Page - (000299) - in Knowledge and Networks

296 Importantly, the definitions above imply that the sum of (14.2) and (14.3) is equal to the numerator of (14.1), in other words, to the overall distance-weighted external reach of city c. Armed with these definitions, we constructed several variables in order to test for the importance of gatekeepers. First, we computed (for each city c and each 5-year time window) the number of inventors who perform the function of gatekeepers. This variable tests whether the mere quantity of gatekeepers can be of relevance for a city’s capacity to recombine and expand its knowledge domains. Second, we computed the share of the overall external reach, which is either directly or indirectly mediated by gatekeepers. Formally: SHREACH GK d d c i n j n ij ind i i GK n j n ij i n c h c h c _ = + = = = ∈ = = ∑ ∑ ∑ ∑ ∑ 1 1 1 1 1 1 1 j n ij h d=∑ 1 1 (14.4) This variable captures to what extent a city’s overall external reach would decrease when all gatekeepers are removed from the city (Borgatti, 2006; Valente & Fujimoto, 2010), in other words, how robust its external relations are to the removal of the links established by gatekeepers or, conversely, how much gatekeepers con- trol the flows of knowledge across cities. Hence, it is suitable for testing whether gatekeepers are fundamental mediators of knowledge exchanges leading to higher technological recombination or whether direct relations (also not mediated by gate- keepers) are more effective. Higher values of the index correspond to cities in which external linkages mostly rely upon gatekeepers. On the other hand, lower values of the index imply that most of the linkages with externally located inventors are direct and do not need any intermediation (i.e., removing the gatekeepers would not diminish substantially the external reach).5 Third, we decomposed the overall distance weighted external reach mediated by gatekeepers (i.e., the numerator of (14.4)) into its two major components, i.e., the direct and the indirect and we computed the following shares: SHDIR GK d d c i i GK n j n ij i n j n ij c h c h _ = = ∈ = = = ∑ ∑ ∑ ∑ 1 1 1 1 1 1 (14.5) 5 For a fuller discussion of this index, see Breschi and Lenzi (2015). S. Breschi and C. Lenzi