1932

Abstract

The development of stochastic models for the analysis of social networks is an important growth area in contemporary statistics. The last few decades have witnessed the rapid development of a variety of statistical models capable of representing the global structure of an observed network in terms of underlying generating mechanisms. The distinctive feature of statistical models for social networks is their ability to represent directly the dependence relations that these mechanisms entail. In this review, we focus on models for single network observations, particularly on the family of exponential random graph models. After defining the models, we discuss issues of model specification, estimation and assessment. We then review model extensions for the analysis of other types of network data, provide an empirical example, and give a selective overview of empirical studies that have adopted the basic model and its many variants. We conclude with an outline of the current analytical challenges.

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2018-03-07
2024-06-16
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Literature Cited

  1. An W. 2016. Fitting ERGMs on big networks. Soc. Sci. Res. 59:107–19 [Google Scholar]
  2. Barndorff-Nielsen O. 1978. Information and Exponential Families in Statistical Theory New York: Wiley [Google Scholar]
  3. Besag J. 1974. Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. B 36:2192–236 [Google Scholar]
  4. Block P, Stadtfeld C, Snijders TAB. 2016. Forms of dependence: comparing SAOMs and ERGMs from basic principles. Sociol. Meth. Res. https://doi.org/10.1177/0049124116672680 [Crossref] [Google Scholar]
  5. Bollobás B. 1985. Random Graphs. New York: Springer [Google Scholar]
  6. Brandes U, Robins G, McCranie A, Wasserman S. 2013. What is network science?. Netw. Sci. 1:11–15 [Google Scholar]
  7. Brennecke J, Rank ON. 2016. The interplay between formal project memberships and informal advice seeking in knowledge-intensive firms: a multilevel network approach. Soc. Netw. 44:307–18 [Google Scholar]
  8. Broekel T, Balland PA, Burger M, van Oort F. 2014. Modeling knowledge networks in economic geography: a discussion of four methods. Ann. Reg. Sci. 53:2423–52 [Google Scholar]
  9. Byshkin M, Stivala A, Mira A, Krause R, Robins G, Lomi A. 2016. Auxiliary parameter MCMC for exponential random graph models. J. Stat. Phys. 165:4740–54 [Google Scholar]
  10. Caimo A, Friel N. 2011. Bayesian inference for exponential random graph models. Soc. Netw. 33:141–55 [Google Scholar]
  11. Caimo A, Friel N. 2013. Bayesian model selection for exponential random graph models. Soc. Netw. 35:111–24 [Google Scholar]
  12. Caimo A, Friel N. 2014. Bergm: Bayesian exponential random graphs in R. J. Stat. Softw. 61:21–25 [Google Scholar]
  13. Caimo A, Lomi A. 2015. Knowledge sharing in organizations: a Bayesian analysis of the role of reciprocity and formal structure. J. Manag. 41:2665–91 [Google Scholar]
  14. Caimo A, Lomi A, Pallotti F. 2017. Bayesian exponential random graph modelling of interhospital patient referral networks. Stat. Med. 36:2902–20 [Google Scholar]
  15. Caimo A, Mira A. 2015. Efficient computational strategies for doubly intractable problems with applications to Bayesian social networks. Stat. Comput. 25:113–25 [Google Scholar]
  16. Chatterjee S, Diaconis P. 2013. Estimating and understanding exponential random graph models. Ann. Stat. 41:52428–61 [Google Scholar]
  17. Chandrasekhar AG, Jackson MO. 2014. Tractable and consistent random graph models NBER Work. Pap. w20276, Natl. Bur. Econ. Res Cambridge, MA: [Google Scholar]
  18. Conaldi G, Lomi A. 2013. The dual network structure of organizational problem solving: a case study on open source software development. Soc. Netw. 35:2237–50 [Google Scholar]
  19. Condon A, Karp RM. 2001. Algorithms for graph partitioning on the planted partition model. Random Struct. Algorithms 18:2116–40 [Google Scholar]
  20. Cranmer SJ, Desmarais BA, Kirkland JH. 2012. Toward a network theory of alliance formation. Int. Interact. 38:3295–324 [Google Scholar]
  21. Cranmer SJ, Heinrich T, Desmarais BA. 2014. Reciprocity and the structural determinants of the international sanctions network. Soc. Netw. 36:15–22 [Google Scholar]
  22. Desmarais BA, Cranmer SJ. 2012. Statistical inference for valued-edge networks: the generalized exponential random graph model. PLOS ONE 7:1e30136 [Google Scholar]
  23. Dickison ME, Magnani M, Rossi L. 2016. Multilayer Social Networks Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  24. Easley D, Kleinberg J. 2010. Networks, Crowds, and Markets: Reasoning About a Highly Connected World Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  25. Erdős P, Rényi A. 1959. On random graphs. Publ. Math. Debrecen 6:290–97 [Google Scholar]
  26. Faming L, Ick HJ, Qifan S, Jun SL. 2016. An adaptive exchange algorithm for sampling from distributions with intractable normalizing constants. J. Am. Stat. Assoc. 111:513377–93 [Google Scholar]
  27. Fienberg SE. 2010. Introduction to papers on the modeling and analysis of network data. Ann. Appl. Stat. 4:11–4 [Google Scholar]
  28. Fienberg SE. 2012. A brief history of statistical models for network analysis and open challenges. J. Comput. Graph. Stat. 21:4825–39 [Google Scholar]
  29. Fienberg SE, Wasserman S. 1981. Categorical data analysis of single sociometric relations. Sociol. Methodol. 12:156–92 [Google Scholar]
  30. Frank O, Strauss D. 1986. Markov graphs. J. Am. Stat. Assoc. 81:395832–42 [Google Scholar]
  31. Gerber ER, Henry AD, Lubell M. 2013. Political homophily and collaboration in regional planning networks. J. Polit. Sci. 57:3598–610 [Google Scholar]
  32. Geyer CJ, Thompson EA. 1992. Constrained Monte Carlo maximum likelihood for dependent data. J. R. Stat. Soc. B 54:3657–99 [Google Scholar]
  33. Gilbert EN. 1959. Random graphs. Ann. Math. Stat. 30:41141–44 [Google Scholar]
  34. Gilbert EN. 1961. Random plane networks. J. Soc. Ind. Appl. Math. 9:533–43 [Google Scholar]
  35. Goldenberg A, Zheng AX, Fienberg SE, Airoldi EM. 2010. A survey of statistical network models. Found. Trends Mach. Learn. 2:2129–233 [Google Scholar]
  36. Gonzalez-Bailon S. 2009. Opening the black box of link formation: social factors underlying the structure of the web. Soc. Netw. 31:4271–80 [Google Scholar]
  37. Goodreau SM, Kitts JA, Morris M. 2009. Birds of a feather, or friend of a friend? Using exponential random graph models to investigate adolescent social networks. Demography 46:1103–25 [Google Scholar]
  38. Green PJ. 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–32 [Google Scholar]
  39. Handcock M. 2002. Statistical models for social networks: inference and degeneracy. Dynamic Social Network Modeling and Analysis R Breiger, K Carley, P Pattison 229–40 Washington, DC: Natl. Acad. Press [Google Scholar]
  40. Handcock M. 2003. Assessing degeneracy in statistical models of social networks Work. Pap. 39, Cent. Stat. Soc. Sci., Univ. Wash. [Google Scholar]
  41. Handcock M, Hunter D, Butts C, Goodreau S, Krivitsky P. et al. 2016. statnet: software tools for the statistical analysis of network data. Statistical software package http://www.statnet.org [Google Scholar]
  42. Handcock M, Hunter D, Butts C, Goodreau S, Morris M. 2008. statnet: software tools for the representation, visualization, analysis and simulation of network data. J. Stat. Softw. 24:11–11 [Google Scholar]
  43. Hanneke S, Fu W, Xing EP. 2010. Discrete temporal models of social networks. Electron. J. Stat. 4:585–605 [Google Scholar]
  44. Hipp JR, Wang C, Butts CT, Jose R, Lakon CM. 2015. Research note: the consequences of different methods for handling missing network data in stochastic actor based models. Soc. Netw. 41:56–71 [Google Scholar]
  45. Hoeting JA, Madigan D, Raftery AE, Volinsky CT. 1999. Bayesian model averaging: a tutorial. Stat. Sci. 14:382–401 [Google Scholar]
  46. Hoff P, Raftery A, Handcock M. 2002. Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97:4601090–98 [Google Scholar]
  47. Holland PW, Leinhardt S. 1976. Local structure in social networks. Sociol. Methodol. 7:11–45 [Google Scholar]
  48. Holland PW, Leinhardt S. 1977. A dynamic model for social networks. J. Math. Sociol. 5:15–20 [Google Scholar]
  49. Holland PW, Leinhardt S. 1981. An exponential family of probability distributions for directed graphs. J. Am. Stat. Assoc. 76:37333–50 [Google Scholar]
  50. Hollway J, Koskinen J. 2016. Multilevel embeddedness: the case of the global fisheries governance complex. Soc. Netw. 44:281–94 [Google Scholar]
  51. Howard M, Cox Pahnke E, Boeker W. 2016. Understanding network formation in strategy research: exponential random graph models. Strat. Manag. J. 37:122–44 [Google Scholar]
  52. Hunter DR. 2007. Curved exponential family models for social networks. Soc. Netw. 29:2216–30 [Google Scholar]
  53. Hunter DR, Goodreau S, Handcock M. 2008. Goodness of fit of social network models. J. Am. Stat. Assoc. 103:248–58 [Google Scholar]
  54. Hunter DR, Handcock M. 2012. Inference in curved exponential family models for networks. J. Comput. Graph. Stat. 15:565–83 [Google Scholar]
  55. Hunter DR, Handcock MS, Butts CT, Goodreau SM, Morris M. 2008. ergm: a package to fit, simulate and diagnose exponential-family models for networks. J. Stat. Softw. 24:31–29 [Google Scholar]
  56. Hunter DR, Krivitsky PN, Schweinberger M. 2012. Computational statistical methods for social network models. J. Comput. Graph. Stat. 21:4856–82 [Google Scholar]
  57. Indlekofer N, Brandes U. 2014. Relative importance of effects in stochastic actor-oriented models. Netw. Sci. 1:3278–304 [Google Scholar]
  58. Kandel DB. 1978. Homophily, selection, and socialization in adolescent friendships. Am. J. Sociol. 84:2427–36 [Google Scholar]
  59. Kass RE, Raftery AE. 1995. Bayes factors. J. Am. Stat. Assoc. 90:773–95 [Google Scholar]
  60. Khanna AS, Goodreau SM, Gorbach PM, Daar E, Little SJ. 2014. Modeling the impact of post-diagnosis behavior change on HIV prevalence in Southern California men who have sex with men (MSM). AIDS Behav 18:8523–31 [Google Scholar]
  61. Kolaczyk ED. 2009. Statistical Analysis of Network Data: Methods and Models New York: Springer [Google Scholar]
  62. Koskinen J, Caimo A, Lomi A. 2015. Simultaneous modeling of initial conditions and time heterogeneity in dynamic networks: an application to foreign direct investments. Netw. Sci. 3:158–77 [Google Scholar]
  63. Koskinen J, Lomi A. 2013. The local structure of globalization. J. Stat. Phys. 151:3–4523–48 [Google Scholar]
  64. Koskinen J, Robins G, Pattison PE. 2010. Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation. Stat. Methodol. 7:3366–84 [Google Scholar]
  65. Koskinen J, Robins G, Wang P, Pattison PE. 2013. Bayesian analysis for partially observed network data, missing ties, attributes and actors. Soc. Netw. 35:4514–27 [Google Scholar]
  66. Krivitsky PN. 2012. Exponential-family random graph models for valued networks. Electron. J. Stat. 6:1100–28 [Google Scholar]
  67. Krivitsky PN. 2016. ergm.count: fit, simulate and diagnose exponential-family models for networks with count edges. Statistical software package http://www.statnet.org [Google Scholar]
  68. Krivitsky PN, Handcock MS. 2014. A separable model for dynamic networks. J. R. Stat. Soc. B 76:129–46 [Google Scholar]
  69. Lauritzen SL. 1996. Graphical Models Oxford, UK: Clarendon [Google Scholar]
  70. Lazega E, Jourda MT, Mounier L, Stofer R. 2008. Catching up with big fish in the big pond? Multi-level network analysis through linked design. Soc. Netw. 30:2159–76 [Google Scholar]
  71. Lazega E, Pattison PE. 1999. Multiplexity, generalized exchange and cooperation in organizations: a case study. Soc. Netw. 21:167–90 [Google Scholar]
  72. Lazega E, Snijders TAB. 2015. Multilevel Network Analysis for the Social Sciences New York: Springer [Google Scholar]
  73. Lazega E, van Duijn M. 1997. Position in formal structure, personal characteristics and choices of advisors in a law firm: a logistic regression model for dyadic network data. Soc. Netw. 19:4375–97 [Google Scholar]
  74. Lehmann EL, Casella G. 1983. Theory of Point Estimation New York: Wiley [Google Scholar]
  75. Lomi A, Lusher D, Pattison PE, Robins G. 2014. The focused organization of advice relations: a study in boundary crossing. Organ. Sci. 25:2438–57 [Google Scholar]
  76. Lomi A, Pallotti F. 2012. Relational collaboration among spatial multipoint competitors. Soc. Netw. 34:1101–11 [Google Scholar]
  77. Lomi A, Pattison PE. 2006. Manufacturing relations: an empirical study of the organization of production across multiple networks. Organ. Sci. 17:3313–32 [Google Scholar]
  78. Lomi A, Robins G, Tranmer M. 2016. Introduction to multilevel social networks. Soc. Netw. 44:266–68 [Google Scholar]
  79. 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. 29:4489–507 [Google Scholar]
  80. Lubell M, Robins G, Wang P. 2014. Network structure and institutional complexity in an ecology of water management games. Ecol. Soc. 19:423 [Google Scholar]
  81. Lusher D, Koskinen J, Robins G. 2012.a Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  82. Lusher D, Robins G, Pattison PE, Lomi A. 2012.b “Trust me”: differences in expressed and perceived trust relations in an organization. Soc. Netw. 34:4410–24 [Google Scholar]
  83. McFarland DA, Moody J, Diehl D, Smith JA, Thomas RJ. 2014. Network ecology and adolescent social structure. Am. Sociol. Rev. 79:61088–121 [Google Scholar]
  84. McPherson M, Smith-Lovin L, Cook JM. 2001. Birds of a feather: homophily in social networks. Annu. Rev. Sociol. 27:1415–44 [Google Scholar]
  85. Melamed D, Breiger RL, Schoon E. 2013. The duality of clusters and statistical interactions. Sociol. Methods Res. 42:141–59 [Google Scholar]
  86. Mele A. 2016. A structural model of dense network formation. Econometrica 85:825–50 [Google Scholar]
  87. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U. 2002. Network motifs: simple building blocks of complex networks. Science 298:5594824–27 [Google Scholar]
  88. Moody J. 2001. Race, school integration, and friendship segregation in America. Am. J. Sociol. 107:3679–716 [Google Scholar]
  89. Morris M, Handcock M, Hunter D. 2008. Specification of exponential-family random graph models: terms and computational aspects. J. Stat. Softw. 24:41548 [Google Scholar]
  90. Murray I, Ghahramani Z, MacKay DJ. 2006. MCMC for doubly-intractable distributions. Proc. 22nd Annu. Conf. Uncertain. Artif. Intell. (UAI)359–66 Corvallis, OR: AUAI [Google Scholar]
  91. Newman ME. 2003. The structure and function of complex networks. SIAM Rev 45:2167–256 [Google Scholar]
  92. Newman ME, Watts DJ, Strogatz SH. 2002. Random graph models of social networks. PNAS 99:suppl. 12566–72 [Google Scholar]
  93. Papachristos AV, Hureau DM, Braga AA. 2013. The corner and the crew: the influence of geography and social networks on gang violence. Am. Sociol. Rev. 78:3417–47 [Google Scholar]
  94. Pattison PE, Robins G. 2002. Neighborhood-based models for social networks. Sociol. Methodol. 32:1301–37 [Google Scholar]
  95. Pattison PE, Robins G, Snijders TAB, Wang P. 2013. Conditional estimation of exponential random graph models from snowball sampling designs. J. Math. Psychol. 57:6284–96 [Google Scholar]
  96. Pattison PE, Wasserman S. 1999. Logit models and logistic regressions for social networks. II. Multivariate relations. Br. J. Math. Stat. Psychol. 52:2169–93 [Google Scholar]
  97. Polyak BT. 1990. A new method of stochastic approximation type. Autom. Remote Control798–107 [Google Scholar]
  98. R Core Team. 2017. R: a language and environment for statistical computing. Vienna: R Found. Stat. Comput https://cran.r-project.org/
  99. Rank ON, Robins G, Pattison PE. 2010. Structural logic of intraorganizational networks. Organ. Sci. 21:3745–64 [Google Scholar]
  100. Rinaldo A, Fienberg SE, Zhou Y. 2009. On the geometry of discrete exponential families with application to exponential random graph models. Electron. J. Stat. 3:446–84 [Google Scholar]
  101. Robbins H, Monro S. 1951. A stochastic approximation method. Ann. Math. Stat. 22:3400–7 [Google Scholar]
  102. Robins G, Alexander M. 2004. Small worlds among interlocking directors: network structure and distance in bipartite graphs. Comput. Math. Organ. Theory 10:169–94 [Google Scholar]
  103. Robins G, Elliott P, Pattison PE. 2001. Network models for social selection processes. Soc. Netw. 23:1–30 [Google Scholar]
  104. Robins G, Pattison P, Wang P. 2009. Closure, connectivity and degree distributions: exponential random graph (p*) models for directed social networks. Soc. Netw. 31:2105–17 [Google Scholar]
  105. Robins G, Pattison P, Wasserman S. 1999. Logit models and logistic regressions for social networks. III. Valued relations. Psychometrika 64:3371–94 [Google Scholar]
  106. Robins G, Pattison P, Woolcock J. 2004. Missing data in networks: exponential random graph (p*) models for networks with non-respondents. Soc. Netw. 26:3257–83 [Google Scholar]
  107. Robins G, Pattison P, Woolcock J. 2005. Small and other worlds: global network structures from local processes. Am. J. Sociol. 110:4894–936 [Google Scholar]
  108. Robins G, Snijders TAB, Wang P, Handcock M, Pattison PE. 2007. Recent developments in exponential random graph (p*) models for social networks. Soc. Netw. 29:2192–215 [Google Scholar]
  109. Ruppert D. 1988. Efficient estimations from a slowly convergent Robbins-Monro process Tech. Rep. 781, Sch. Oper. Res. Ind. Eng., Cornell Univ. [Google Scholar]
  110. Salter-Townshend M, White A, Gollini I, Murphy TB. 2012. Review of statistical network analysis: models, algorithms, and software. Stat. Anal. Data. Min. 5:4243–64 [Google Scholar]
  111. Saul ZM, Filkov V. 2007. Exploring biological network structure using exponential random graph models. Bioinformatics 23:192604–11 [Google Scholar]
  112. Schweinberger M. 2012. Instability, sensitivity, and degeneracy of discrete exponential families. J. Am. Stat. Assoc. 106:4961361–70 [Google Scholar]
  113. Smith S, Van Tubergen F, Maas I, McFarland DA. 2016. Ethnic composition and friendship segregation: differential effects for adolescent natives and immigrants. Am. J. Sociol. 121:41223–72 [Google Scholar]
  114. Snijders TAB. 1996. Stochastic actor-oriented models for network change. J. Math. Sociol. 21:1–2149–72 [Google Scholar]
  115. Snijders TAB. 2002. Markov chain Monte Carlo estimation of exponential random graph models. J. Soc. Struct. 3:21–40 [Google Scholar]
  116. Snijders TAB. 2004. Explained variation in dynamic network models. Math. Sci. Hum. 168:31–42 [Google Scholar]
  117. Snijders TAB. 2005. Models for longitudinal network data. Models and Methods in Social Network Analysis PJ Carrington, J Scott, S Wasserman 215–47 Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  118. Snijders TAB. 2011. Statistical models for social networks. Annu. Rev. Sociol. 37:131–53 [Google Scholar]
  119. Snijders TAB. 2017. Stochastic actor-oriented models for network dynamics. Annu. Rev. Stat. Appl. 4:343–63 [Google Scholar]
  120. Snijders TAB, Pattison PE, Robins G, Handcock M. 2006. New specifications for exponential random graph models. Sociol. Methodol. 36:199–153 [Google Scholar]
  121. Solomonoff R, Rapoport A. 1951. Connectivity of random nets. Bull. Math. Biophys. 13:2107–17 [Google Scholar]
  122. Srivastava SB, Banaji MR. 2011. Culture, cognition, and collaborative networks in organizations. Am. Sociol. Rev. 76:2207–33 [Google Scholar]
  123. Steglich C, Snijders TAB, Pearson M. 2010. Dynamic networks and behavior: separating selection from influence. Sociol. Methodol. 40:1329–93 [Google Scholar]
  124. Stivala AD, Koskinen JH, Rolls DA, Wang P, Robins G. 2016. Snowball sampling for estimating exponential random graph models for large networks. Soc. Netw. 47:167–88 [Google Scholar]
  125. Strauss D, Ikeda M. 1990. Pseudolikelihood estimation for social networks. J. Am. Stat. Assoc. 85:409204–12 [Google Scholar]
  126. Tierney L. 1994. Markov chains for exploring posterior distributions. Ann. Stat. 22:1701–28 [Google Scholar]
  127. Tierney L, Mira A. 1999. Some adaptive Monte Carlo methods for Bayesian inference. Stat. Med. 18:2507–15 [Google Scholar]
  128. Thiemichen S, Friel N, Caimo A, Kauermann G. 2016. Bayesian exponential random graph models with nodal random effects. Soc. Netw. 46:11–28 [Google Scholar]
  129. Thiemichen S, Kauermann G. 2017. Stable exponential random graph models with non-parametric components for large dense networks. Soc. Netw. 49:67–80 [Google Scholar]
  130. Trapido D. 2013. Dual signals: how competition makes or breaks interfirm social ties. Organ. Sci. 24:2498–512 [Google Scholar]
  131. van Duijn MA, Gile KJ, Handcook M. 2009. A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Soc. Netw. 31:152–62 [Google Scholar]
  132. van Duijn Snijders MA TAB, Zijlstra BJ. 2004. p2: a random effects model with covariates for directed graphs. Stat. Neerl. 58:2234–54 [Google Scholar]
  133. Vinciotti V, Wit E. 2017. Preface to the themed issue on “Statistical network science and its applications. J. R. Stat. Soc. C 66:3451–53 [Google Scholar]
  134. Wang P. 2012. Exponential random graph model extensions: Models for multiple networks and bipartite networks. Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications D Lusher, J Koskinen, G Robins 115–29 Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  135. Wang P, Pattison PE, Robins G. 2013.a Exponential random graph model specifications for bipartite networks—a dependence hierarchy. Soc. Netw. 35:2211–22 [Google Scholar]
  136. Wang P, Robins G, Matous P. 2016. Multilevel network analysis using ERGM and its extension. Multilevel Network Analysis for the Social Sciences E Lazega, TAB Snijders 125–43 New York: Springer [Google Scholar]
  137. Wang P, Robins G, Pattison PE. 2009.a PNet: a program for the simulation and estimation of exponential random graph models. Software for statistical analysis of social network data Melbourne Sch. Psychol. Sci Univ. Melbourne: [Google Scholar]
  138. Wang P, Robins G, Pattison PE, Koskinen J. 2014. MPNet: program for the simulation and estimation of (p*) exponential random graph models for multilevel networks. Software for statistical analysis of social network data Melbourne Sch. Psychol. Sci Univ. Melbourne: [Google Scholar]
  139. Wang P, Robins G, Pattison PE, Lazega E. 2013.b Exponential random graph models for multilevel networks. Soc. Netw. 35:196–115 [Google Scholar]
  140. Wang P, Sharpe K, Robins G, Pattison PE. 2009.b Exponential random graph (p*) models for affiliation networks. Soc. Netw. 31:112–25 [Google Scholar]
  141. Wasserman S. 1978. Models for binary directed graphs and their applications. Adv. Appl. Probab. 10:4803–18 [Google Scholar]
  142. Wasserman S. 1980.a A stochastic model for directed graphs with transition rates determined by reciprocity. Sociol. Methodol. 11:392–412 [Google Scholar]
  143. Wasserman S. 1980.b Analyzing social networks as stochastic processes. J. Am. Stat. Assoc. 75:370280–94 [Google Scholar]
  144. Wasserman S, Faust K. 1994. Social Network Analysis: Methods and Applications Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  145. Wasserman S, Pattison P. 1996. Logit models and logistic regressions for social networks. I. An introduction to Markov graphs and p*. Psychometrika 61:3401–25 [Google Scholar]
  146. Wasserman S, Robins G, Steinley D. 2007. Statistical models for networks: a brief review of some recent research. Statistical Network Analysis: Models, Issues, and New Directions E Blei, S Goldenberg, E Zheng 45–56 Berlin: Springer [Google Scholar]
  147. Whittaker J. 2009. Graphical Models in Applied Multivariate Statistics New York: Wiley [Google Scholar]
  148. Wimmer A, Lewis K. 2010. Beyond and below racial homophily: ERG models of a friendship network documented on Facebook. Am. J. Sociol. 116:2583–642 [Google Scholar]
  149. Zappa P, Lomi A. 2015. The analysis of multilevel networks in organizations models and empirical tests. Organ. Res. Methods 18:3542–69 [Google Scholar]
  150. Zappa P, Robins G. 2016. Organizational learning across multi-level networks. Soc. Netw. 44:295–306 [Google Scholar]
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