The importance of network structures for the transmission of knowledge and the diffusion of technological change has been recently emphasized in economic geography. Since network structures drive the innovative and economic performance of actors in regional contexts, it is crucial to explain how networks form and evolve over time and how they facilitate inter-organizational learning and knowledge transfer. The analysis of relational dependent variables, however, requires specific statistical procedures. In this paper, we discuss four different models that have been used in economic geography to explain the spatial context of network structures and their dynamics. First, we review gravity models and their recent extensions and modifications to deal with the specific characteristics of networked (individual level) relations. Second, we discuss the quadratic assignment procedure that has been developed in mathematical sociology for diminishing the bias induced by network dependencies. Third, we present exponential random graph models that not only allow dependence between observations, but also model such network dependencies explicitly. Finally, we deal with dynamic networks, by introducing stochastic actor-oriented models. Strengths and weaknesses of the different approach are discussed together with domains of applicability the geography of innovation studies.
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A burgeoning literature starts to integrate the geographical dimension in sociology and network science: see for instance the special issue 34.1 in Social Networks of January 2012 on Capturing Context: Integrating Spatial and Social Network Analysis, edited by Jimi Adams, Katherine Faust and Gina Lovasi.
See the special issue 43.3 in The Annals of Regional Science of September 2009 on Embedding Network Analysis in Spatial Studies of Innovation, edited by Edward Bergman.
For an early overview of studies that applied the gravity model in economic geography, see Lukermann and Porter (1960).
However, the term “gravity model” is not often used when studies are conducted at the micro-level. Rather scholars research the effect of geographical proximity on network formation.
In practice, it would be possible to estimate the gravity model with these techniques.
Please note that we only discuss problems specifically pertaining to network data. Other problems related to, for example, the fact that the outcome is not always a continuous numeric variable and the many zeros in the network (e.g., Helpman et al. 2008; Burger et al. 2009b) and causality (e.g., Egger 2004) are discussed elsewhere in the literature. Although these are problems that all empirical researchers are facing, a discussion of these issues is beyond the scope of this paper.
Another (non-spatial) method that controls for the network structure but is not often used in the gravity model literature is the multiple regression quadratic assignment procedure (MRQAP). A more elaborate discussion of this method can be found in the next section.
For a more elaborate critique on the use of remoteness indices, see Anderson and Wincoop (2003).
Accordingly, MRQAP is rather a particular permutation method for hypothesis testing and not a model on its own. However, we will refer to it as model in the following to keep a consistent terminology.
See Lusher et al. (2013) for a more detailed introduction to ERGM.
More details can be found in Robins et al. (2007).
This class of models is often referred to directly as SIENA models. SIENA stands for “Simulation Investigation for Empirical Network Analysis.” The RSiena package is implemented in the R language and can be downloaded from the CRAN website: http://cran.r-project.org/web/packages/RSiena/.
See Liu et al. (2013) for an example of how GM can be used to model regional networks.
As pointed out by one of the referee, it is possible to avoid complete cliquishness or to go beyond assuming symmetric ties in two-mode networks if researchers have detailed data on the level of involvement/learning of actors in a given event.
In SAOM, it is assumed that all agency ruling the dynamics of the network comes from the actors of the first mode of the two-mode network (Snijders et al. 2013). As a result, the second mode is passive and cannot decide to establish a link with the first mode. Besides, no coordination is possible between the first and second mode.
Almeida P, Kogut B (1999) Localization of knowledge and the mobility of engineers in regional networks. Manag Sci 45:905–917
Anderson JE, van Wincoop E (2003) Gravity with gravitas: a solution to the border puzzle. Am Econ Rev 93(1):170–192
Autant-Bernard C, Billand P, Frachisse D, Massard N (2007) Social distance versus spatial distance in R&D cooperation: empirical evidence from European collaboration choices in micro and nanotechnologies. Pap Reg Sci 86:495–519
Balland PA (2012) Proximity and the evolution of collaboration networks: evidence from research and development projects within the global navigation satellite system (GNSS) industry. Reg Stud 46:741–756
Balland PA, Belso-Martínez JA, Morrison A (2014) The dynamics of technological and business networks in industrial clusters: embeddedness, status or proximity? Pap Evolut Econ Geogr 14:12
Balland PA, de Vaan M, Boschma R (2013) The dynamics of interfirm networks along the industry life cycle: the case of the global video games industry 1987–2007. J Econ Geogr (forthcoming)
Barca F, McCann P, Rodriguez-Pose A (2012) The case for regional development intervention: place-based versus place-neutral approaches. J Reg Sci 52:134–152
Behrens K, Ertur C, Koch W (2012) Dual’ gravity: using spatial econometrics to control for multilateral resistance. J Appl Econom 27(5):773–794
Bell GG (2005) Clusters, networks, and firm innovativeness. Strateg Manag J 24:287–295
Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc Ser B (Methodological) 36(2):192–236
Besag J (1975) Statistical analysis of non-lattice data. J R Stat Soc Ser D (The Statistician) 24:179–195
Bikker JA (2010) An extended gravity model with substitution applied to international trade flows. In: van Bergeijk PAG, Brakman S (eds) The gravity model in international trade: advances and applications. Cambridge University Press, Cambridge, pp 135–162
Boschma R (2005) Proximity and innovation: a critical assessment. Reg Stud 39:61–74
Boschma R, Balland PA, Kogler D (2011) A relational approach to knowledge spillovers in biotech. Network structures as drivers of inter-organizational citation patterns. Papers in evolutionary economic geography 11.20, Utrecht University, Utrecht
Boschma R, Frenken K (2006) Why is economic geography not an evolutionary science? Towards an evolutionary economic geography. J Econ Geogr 6:273–302
Breschi S, Lissoni F (2009) Mobility of skilled workers and co-invention networks: an anatomy of localized knowledge flows. J Econ Geogr 9:439–468
Broekel T, Hartog M (2013a) Explaining the structure of inter-organizational networks using exponential random graph models: does proximity matter? Ind Innov 20(3):277–295
Broekel T, Hartog M (2013b) Determinants of cross-regional R and D collaboration networks: an application of exponential random graph models. Working Papers on Innovation and Space, #13.4
Broekel T, Boschma RA (2012) Knowledge networks in the Dutch aviation industry: the proximity paradox. J Econ Geogr 12:409–433
Burger MJ, Van Oort FG, Frenken K, Van der Knaap B (2009a) Networks and economic agglomerations: introduction to the special issue. Tijdschr voor Econ Soc Geogr 100:139–144
Burger MJ, Van Oort FG, Linders GJM (2009b) On the specification of the gravity model of trade: zeros, excess zeros and zero-inflated estimation. Spat Econ Anal 4:167–190
Cantner U, Graf H (2006) The network of innovators in Jena: an application of social network analysis. Res Policy 35(4):463–480
Carey HC (1858) Princ Soc Sci. Lippincott, Philadelphia
Castellani D, Palmero AJ, Zanfei A (2013) How remote are R&D labs? Distance factors and international innovative activities. J Int Bus Stud 44:649–675
Coe DT, Subramanian A, Tamirisa NT (2007) The missing globalization puzzle: evidence of the declining importance of distance. IMF Staff Pap 54:34–58
Cranmer SJ, Desmarais BA (2011) Inferential network analysis with exponential random graph models. Polit Anal 19(1):66–86
Daraganova G, Pattison P, Koskinen J, Mitchell B, Bill A, Watts M, Baum S (2012) Networks and geography: modelling community network structures as the outcome of both spatial and network processes. Soc Netw 34:6–17
De Federico A (2004) L’analyse longitudinal de réseaux sociaux totaux avec SIENA: méthode, discussion et application. Bull Méthodol Sociol 84:5–39
De Graaff T, Boter J, Rouwendal J (2009) On spatial differences in the attractiveness of Dutch museums. Environ Plan A 41:2778–2797
De Groot HLF, Poot J, Smit MJ (2009) Agglomeration externalities, innovation and regional growth: theoretical perspectives and meta-analysis. In: Capello R, Nijkamp P (eds) Handb Reg Growth Dev Theor. Edward Elgar, Cheltenham
Dekker D, Krackhardt D, Snijders TAB (2007) Sensitivity of MRQAP tests to collinearity and autocorrelation conditions. Psychometrika 72:564–581
Egger P (2004) On the Problem of Endogenous Unobserved Effects in the Estimation of Gravity Models. Journal of Economic Integration 19:182–191
Erdös P, Rènyi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17–61
Feenstra RC (2004) Advanced international trade: theory and evidence. Princeton University Press, Princeton
Fienberg S, Wasserman S (1981) An exponential family of probability distributions for directed graphs: comment. J Am Stat Assoc 76:54–57
Fischer MM, Griffith DA (2008) Modeling spatial autocorrelation in spatial interaction data: an application to patent citation data in the European Union. J Reg Sci 48:969–989
Fischer MM, Scherngell T, Jansberger E (2006) The geography of knowledge spillovers between high-technology firms in Europe: evidence from a spatial interaction modeling perspective. Geogr Anal 38:288–309
Frankel JA, Wei SJ (1998) Regionalization of world trade and currencies: economic and politics. In: Frankel JA (ed) The regionalization of the world economy. University of Chicago Press. National Bureau of Economic Research Project Report, pp 189–226
Fowler JH, Dawes CT, Chistakis NA (2009) Model of genetic variation in human social networks. Proc Nat Acad Sci 206:1720–1724
Giuliani E (2013) Network dynamics in regional clusters: evidence from Chile. Res Policy 42:1406–1419
Glückler J (2007) Economic geography and the evolution of networks. J Econ Geogr 7:619–634
Gómez-Herrera E (2013) Comparing alternative methods to estimate gravity models of bilateral trade. Empir Econ 44(3):1087–1111
Goodreau SM, Handcock MS, Hunter DR, Butts CT, Morris M (2008) A statnet Tutorial. J Stat Softw 24(9):1–27
Grabher G (2006) Trading routes, bypasses, and risky intersections: mapping the travels of “networks” between economic sociology and economic geography. Prog Hum Geogr 30:163–189
Graf H (2010) Gatekeepers in Regional Networks of Innovation. Camb J Econ (online account, doi:10.1093/cje/beq001), pp 173–198
Griffith DA (2007) Spatial structure and spatial interaction: 25 years later. Rev Reg Stud 37:28–38
Hagedoorn J (2002) Inter-firm R&D partnerships: an overview of major trends and patterns since 1960. Res Policy 31:477–492
Hanneke S, Xing EP (2007) Discrete temporal models of social networks. In: Statistical network analysis: models, issues, and new directions. Lecture Notes in Computer Science, vol 4503. Springer, Berlin, Germany, pp 115–125
Hazir CS, Autant-Bernard C (2012) Using affiliation networks to study the determinants of multilateral research cooperation: some empirical evidence from EU framework programs in biotechnology. GATE Working Paper No. 1212
Head K, Mayer T (2000) Non-Europe: the magnitude and causes of market fragmentation in the EU. Rev World Econ 136:285–314
Head K, Mayer T (2014) Gravity equations: workhorse, toolkit, and cookbook. In: Gopinath G, Helpman E, Rogoff K, Handbook of international economics, vol 4. North Holland, pp 131–195
Helliwell J (1997) National borders, trade and migration. Pac Econ Rev 2:165–185
Helpman E, Melitz M, Rubinstein Y (2008) Estimating trade flows: trading partners and trading volumes. Q J Econ 123:441–487
Hoekman J, Frenken K, Van Oort FG (2009) The geography of collaborative knowledge production in Europe. Ann Reg Sci 43:721–738
Holland PW, Leinhardt S (1981) An exponential family of probability distributions for directed graphs (with discussion). J Am Stat Assoc 76:33–65
Hubert LJ (1987) Assignment methods in combinatorial data analysis. Marcel Dekker, New York
Hubert L, Schulz J (1976) Quadratic assignment as a general data analysis strategy. Br J Math Stat Psychol 29(2):1900241
Hunter DR (2007) Curved exponential family models for social networks. Soc Netw 29:216–230
Hunter DR, Goodreau SM, Handcock MS (2008) Goodness of fit for social network models. J Am Stat Assoc 103:248–258
Isard W (1956) Location and space-economy. MIT Press, Cambridge
Jackson MO, Rogers BW (2007) Meeting strangers and friends of friends: how random are social networks? Am Econ Rev 97:890–915
Krackhardt D (1987) QAP partialling as a test of spuriousness. Soc Netw 9:171–186
Krackhardt D (1988) Predicting with networks: nonparametric multiple regression analysis of dyadic data. Soc Netw 10:359–381
Krivitsky PN (2012) Exponential-family random graph models for valued networks. Electr J Stat 6:1100–1128
Krivitsky PN, Goodrea SM (2012) STERGM—separable temporal ERGMs for modeling discrete relational dynamics with statnet, STERGM Tutorial: https://statnet.csde.washington.edu/trac/raw-attachment/wiki/Resources/STERGMtutorial.pdf (21.11.2013)
Krivitsky PN, Handcock MS (2013) A separable model for dynamic networks. JRSS Ser B (Statistical Methodology) 76(1):29–46
LeSage JP, Fischer MM (2010) Spatial econometric modeling of origin-destination flows. In: Fischer MM, Getis A (eds) Handbook of applied spatial analysis. Springer, Berlin, pp 409–433
LeSage JP, Pace RK (2008) Spatial econometric modeling of origin-destination flows. J Reg Sci 48:941–967
LeSage JP, Pace RK (2009) Introduction to Spatial Econometrics. CRC Press, Cleveland
LeSage JP, Polasek W (2008) Incorporating transportation network structure in spatial econometric models of commodity flows. Spat Econ Anal 3:225–245
Lill E (1891) Das Reisegesetz und seine Anwendung auf den Eisenbahnverkehr, Vienna
Linders G-J, Patuelli R, Griffith DA (2010) The space of gravity: spatial filtering estimation of a gravity model for bilateral trade. Mimeo
Lindgren KO (2010) Dyadic regression in the presence of heteroscedasticity—an assessment of alternative approaches. Soc Netw 32:279–289
Liu WH, Lin YC (2005) Foreign patent rights and high-tech exports: evidence from Taiwan. Appl Econ 37:1534–1555
Liu X, Derudder B, Liu Y (2013) Regional geographies of intercity corporate networks: The use of exponential random graph models to assess regional network-formation, Papers in Regional Science (forthcoming)
Lerner J, Indlekofer N, Nick B, Brandes U (2013) Conditional independence in dynamic networks. J Math Psychol 57(6):275–283
Lospinoso JA, Snijders TAB (2011) Goodness of fit for stochastic actor oriented models. Working paper
Lubbers MJ, Snijders TAB (2007) A comparison of various approaches to the exponential random graph model: a reanalysis of 102 student networks in school classes. Soc Netw 29:489–507
Lukermann F, Porter PW (1960) Gravity and potential models in economic geography. Ann Assoc Am Geogr 50:493–504
Lusher D, Koskinen J, Robins G, Lusher D, Koskinen J, Robins G (2013) Exponential random graph models for social networks. Structural analysis in the social sciences. Cambridge University Press, New York
Maggioni MA, Uberti TE (2007) Inter-regional knowledge flows in Europe: an econometric analysis. In: Frenken K (ed) Applied evolutionary economics and economic geography. Edward Elgar, Cheltenham
Maggioni M, Uberti E (2011) Networks and geography in the economics of knowledge flows. Qual Quant 45 (forthcoming)
Maggioni MA, Nosvelli M, Uberti TE (2007) Space versus networks in the geography of innovation: a European analysis. Pap Reg Sci 86(3):471–493
Mantel N (1967) The detection of disease clustering and a generalized regression approach. Cancer Res 27(2 Part 1):209–220
Mantel N, Valand RS (1970) A technique of nonparametric multivariate analysis. Biometrics 26:547–558
Melo PC, Graham DJ, Noland RB (2009) A meta-analysis of estimates of urban agglomeration economies. Reg Sci Urban Econ 39:332–342
Miguélez E, Moreno R (2013) Research networks and inventors’ mobility as drivers of innovation: evidence from Europe. Reg Stud 47(10):1668–1685
Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256
Newman M, Barabási AL, Watts DJ (2006) The structure and dynamics of networks. Princeton University Press, Princeton
Opsahl T (2013) Triadic closure in two-mode networks: redefining the global and local clustering coefficients. Soc Netw 35 doi:10.1016/j.socnet.2011.07.001
Ove F, Strauss D (1986) Markov Graphs. J Am Stat Assoc 81(395):832–842
Park J, Newman MEJ (2004) The Statistical Mechanics of Networks. Physical Review E 70(6)
Pattison P, Wasserman S (1999) Logit models and logistic regressions for social networks. II. Multivariate relations. Br J Math Stat Psychol 52:169–194
Peri G (2005) Determinants of knowledge flows and their effect on innovation. Rev Econ Stat 87:308–322
Ponds R, Van Oort F, Frenken K (2007) The geographical and institutional proximity of research collaboration. Pap Reg Sci 86:423–444
Ravenstein EG (1885) The laws of migration. J Stat Soc 48:167–235
Reilly W E (1931) The law of retail gravitation, Knickerbocker Press, New York
Ripley RM, Snijders TAB, Preciado Lopez P (2011) Manual for RSiena, University of Oxford, Department of Statistics, Nuffield College, October 7
Robins G, Morris M, Pattison P, Kalish Y, Lusher D (2007) An introduction to exponential random graph (p*) models for social networks. Soc Netw 29(2):173–191
Saul ZM, Filkov V (2007) Exploring biological network structure using exponential random graph models. Bioinformatics 23(19):2604–2611
Scherngell T, Barber MJ (2009) Spatial interaction modelling of cross-region R&D collaborations: empirical evidence from the 5th EU Framework Programme. Pap Reg Sci 88:531–546
Scherngell T, Lata R (2013) Towards an integrated European Research Area? Findings from Eigenvector spatially filtered spatial interaction models using European Framework Programme data. Pap Reg Sci 92:555–577
Schneider JW, Borlund P (2007) Matrix comparison, part 2: measuring the resemblance between proximity measures or ordination results by use of the mantel and procrustes statistics. J Am Soc Inf Sci Technol 58(11):1596–1609
Snijders TAB (2001) The statistical evaluation of social network dynamics. In: Sobel M, Becker M (eds) Sociological Methodology. Basil Blackwell, Boston and London, pp 361–395
Snijders TAB (2002) Markov Chain Monte Carlo Estimation of Exponential Random Graph Models. J Soc Struct 3(2)
Snijders TAB (2008) Longitudinal methods of network analysis. In: Meyers B (ed) Encyclopedia of complexity and system science. Springer, Berlin
Snijders TAB (2011) Statistical models for social networks. Ann Rev Sociol 37:131–153
Snijders TAB, van de Bunt G, Steglich C (2010a) Introduction to stochastic actor-based models for network dynamics. Soc Netw 32(1):44–60
Snijders TAB, Steglich C, Schweinberger M, Huisman M (2010b) Manual for SIENA version 3.2. Provisional version. Oxford: University of Oxford, Department of Statistics
Snijders TAB, Pattison PE, Robins G, Handcock MS (2006) New specifications for exponential random graph models. Sociol Methodol 36:99–153
Snijders TAB, Lomi A, Torló VJ (2013) A model for the multiplex dynamics of two-mode and one-mode networks, with an application to employment preference, friendship, and advice. Soc Netw 35(2):265–276
Solomonoff R, Rapoport A (1951) Connectivity of random nets. Bull Math Biophys 13:107–117
Steglich CEG, Snijders TAB, West P (2006) Applying SIENA: An illustrative analysis of the co-evolution of adolescents’ friendship networks, taste in music, and alcohol consumption. Methodology 2:48–56
Steglich C, Snijders TAB, Pearson M (2010) Dynamic networks and behavior: separating selection from influence. Sociol Methodol 40:329–393
Stewart JQ (1948) Demographic gravitation: evidence and application. Sociometry 1:31–58
Stouffer SA (1940) Intervening opportunities: a theory relating mobility and distance. Am Sociol Rev 5:845–867
Sunley P (2008) Relational economic geography: a partial understanding or a new paradigm? Econ Geogr 84:1–26
Ter Wal ALJ (2013) The spatial dynamics of the inventor network in German biotechnology: geographical proximity versus triadic closure. J Econ Geogr (forthcoming)
Ter Wal A, Boschma R (2009) Applying social network analysis in economic geography: theoretical and methodological issues. Ann Reg Sci 43:739–756
Thissen M, Van Oort F, Diodato D, Ruijs A (2013) Regional competitiveness and smart specialization in Europe. Place-based development in international economic networks. Edward Elgar, Cheltenham
Tinbergen J (1962) Shaping the world economy. Twentieth Century Fund, New York
Van de Bunt G, Groenewegen P (2007) An actor-oriented dynamic network approach: the case of interorganizational network evolution. Organ Res Methods 10(3):463
Van der Leij MJ (2011) Experimenting with buddies. Science 334:1220–1221
Van Duijn MAJ, Gile KJ, Handcock MS (2009) A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Soc Netw 31(1):52–62
Van Oort F, Lambooy J (2013) Cities, knowledge and innovation. Chapter. In: Fischer M, Nijkamp P (eds) Handbook of regional science. Springer, Berlin, pp 475–488
Wagner D, Head K, Ries J (2002) Immigration and the trade of provinces. Scott J Polit Econ 49:507–525
Wang P, Sharpe K, Robins G, Pattison P (2009) Exponential random graph (p*) models for affiliation networks. Soc Netw 31(1):12–25
Wasserman S, Pattison P (1996) Logit models and logistic regressions for social networks. I. An introduction to Markov graphs and p*. Psychometrika 61:401–425
The authors would like to thank two anonymous referees and Tom Snijders for very helpful comments and suggestions. Of course, all remaining errors are ours.
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Broekel, T., Balland, PA., Burger, M. et al. Modeling knowledge networks in economic geography: a discussion of four methods. Ann Reg Sci 53, 423–452 (2014). https://doi.org/10.1007/s00168-014-0616-2