Social Networks as Determinants of Knowledge Diffusion Patterns

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1 Social Networks as Determinants of Knowledge Diffusion Patterns Jasjit Singh Harvard Business School and Department of Economics Jan 9, 2004 Abstract: This paper examines if social networks drive diffusion of knowledge, and help explain observed patterns of knowledge diffusion. When an inventor joins a new team, she might bring knowledge from her past innovations, and from innovations by others she has direct or indirect collaborative and social ties with. To capture this in an empirical model, I construct a social proximity graph of inventing teams for all US Patent Office patents from The graph is defined to have an edge between any two teams with a common inventor. Teams with socially linked inventors have nodes belonging to the same connected component of this graph. The strength of their social link, or the social distance between the teams, is given by the number of intermediate nodes on the minimum path between the two. This construct is then used to explain knowledge flows between teams, as measured using citations among over half a million patents from I use a weighted maximum likelihood approach, based on choice-based sampling, to estimate a model for probability of citation between two patents. The existence of a social link between two teams is found to be associated with higher probability of knowledge flow, with the probability decreasing with higher social distance. I also find evidence that social links help explain geographic localization of knowledge spillovers. In particular, conditional on having a close social link, the probability of knowledge flow between two innovating teams is found to be independent of whether they are located in the same geographic region or not. JEL Codes: F2, O3, R1, L0, M2 Keywords: Social Networks, Mobility, Knowledge spillovers, Innovation, Technology diffusion jasjit@jasjitsingh.com. Please check for latest version. I would like to thank Ajay Agrawal, Bharat Anand, Richard Caves, Ken Corts, Lee Fleming, Robert Gibbons, Heather Haveman, Witold Henisz, Tarun Khanna, Steven Klepper, Josh Lerner, Megan MacGarvie, Jordan Siegel, Olav Sorenson, Peter Thompson, Dennis Yao and seminar participants at Harvard, MIT, NBER, CCC Doctoral Consortium and Academy of Management BPS Consortium for helpful discussions and comments. I also thank Division of Research at HBS for funding. Errors remain my own Jasjit Singh

2 1. Introduction and Related Literature A large empirical literature shows that knowledge spillovers tend to be geographically localized. 1 The explanation often given is that knowledge has a tacit element, which cannot be easily codified for transmission over large distances (Arrow, 1969; Teece, 1977). But how should we formally think about diffusion of such knowledge? This paper suggests a way of modeling how transmission of knowledge may be facilitated by the existence of a social link between two teams of inventors. Such a link could be the result of movement of an inventor from one team to the other, or direct collaborative ties between inventors from the two teams, or indirect collaborative ties that link inventors in the two teams through a series of intermediate individuals. The first goal of this paper is to empirically examine knowledge diffusion through such social links. Specifically, I investigate whether the probability of knowledge diffusion between two teams of inventors, as measured using patent citations, decreases with increase in the social distance between them. The second goal of this paper is to study how social links and geographic location interact in determining knowledge diffusion. In particular, I look for evidence on whether geographic localization of social networks, combined with their positive effect on knowledge diffusion, can help explain why knowledge spillovers tend to be geographically localized. How is the tacit knowledge embodied in people propagated? Mobility of individuals with the relevant experience has been shown to be one possible mechanism of knowledge diffusion to existing firms (Saxenian, 1994; Almeida and Kogut, 1999; Rosenkopf and Almeida, 2003) as well as start-ups (Klepper, 2001; Gompers, Lerner and Scharfstein, 2002). However, even in the absence of direct mobility of individuals, knowledge can diffuse between teams that have social ties. Zucker, Darby and co-authors (1998, 2001) as well as Stuart and Sorenson (2003) show that biotechnology startups draw upon the entrepreneur s social ties with star scientists working at universities and research laboratories. Stolpe (2001) argues that social ties resulting from the persistence of past collaborative links between individuals can lead to knowledge diffusion. However, he does not find empirical support for this claim in the liquid crystal display technology sector, possibly because knowledge is easy to codify in this sector or because the sample of collaborations he uses is too small (65). Agrawal, Cockburn and McHale (2003) show that even inventors who change geographic regions continue to be cited heavily by former 1 See, for example, Jaffe, Trajtenberg and Henderson (1993), Branstetter (2001) and Keller (2002). 1

3 collaborators in their original region, reflecting that social ties resulting from past collaborations can be a source of knowledge flows even across regions. I extend this stream of research by considering not just direct social ties but also indirect social ties as an explanation for micro-level knowledge flows. 2 For example, if an inventor X has collaborated with inventor Y, and Y has also collaborated with Z, Z might learn indirectly about X s work through her social tie with Y. There is a long history of sociology literature that studies diffusion of information through such social links (e.g., Ryan and Gross, 1943; Coleman, Katz and Mendel, 1966; Granovetter, 1973; Burt, 1987, 1992; Rogers, 1995; Valente, 1995). 3 With this literature as a motivation, my paper tests if interpersonal contact between teams can explain how new innovations arise by building upon existing knowledge transmitted through the social network of inventors. 4 I then extend this analysis to study if the observed link between geography and knowledge flows could result from the geographic distribution of social ties responsible for knowledge transmission. To explore knowledge diffusion resulting from both direct and indirect social ties between teams of inventors, I use social network analysis (Wasserman and Faust, 1994). Unlike previous studies that typically look at the effect of either inventor mobility or past collaboration in isolation, I provide a general framework called the social proximity graph where these two are just special cases of a measure of social distance between the innovating teams. Social distance between two teams of inventors is defined as the number of intermediate people through which knowledge has to be transmitted in order to diffuse from one team to the other. The social distance is therefore 0 when the two teams have a common inventor, 1 when an inventor from one team has a collaborative tie with someone from the other team, 2 when the two teams have inventors with collaborative ties with a common person, and so on. I then test two hypotheses: First, if knowledge is indeed transmitted though people, the probability of 2 Following common convention in social network analysis, my use of the phrase social ties should be interpreted broadly as representing any kind of relationship between individuals (formal or personal), and not just one or the other. As discussed later, the data on social ties that I use is based on past collaborations on patents, implying that the measured social ties will be a subset of the actual social ties. 3 Most of this literature takes existing social ties as given, and studies the effect of these ties. My paper follows a similar approach. Coleman (1988) describes how such social networks can arise through deliberate investment in social capital by rational actors. Likewise, economists have also started emphasizing the role of deliberate social capital investments in determining economic outcomes (e.g., Glaeser, Laibson and Sacerdote, 2002). 4 Recent papers on the small worlds phenomenon also study collaborative networks similar to my networks of inventors (Watts and Strogatz, 1998; Newman, 2001). But their primary interest is studying the structure of the network itself rather than measuring knowledge flow through these networks. 2

4 knowledge diffusion should be higher between innovating teams with a social link than between teams that are not socially connected. Second, if some knowledge gets dissipated at each intermediate step so that closer social links are more efficient for knowledge diffusion, the probability of knowledge diffusion should be a decreasing function of the social distance between the two teams. To be able to test the above hypotheses, I measure knowledge flows using patent citations among around half a million US Patent Office (USPTO) patents from approximately 3,000 firms, covering years 1986 to 1995 and all manufacturing sectors. I use data on all patents since 1975 to construct backward-looking social proximity graphs for each year from 1986 to In particular, the graph for year t is used to measure social distance from older patents to those for year t. I use a novel citation-level regression framework to test whether social distance helps predict probability of citation in a sample consisting of both actual citations, and a control set of potential citations constructed using a 3-digit technology class match. Unlike the simple regression approach commonly used in existing papers employing such matched samples, I use a weighted maximum likelihood approach (Manski and Lerman, 1977) to avoid estimation biases from choice-based sampling, i.e., from sampling patent pairs with actual citations (the ones ) at a much higher rate than sampling other potential citations (the zeroes ). Another methodological issue is how to control for variation in the propensity to cite arising from differences in technological relatedness of pairs of patents. Thompson and Fox-Kean (2003) have shown that just the matching based on coarse 3-digit technological classification of patents does not suffice. To remedy this, I use information on the 9-digit technological classification of patents, while also taking into account not just the primary technological class (typically used in existing studies) but also additional technological classes for a patent as reported by USPTO. The analysis reveals strong empirical support for both the hypotheses outlined earlier, highlighting the importance of social networks for knowledge diffusion. The second part of the paper explores whether inventor mobility and social networks can help explain localization of knowledge spillovers. A large empirical literature, starting with Jaffe, Trajtenberg and Henderson (1993), shows that knowledge flows are greater within the same region than across regions. Could this be because inventor mobility and direct or indirect social ties, which promote knowledge flows, also tend to be local? In my regression analysis, I find that introducing control variables for social distance leads to a statistically significant drop 3

5 in the marginal effect of co-location of patents on probability of citation between them. This suggests that social links could indeed be a mechanism driving the observed localization of knowledge flows. However, although the drop in this marginal effect is quite large when both inter-firm and intra-firm citations are considered, it becomes much smaller in a sub-sample with only inter-firm citations. One plausible explanation is that errors in variables resulting from the noisiness of patent data leads to the detected effect being small. To confirm this, I repeat the analysis for the biotechnology industry, since patent-based measures of innovation and knowledge flows are likely to be least noisy in this setting (Levin, Klevorick, Nelson and Winter, 1987). I find that controlling for social distance now leads to a much larger drop in the estimated effect of co-location even in inter-firm settings, showing the importance of inventor mobility and social networks in explaining localization of knowledge flows. In their study of patent citations from 366 Italian patents, Breschi and Lissoni (2002) examine interaction effects of geographic co-location and social connectedness on knowledge flows. They find the association between patent citations and geographic co-location to be greater for socially connected patents. However, they do not distinguish between patent pairs with different social distances between them. I find that this distinction turns out to be quite important: For socially close patents, geographical co-location has very little or no effect on citation probability after conditioning on social distance. Thus, close relationships seem to be a substitute for geographic proximity. In contrast, for patent pairs with large social distances or no social link at all, geographic co-location continues to be associated with higher citation probability. One explanation could be the existence of other kinds of network relationships that substitute for collaborative links captured by patent data. If these unmeasured relationships are geographically localized, and also lead to greater knowledge flows, geographic co-location would indeed turn out to be significant for cases where two patents are not socially close using patent-based measures. The paper is organized as follows. Section 2 describes the patent citation data as well as the data on inventors. Section 3 formally defines social proximity graph and social distance, which I subsequently use to study the effect of inventor mobility and social networks on knowledge flows between innovating teams. Section 4 introduces a citation-level regression framework based on choice-based sampling in order to estimate a model for the probability of knowledge flow as captured using patent citations. Section 5 reports the 4

6 analysis of the association between social distance and probability of knowledge flows. Section 6 investigates whether social networks are able to explain geographic localization of knowledge flows. Section 7 offers concluding thoughts. 2. Patent Data I use patent citations to measure micro-level knowledge flows across innovations. Since patent citations are intended to delimit the property rights of the citing patent by listing all relevant prior art, they leave behind a trail of how new innovations build upon existing ones. In contrast with citations made by academic papers, the inventor of a patent has an incentive not to include unnecessary citations (as it only reduces the scope of his patent), and yet is legally bound to cite relevant prior art for the innovation (with the possibility of legal prosecution if found to have deliberately omitted prior work he or she builds upon). In addition, an important job of the patent examiner is to ensure that the inventor does not deliberately omit any relevant prior art, increasing the objectivity of patent citations. Sometimes a patent examiner might add relevant citations of which the original inventor was unaware, adding noise to patent citations as a measure of knowledge flows. Nevertheless, as recent studies (Jaffe, Trajtenberg and Fogarty, 2000; Duguet and MacGarvie, 2002) have verified through a direct comparison of patent citations with data on knowledge flows reported directly by inventors of these patents, the correlation between patent citations and actual knowledge flows is fairly high. This justifies the use of patent citation data as a reasonable, though noisy, proxy for knowledge flows USPTO Patents Patents from different patent offices are not comparable to each other because of differences in patent breadth, patenting costs, citation practice, approval requirements and enforcement rules across countries. Therefore, it is common practice in academic research to use data from a single patent-granting country in order to standardize the unit of innovation. 5 Firms rely not just on patents but also on other mechanisms such as secrecy, complementary sales, service capabilities, and quicker lead times for appropriating returns from innovation (Levin, Klevorick, Nelson and Winter, 1987). In addition, when patents do get used, it is sometimes to block the development of a substitute or a threat to force rivals into negotiations. Nevertheless, patents have been shown to be a reasonable, though noisy, measure for innovation and are widely used as such in research (Griliches, 1990). It is worth emphasizing that I do not use raw patent or citation counts for cross-country comparisons, so there is no bias from differences in propensity to patent across different types of inventors. I only use US patent data to track knowledge flows conditional on the given set of innovations as embodied in US patents. 5

7 Following this practice, I use a dataset on US patents that I obtain by combining raw data from the US Patent Office with an NBER dataset described in Hall, Jaffe and Trajtenberg (2002). Since the US is the largest and technologically most advanced market in the world, any innovation made with a global market in mind has a high probability of being patented with the USPTO. In addition, the USPTO maintains much cleaner and detailed data on inventors than any other patent office. 6 It is worth emphasizing that I do not use raw patent or citation counts for cross-country comparisons, so differences in the propensity to patent across different types of inventors (e.g., US inventors being more likely to patent in the US than non-us inventors) are not a source of bias for my results. I only use US patent data to track knowledge flows conditional on the given distribution of innovations as embodied in US patents. As described later, I use country fixed effects to control for differences across countries in the propensity of inventors to cite US patents. In a related paper (Singh, 2003), I use European Patent Office (EPO) data to analyze of the extent of possible biases from using USPTO patent citations to compare knowledge diffusion across different countries and firms. The basic finding is that, although there are some possible biases, the magnitude of such biases is small for data from the most developed countries Inventors A record in the patent database includes not just information on the assignee firm and the application year for the patent, but also the names and addresses of individuals who made the innovation. Thus, by analyzing these data, one can infer mobility of inventors across project teams as well as the history of their past collaborations. A big challenge in doing so, however, is correctly identifying the inventors themselves. In order to decide when the inventors listed for different patents are actually the same person, I use available information on the first, middle and last names of the inventors, and on the technological characteristics of their patents. The simplest approach would have been to use just first and last names to identify inventors, but matching on these two fields alone would often lead to several inventors getting bunched together as one. The middle name 6 USPTO uses first-to-invent rule, i.e., patents are applied for by inventors, who then typically assign them to their employer as a part of their employment contract. In contrast, the first-to-file system followed by European Patent Office (EPO) and some other patent offices involves companies applying directly for patents in their own names, and simply listing the inventors in the document. This, at least partly, explains why quality of inventor data is much better for USPTO data. 6

8 information is potentially useful in distinguishing these, but is erratic - the field is sometimes blank even for people with middle names, or may contain just the middle initial instead of the full middle name. I experimented with several methods to arrive at reasonable compromise between generating too many false positives (i.e. two different inventors being incorrectly identified as being the same) and too many false negatives (i.e. records of the same inventor being incorrectly identified as belonging to two different inventors). In the end, I found the following algorithm to work well. Two different records are taken as belonging to the same inventor if and only if the following three conditions are satisfied: 1. The first and last names match exactly for the two records. 2. The middle initials, if available for both records, are the same. 3. When the middle initial field is blank in at least one of the two records, the records should overlap on at least one of the primary or additional technology "subcategories". 7 Using just the first two criteria would have identified around 1.3 million distinct inventors. The third condition makes the matching criteria more stringent, leading to around 1.7 million inventors in the entire USPTO database from 1975 to As mentioned above, there are bound to be errors in identifying inventors. However, unless there was any reason to believe that these are systematic, this should only cause an attenuation bias that understates the effect of inventor mobility and social networks on probability of knowledge diffusion Patent Citations among Firms This paper aims to study the effect of inventor mobility and networks on knowledge diffusion within and between firms. Therefore, observations involving patents assigned to individuals and non-firm organizations were not included in the data on patent citations. 9 In 7 The subcategory definition is taken from Hall, Jaffe and Trajtenberg (2002), who divide the 418 US patent classes into 38 different subcategories. I also tried using the US patent class itself as a requirement in (3) instead, but that turns out to be too conservative as the same inventor (as determined by exact match along other fields including the address) very often has patents with non-overlapping US patent classes, leading to too many "false negatives" in the matching. 8 I also tried to rule out more false positives by checking for overlap of patent citations, but that turned out to be too conservative as citations of different patents of even the same inventor are actually not as closely overlapping as I had expected. A match on street address and/or on patent assignee is also not useful since mobility across location and firms is a central issue being studied here! 9 Note that such patents were still included in the definition of social proximity graph as described below since, for example, two teams of inventors working for firms might be linked through an inventor from a university, research laboratory or government body. 7

9 addition, since I distinguish between citations within firms and across firms, it becomes important to correctly identify whether a citation between two distinct patent assignees is actually one between two subsidiaries of the same firm or a citation between distinct firms. Original assignee information in patents does not always refer to the corporate parent, often pointing instead to the name of a subsidiary listed as a distinct assignee in the data. Therefore, the data had to be extensively cleaned. Through a process described in detail in Singh (2003), I did this using a combination of CUSIP information from 1989, Stopford s 1992 Directory of Multinationals, Dun and Bradstreet s Who Owns Whom directories from 1991, Internet sources, keyword search and manual checking. In the process, I examined about 10,000 patent assignees and identified about 4,400 unique organizations, about 3,000 of which are firms, 400 are government-affiliated bodies, 550 are research institutes and 450 are universities. The 3,000 or so firms account for around half a million patents from , or around half of the overall USPTO patents for this period. I restricted my analysis to patent citations within the set of these firms, looking only at citing and cited patents between 1986 and 1995 according to their application dates. 10 Restricting the observations to this 10-year window has two advantages. First, I can be reasonably sure of the parent firm information (which is inferred from data sources from around 1990, as mentioned above). Second, I can use the pre-1985 data to infer historical ties that existed even in the beginning of my time window. This led to about 1.3 million citations between patents in the included set, about 1 million of these being between patents of different firms and 0.3 million being self-citations by firms. I separately study the effect of inventor mobility and social networks on knowledge diffusion for two samples: one that includes selfcitations by firms, and the other that excludes them. 3. Measuring Social Distance between Innovating Teams Data on patents and inventors can be seen as an instance of what is called an affiliation network in social network analysis (Wasserman and Faust, 1994, Chapter 8). This affiliation network consists of two kinds of nodes: the inventors (the actors ), and the patents (the events ). One way to represent the network is using an affiliation matrix A = {a ij }. In this matrix, element a ij is 1 if the ith inventor is on the innovating team for the jth patent, and is 10 It is worth re-emphasizing that, even though patent citations are analyzed for only the 3,000 or so selected firms and only for year 1985 onwards, the social ties used to explain these citations are inferred using patents filed by all 8

10 0 otherwise. Figure 1 gives an example affiliation matrix, with 7 inventors A, B, C, D, E, F and G, and 7 patents P1, P2, P3, P4, P5, P6 and P7. A value of 1 for element (A, P1) and 0 for element (C, P1), for example, implies that A is one of the inventors for patent P1, but C is not. The first step in measuring social relatedness is to derive an undirected graph that I call the social proximity graph. In the social proximity graph for year t, I include as nodes all patents with application years less than or equal to t. There is an edge between patents X and Y if and only if the two have a common inventor. 11 This common inventor serves as a bridge of knowledge flow between the two teams. Any two patents not linked via a direct edge would still have a path between them as long as the two teams are linked via a series of collaborations. In other words, two innovating teams can be seen as being socially related if they belong to the same connected component of the social proximity graph. To measure the strength of the social link, the social distance between any two related patents can be defined as the number of intermediate nodes in the minimum path (the geodesic) between the two, or one less than the minimum path length. This minimum path can be interpreted as a sequence of people through which knowledge can be transmitted in order to diffuse from one innovation to another. The underlying model behind this is the one that is commonly assumed in social network analysis of collaborative networks: the direct social tie between two persons persists far beyond the date of their recorded collaboration (Newman, 2001; Stolpe, 2001; Breschi & Lissoni, 2002; Agrawal, Cockburn and McHale, 2003; Fleming, Juda and King, 2003). Thus, through a sequence of such social ties, the knowledge embodied in inventors for a patent can get transmitted to other teams that are socially related to it. To make the above discussion clearer, let us analyze the affiliation network from Figure 1. Figure 2 constructs corresponding social proximity graphs for 1987, 1988, 1989 and These graphs are then used to calculate social distances between pairs of innovating teams, as reported in Figure 3. Let us start with the graph for 1987 from Figure 2(a). Since there is a direct link between the teams for patents P1 and P2 through their common inventor inventors (whether unassigned or assigned to any firm or organization) and for year 1975 onwards. 11 This social proximity graph is analogous to the networks of individuals defined in the Small Worlds literature (Watts and Strogatz, 1998; Newman, 2001; Fleming, Juda and King, 2003). That literature, however, uses nodes to represent individuals instead of teams, with edges between two individuals if they have ever collaborated on the same team. Conceptually, the two approaches are equivalent (Wasserman and Faust, 1994, Chapter 8). In my 9

11 A, the social distance for P1 Ø P2 is 0. Note that knowledge flows only make sense from an innovation that happens earlier to one that happens later. Social distance is therefore left undefined for P2 Ø P1, P1 Ø P1 and P2 Ø P2 in Figure 3. Looking at year 1988 in Figure 2(b), a similar reasoning gives a social distance of 0 for P2 Ø P3. While there is no direct link between P1 and P3, knowledge from P1 can flow to P3 indirectly by being passed from A to C (who collaborate for P2). Therefore, social distance is 1 for P1 Ø P3. Let us now look at year 1989 in Figure 2(c). The above definition of social distance gives a value of 1 for P2 Ø P4, since there is a path P2 Ø P1 Ø P4. Does this make sense even though P1 precedes P2 in time? If the year of a recorded collaboration were literally the only time when knowledge could be passed between the collaborating inventors, the social proximity graph would have to be a directed graph, with the edges being unidirectional from the earlier patent to the later patent. In other words, the application year of every intermediate patent on the minimum path would have to exceed that of the one preceding it, and there would thus be no path from P2 to P4. However, if A and B continue to maintain social ties after their collaboration in 1986, B (who is the inventor for P4) can build upon knowledge of P2 that she may gain through her social ties with A. Therefore, node P1 should not be narrowly interpreted as just a collaboration in 1986, but more broadly as an ongoing social tie that started in With this interpretation, knowledge can flow backwards along the link P1 Ø P2, and then on to the link P2 Ø P4. Likewise, knowledge from P3 could be passed by C to A, and then further from A to B. In other words, knowledge flows through the chain of social ties P3 Ø P2 Ø P1 Ø P4, making the social distance P3 Ø P4 to be 2. Other pair-wise social distances can be calculated following a similar logic. Note that, in measuring the social distance relevant for a knowledge flow to an innovating team, we need to use the social proximity graph as it existed in the year in which the innovation took place. For example, the correct value of social distance from P3 to P6 is infinity (since P6 took place in 1989, and P3 and P6 are not even in the same connected component in 1989) and not 2 (as an incorrect interpretation of the 1990 graph might suggest). As the above example illustrates, the social proximity graph relevant for the new innovations made in year t should use data on patents with application years up to year t. setting, it is just more natural to define the collaborating teams as nodes since knowledge flows, as measured using patent citations, are from one team to another. 10

12 Therefore, for my analysis, I use separate social proximity graphs for t=1986 through t=1995 to cover all the years for which I analyze knowledge flows. Following the convention in related literature, I define this graph cumulatively, i.e., including all innovations from the first year that I have the data for (1975) until year t. 12 There are two practical issues in doing actual analysis using social distance as defined above. First, one would like flexibility of functional form in studying the effect of social distance on knowledge diffusion, and would like to avoid mixing what seem like apples and oranges into a single cardinal measure (e.g. the common inventor case with distance=0, and the past collaboration case with distance=1). Second, because of the large graph size of around 0.5 million nodes, computing exact pair-wise social distances for millions of pairs (representing actual or potential citations) is impractical. 13 However, it is still practical to classify all observations into the following five mutually exclusive and exhaustive categories based on whether the social distance is 0, 1, 2, any finite value greater 2, or infinity: 14 Common inventor: There is a common inventor between the citing and cited patents, i.e., the social distance is 0. Past collaboration: An inventor of the citing patent has collaborated with an inventor of the cited patent. This corresponds to a social distance of 1. Common past collaborator: An inventor of the citing patent and an inventor of the cited patent have both separately collaborated with the same person. The social distance for this case is In order to address the resulting incomparability of the derived social distance measures for different years, I use year fixed effects in all regressions. If we believed that ties go stale a few years after a joint collaboration, an alternative would be to use a rolling time window, e.g., use patents from years t-10 to t in defining the social proximity graph in year t. 13 Wasserman and Faust (1994, chapter 4) suggest the following for computing pair-wise distances: Represent the graph using a matrix X, with element x ij being 1 if there is an edge between nodes i and j, 0 otherwise. The distance between nodes i and j is the smallest number p such that the p th power matrix of X (i.e. p-1 multiplications of X into X) has a non-zero entry for row i and column j. Unfortunately, this is impractical when number of nodes is large since the power matrices quickly become non-sparse. Graph theorists have offered alternate methods, but they are also impractical since they require O(N 3 ) computational resources (Cormen, Leiserson and Rivest, chapter 26). 14 I explicitly find out all pairs with a minimum path length of 1, 2 or 3 (i.e. a social distance of 0, 1or 2) by calculating the first 3 power matrices mentioned above, since these matrices are sparse enough to be computationally manageable. Also, I am able to distinguish between having a more indirect social link (with social distance > 2 but finite) and not having a social link at all (i.e., social distance of infinity) since it is computationally practical to find all connected components of a graph. This gives the five cases. 11

13 Indirect social link: None of the above hold, but the two patents still belong to the same connected component of the social proximity graph. This corresponds to a social distance that is finite, but more than 2. No social link: The patents do not belong to the same connected component. In other words, the social distance is infinity. The first goal of the paper is to formally test if knowledge is propagated through people. Formally, I first test if the probability of citation is higher between patents that are socially related than between patents that are not socially related. Further, we might expect that the knowledge diffusion effect of social relatedness weakens as the number of intermediate ties needed increases. This leads to a test that the probability of citation is decreasing for the five successive cases described above: common inventor, past collaboration, common past collaborator, indirect social link and no social link. After establishing these results first, the paper will then return to the issue of studying if geographic localization of knowledge flows can be explained by social networks of inventors being localized. 4. Regression Framework Imagine that the probability that a patent K cites a patent k is given by a citation function P(K, k). Our interest lies in testing the hypothesis that P(K, k) is higher for socially related patents, and is a decreasing function of the social distance between the two patents. However, in testing these hypotheses, it is crucial to control for other factors that affect the propensity to cite. As discussed below, not having appropriate controls could easily lead to an omitted variable bias. I now present a citation-level regression framework that makes such controls possible Choice-Based Sampling 15 Since the number of potentially citing and cited patents can be of the order of a million, the number of all possible dyads (K, k) can be of the order of a trillion. In principle, we could take a random sample of patent dyads from the population of all possible dyads. We 15 Because of similar methodology, the discussion here overlaps significantly with that in Singh (2003). 12

14 could then define a binary variable y that equals 1 if the citation actually takes place, and 0 otherwise, and estimate the citation function by assuming that it can be approximated using a logistic functional form. In other words, the dichotomous dependent variable y would be taken as a Bernoulli outcome that takes a value 1 for observation i with the probability 1 Pr( y = 1 x = xi ) = Λ( xi β ) = 1+ e x i β where x i is the vector of covariates and β is the vector of parameters to be estimated. However, an estimation approach based on random sampling of patent pairs is not practical because actual citations are very rare. There are only about seven actual citations for every one million potential citations, making estimation impossible even with large samples. From an informational point of view, it would be desirable to have a higher fraction of observations with y = 1 in the sample. 16 An alternative approach is to use a choice-based sampling procedure that deliberately oversamples the patent pairs with actual citations, i.e., with y = In this approach, the sample is formed by taking a fraction α of the population s dyads with y = 0, and a fraction γ of the dyads with y = 1, where α is much smaller than γ. Since this stratification is done on the dependent variable, however, using the usual logistic estimates would lead to a selection bias. If we were sure that the underlying functional form is indeed logistic, it is possible to make a correction for this bias. 18 The efficiency of the correction, however, depends crucially on the assumption that the logit functional form is not misspecified (Manski and Lerman, 1977; Cosslet, 1981). Since there is no reason to expect that the functional form would hold exactly, a better option is to use the weighted exogenous 16 ' The asymptotic covariance matrix for the MLE for logit is given by Λ (1 Λ ) x x (see Greene, 2003, p. 672). If the logit model has some explanatory power, L i is larger (i.e. closer to 0.5 for rare events) when y i =1. Thus L i (1-L i ) is larger for these observations, implying that having a higher fraction of 1 s in the sample would, other things being equal, reduce variance. 17 For a general discussion of choice-based sampling, see Amemiya (1985, pp ), Greene (2003, p. 673) or King and Zeng (2001). Sorenson and Fleming (2001) have also used this technique for predicting patent citations. 18 The probability of a citation conditional on the dyad being in the sample flows from Bayes rule: Λ ' i = γ Λ i γ Λ + α = γ i βx i γ ( 1 Λ i ) γ + αe (ln + βx i ) α = 1+ e 1 This differs from Λ i as there is now an extra term ln(γ/α) in the exponent, leading to a bias. However, since the functional form is still logistic, a simple estimation strategy is to use the usual logit-based maximum likelihood estimation, and then subtract ln(γ/α) from the estimate for the constant term. n i= 1 i i i i 1 13

15 sampling maximum likelihood (WESML) estimator suggested by Manski and Lerman (1977). The central idea is to explicitly recognize the difference in sampling of 0 s and 1 s by weighting each term in the log likelihood function by the inverse of the ex ante probability of inclusion of the corresponding observation in the sample. In other words, each sample observation is weighted by the number of elements it represents from the overall population in order to make the choice-based sample simulate a random exogenous sample. 19 The WESML estimator is then obtained by maximizing the following weighted pseudolikelihood function: ln L w = 1 γ 1 ln( Λ i ) + α { y = 1} { y = 0} i i ln(1 Λ ) i = n i= 1 w ln(1 + e i (1 2 yi ) xiβ ) where w i = ( 1/ γ ) yi + (1/ α)(1 yi ). In addition, the appropriate estimator of the asymptotic covariance matrix is White s robust sandwich estimator used for pseudo-maximum likelihood estimation. Since the same citing patent can occur in multiple observations, the standard errors should be calculated without assuming independence across these observations Sample Construction While the above sampling design is an improvement over random sampling, it can be improved further. In particular, the above approach still samples all y = 0 observations with equal probability, irrespective of their relevance. To see why that might be an issue, recall that the technological similarity of two patents is particularly relevant for the probability of citation. Therefore, to estimate other coefficients efficiently, we would ideally like to have a sample with 19 For intuition, let the joint probability density be g(x,y) for the sample, and g*(x,y) for the population. Let the fraction of elements with y = j be f(j) in the sample, and f*(j) in the population (j = 0,1). Let n and N be sample size and population size respectively, and n j and N j be the number with y = j. Using conditional probability rules, g * ( x, j) f ( j) g * ( x, j)( n j / n) N / n g( x, j) = Pr( x y = j) f ( j) = = = g * ( x, j) f * ( j) N / N w( j) j where w(j) = N j /n j is the reciprocal of the sampling rate for observations with y = j. Let P(y i ) be the probability of y = y i conditional on x = x i in the population. Then, the expected value of the weighted likelihood function is n n n N / n N E ln Lw w( yi )[ln P( yi )] g( x, yi ) dx = = w( yi )[ln P( yi )] g * ( x, yi ) dx = [ln P( yi )] g * ( x, yi ) dx i 1 i 1 w( yi ) n = = i= 1 Thus, ignoring the constant scaling factor N/n, the expected value of the weighted log likelihood equals the expected log likelihood for the same sample resulting through random exogenous sampling from the population. As shown formally in Amemiya (1985, section 9.5.2), this ensures consistency of WESML estimation. 14

16 sufficient variation in the dependent variable within groups of patents with comparable degree of technological similarity. However, randomly drawn y = 0 observations typically have patent dyads that are not technologically related while y = 1 observations are very often technologically related. One way to improve the estimation is to extend the basic choice-based sampling design to also allow matching on the technological class of the citing and cited patents. The weighted likelihood function described above now has to be generalized since the probability of a y = 0 pair getting selected depends on the technological classes of the two patents. As an appendix in Singh (2003) shows, this generalization involves defining the weight attached to a y = 0 observation as the reciprocal of the ex ante probability of a y = 0 population pair with the same respective technological cell (i.e. combination of technological classes for the citing and cited patents) being selected into the sample. I constructed the sample as follows. Each of the 1.3 million citations was matched with 10 control patent pairs such that the selected patents in each pair belonged to the same technology cell as the original citation, i.e., had the same respective technological classes as the patents in the original citation. Out of these, all of the potential citations from a firm to itself in the same location, and also all potential citations with the citing application year being later than the cited patent year were then dropped. In addition, at least one representative control citation was included even for cells with no y = 1 observations. This procedure led to a total of 7.6 million control citations in the sample. Each of these was then weighted by the inverse of the fraction of y = 0 observations from its technology cell included in the sample. In addition, each of the original citations was assigned a weight of one since all actual citations from the population had been included in the sample Control Variables for Propensity to Cite In testing the impact of social distance of a pair of patents on the probability of citation, we need to control for other factors that might affect the propensity to cite between two patents. The first issue is time effects. It is well documented that the citation probability increases initially as the time lag between the citing and cited patents increases, but then starts to fall after a certain point (Jaffe and Trajtenberg, 2002). Since the exact shape of this lag function is not central to this paper, but the lag effect does needs to be controlled for, I introduce dummy variables for the number of years of lag between the citing and cited patent. 15

17 In addition, since the probability of citations may depend on the citing year (Hall, Jaffe and Trajtenberg, 2002), additional dummy variables are used to capture the citing year fixed effects. The next issue is that innovators in different countries might have a different propensity to cite patents registered with the USPTO. For example, a USPTO patent filed by a Japanese firm might not necessarily cite the USPTO patent for the innovation it builds upon, but might instead cite the corresponding Japanese patent for that innovation. In order to avoid possible biases arising from this, all regressions include fixed effects for the citing country. Another critical issue affecting propensity to cite is the technological characteristics and relatedness of the citing and cited patent. First, patents in different industry categories have different propensities to patent. To address this, I include fixed effects for the broad technological category of the citing patent, as defined in the NBER database. Second, patents with similar technological characteristics are more likely to cite each other, an issue that has not been adequately addressed in existing studies on knowledge diffusion. Since closely related inventors are also more likely to work in similar technologies, inadequate technology controls can lead to an omitted variable bias. In order to operationalize technological similarity between two patents, I define dummy variables for the same broad technological category, the same technological subcategory, the same 3-digit primary class and the same 9- digit primary class. This gives a set of hierarchical controls for technological relatedness of patents. However, the designation of one of the patent s subclasses as primary can often be almost arbitrary. Therefore, I also include another dummy variable that captures whether at least one of the secondary subclasses of a patent is the same as the primary or one of the secondary subclasses for the other patent in the pair. Using such careful controls for propensity to cite is an important contribution of this paper. As Thompson and Fox-Kean (2003) show, not using such controls has led to significant biases in the results reported in existing studies on knowledge diffusion Existing regression-based studies do not typically capture knowledge flows at the individual citation-level. Instead, they use a more aggregate dependent variable like the number of citations at the level of a country, a firm or some other level of aggregation (Jaffe and Trajtenberg, 2002). These models control for propensity to cite by including a measure of average technological distance between the aggregate sets of citing and cited patents, and ignore within-set heterogeneity. Since this technological distance is measured only at the coarse 2 or 3-digit technology classification level, the issue of biases discussed earlier still remains. To see this, imagine that there are no localized knowledge spillovers, and that technological distance as measured at the 3-digit level is the same for all observations. Even so, citing and cited sets of patents with a higher fraction of patent pairs belonging to the same 16

18 5. Results 5.1. Summary Statistics Table 1 defines the variables used in the analysis. In particular, based on the earlier discussion, common inventor, past collaboration, common past collaborator, indirect social link and no social link are dummy variables used to capture a social distance of 0, 1, 2, a finite value more than 2, and infinity respectively. For all cases where two patents belong to the same connected component (i.e. have a finite path between them) in the social proximity graph, exactly one of first four dummy variables would have value 1. Table 2 reports summary statistics for the entire sample as well as for the sub-sample that excludes self-citations from a firm to itself. For the entire sample, the fraction of pairs belonging to the same connected component is 58% for pairs with citations, and only 44% for pairs with no citation, consistent with the hypothesis that connectedness leads to higher probability of citation. The inequality continues to hold true for the sub-sample without self-citations, where the fraction of pairs belonging to the same connected component is 50% for pairs with citations, and only 44% for pairs with no citation 5.2. Regression Analysis The summary statistics do not control for differences in other factors that affect the probability of a citation. To analyze the data more carefully, I now turn to regression analysis based on the methodology discussed in the previous section. The results of weighted logit regressions (WESML) appear in Table 3. Consistent with the summary statistics, the estimates for common inventor, past collaboration, common past collaborator and indirect social link in column (1) are all positive and significant. In addition, the magnitude of this effect is highest for common inventor, followed by past collaboration, then by common past collaborator, and finally by indirect social link. In other words, the probability of citation falls as the social distance for the pair of patents increases. Note that the reference group for comparison is the pairs of patents that are not connected at all in the social proximity graph. The mean predicted value of patent citation is about 7.4 in a million. Using this value, the marginal effects can be detailed 9-digit technology will have a higher fraction of citations and also a higher co-location of citing and cited patents (the latter resulting purely from technological specialization of regions), leading to a spurious correlation. 17