three dimensions are the (g)eographical, (t)echnological, and (o)rganizational; synergy will be denoted as TGTO and measured in millibits with a minus sign.

Decomposition in terms of U.S. states
First we decompose the U.S. in terms of its 50+ states. Figure 1 shows the percentages of synergy contributions of states.  The assumption of a national innovation system in the U.S. is therefore not supported by our results. We proceed in the next section with the sector-based decomposition. Does one find similar patterns when focusing on high-tech manufacturing or knowledge-intensive services? Or do we observe specialization among states and regions?

Sectorial decomposition at the level of states
As noted, ΔΤ values can be compared as percentages of contributions to the national synergy after normalization. Let us compare the four sectors specified in  The high correlations in Table 5 lead to the conclusion that the distributions over the states for the various sectorial decompositions are not significantly different from one another or from the overall distribution of the synergy over the states. Thus, the synergy contribution is state-specific; the sectors only modulate the state-specific averages.

CBSA and CSA
Of the 940 CBSA distinguished in the data, 446 contribute to the national synergy. Figure 3 shows a map with these CBSA in shades according to their contribution; Figure 4 provides the corresponding map for 139 (of the 165) CSA which contribute to the synergy at this level. 17 The maps show in a bird eye's view that synergy is more concentrated in CSA than CBSA. Not only are the values (expressed as percentages contribution) higher, but the concentration in the northeast (New York-Philadelphia and New England) are more pronounced. The region of LA is clearly indicated in Figure 4, but less so in Figure 3. SF is not indicated as a metropolitan CBSA, but it is as part of the CSA of the Bay area and Silicon Valley.    Table 6: 20 CBSA (left-column) and CSA (right column) contributing most to the national synergy. Table 6 lists the top-20 CBSA and CSA in terms of contributions to the synergy in these two domains. The ranking is relatively robust. The sectorial decomposition, however, nuances the picture. Figure 5 shows that the metropolitan regions of New York and Boston deviate by having no synergy contributions from medium-high-tech manufacturing. LA excels in HTM, but is also strongest in MHTM. The (Spearman) rank-order correlations in Table 7 indicate that the ranks vary among sectors. 18 This variation is not among the highest rankings (Table 8), but in the middle range. Nevertheless, the NY-NJ-PA district makes the most significant contribution to the national synergy (8.65%) among all CBSA ( Figure 5). This confirms Feldman & Florida's (1994) observation about the contribution of New Jersey to the national geography of innovation. 18 The Pearson correlations among the four sectorial groups are all above .99.

Silicon Valley and the Bay area
Silicon Valley is located southeast of the San Francisco area. The CSA of the Bay area is named "San Jose-San Francisco-Oakland, CA" and is composed of seven CBSA. "San Jose-Sunnyvale-Santa Clara, CA" is the CBSA which covers Silicon Valley itself.  More than 80% of the synergy in the CSA is generated by the seven CBSA; that is, within the area. The contributions to the synergy are not sector-specific. However, the last lines of Table 9 teach us that 1.40% of the companies in HTM contribute 3.43% to the synergy in this region, whereas HTKIS and KIS contribute proportionally less than expected.
Let us further decompose. The data of the CBSA "San Jose-Sunnyvale-Santa Clara, CA" contains two zip-codes at the two-digit level: 94 and 95. Companies with zipcode 95 (n = 36,330) generate 43.27% of the synergy in this CBSA; 3.95% is generated by 13,184 companies with zipcode 94. Further decomposition of the Valley is possible in terms of cities or companies.
Using company names, however, one obtains maximum entropy in the geographical dimension because all companies have unique names. Decomposition in terms of cities leads to subsets which have the city as a constant and consequently zero entropy in the geographical dimension.
In the latter case, the redundancy is necessarily zero and in the former model uncertainty prevails to such an extent that TUIG is part of the maximum entropy and therefore necessarily positive (TUIG > 0). In both these cases, no synergy can be measured for methodological reasons. In other words, this methodology cannot be used for the lowest level because there is either no variance in a single city name or maximum entropy when using unique company names.
In Silicon Valley (SV), HTM contributes more to the synergy than MHTM and HTKIS more than KIS, although both KIS and HTKIS contribute less than expected. HTKIS, for example, contributes 4.05% to the synergy with 6.06% of the companies, whereas HTM contributes 5.20% with 2.90% of the companies. In other words, there is much more KIS (and HTKIS) than HTM and MHTM in Silicon Valley, but the synergy contribution of manufacturing is much higher than that of the knowledge-intensive services.

Discussion and limitations
We used synergy among the distributions of sectors, company addresses, and size-classes as an indicator of innovation-systemness to study the U.S. at various levels of aggregation. Obviously, the main limitation of this study is the use of ORBIS data. We have no access to how the data is collected; the database is private property. Still, it is probably the best data currently available for this type of study. As noted, ZIP codes vary over geographical regions; however, in reference to the other two dimensions, the distribution of ZIP codes indicates local constraints (such as infrastructure) operating as a selection environment. In the case of NACE codes, the alternative of NIAC would be an option, and other schemes could be used for the scaling of companies in terms of size-classes. Most importantly, the definition of what counts as a company in the database is beyond our control.
The results thus provide us only with a window, and we do not wish to deny that other approaches are possible and perhaps even more fruitful. However, we improve on other approaches by moving beyond a political definition of innovation systems to an empirical one which can be tested by using synergy as a measure of systemness (cf. Griliches, 1994).
Systemness can then also be rejected as a fruitful hypothesis at specific levels of aggregation and/or within sectors. Our results, for example, do not indicate the systemness of the U.S. innovation system at the national level.
Given the proviso of the methodological constraints of the study, our analysis suggests that the states, and not the nation or regions, are the most relevant innovation environments in the U.S.
To the extent that states are the relevant geographical entities for innovation, significant policy considerations should follow. Note that our conclusions do not imply volition or initiative on the part of state governments; these are input data. We are just reporting empirical findings about systemic configurations. In the past, state initiatives have often been evaluated as ineffective or incompetent compared with initiatives at the level of metropolitan regions (Agrawal, Cockburn, Galasso, & Oettl, 2014;Bartik, 2017). Even so, states have a long history of creating baskets of incentives, training, and investment programs to grow industry (Shapira & Youtie, 2010). Our results indicate synergy in the knowledge base of specific states along the East Coast (New Jersey, Massachusetts, New York, and Pennsylvania), in California, and, in the case of HTKIS, Texas.
The regions measured as CBSAs are too small to comprise innovation systems; the innovation systems spill over the boundaries of these units of analysis. As could be expected, CSAscombining contiguous CBSAs-are more appropriate units of analysis in terms of the development of synergy. The decomposition in terms of sectors shows specialization among states and regions, but does not change the main pattern other than modulating it. The overall picture is one of concentration of high-tech and dispersed specialization at many different locations. Knowledge-intensive services are dominant, but do not contribute to the synergy above expectation.
Focusing on California, three regions are most relevant for the discussion: LA with synergy in manufacturing (both HTM and MHTM), San Francisco with synergy in KIS and HTKIS, and Silicon Valley with mainly KIS in the portfolio but manufacturing as the generator of synergy.
The services in Silicon Valley are not contributing to synergy in the region but operating at national and global levels. While these conclusions may not be surprising from the perspective of hindsight, ex ante it would have been difficult to specify the nuances in such detail without a quantitative analysis.