Advertisement

Introduction to Social Network Analysis

  • Alistair James O’MalleyEmail author
  • Jukka-Pekka Onnela
Reference work entry
Part of the Health Services Research book series (HEALTHSR)

Abstract

This chapter introduces statistical methods used in the analysis of social networks and in the rapidly evolving parallel-field of network science. Although several instances of social network analysis in health services research have appeared recently, the majority involve only the most basic methods and thus scratch the surface of what might be accomplished. Cutting-edge methods using relevant examples and illustrations in health services research are provided.

Keywords

Dyad Homophily Induction Network science Peer-effect Relationship Social network 

Notes

Acknowledgments

The time and effort of Dr. O’Malley and Dr. Onnela on researching and developing this chapter was supported by NIH/NIA grant P01 AG031093 and Robert Wood Johnson Award #58729. The authors thank Mischa Haider, Brian Neelon, and Bruce E Landon for reviewing an early draft of the manuscript and providing several useful comments and suggestions.

References

  1. Airoldi EM, Fienberg SE, Xing EP. Mixed membership stochastic blockmodels. J Mach Learn Res. 2008;9:1981–2014.PubMedPubMedCentralGoogle Scholar
  2. Anselin L. Spatial econometrics: methods and models. Dordrecht: Kluwer; 1988.CrossRefGoogle Scholar
  3. Barabasi A-L, Albert R. Emergence of scaling in random networks. Science. 1999;286:509–12. http://www.sciencemag.org/content/286/5439/509.abstract
  4. Barabasi A-L, Albert R, Jeong H. Mean-field theory for scale-free random networks. Phys A Stat Mech Appl. 1999;272:173–87. http://www.sciencedirect.com/science/article/pii/S0378437199002915.CrossRefGoogle Scholar
  5. Barnett ML, Landon BE, O’Malley AJ, Keating NL, Christakis NA. Mapping physician networks with self-reported and administrative data. Health Serv Res. 2011;46:1592–609.PubMedPubMedCentralCrossRefGoogle Scholar
  6. Barnett ML, Christakis NA, O’Malley AJ, Onnela J-P, Keating NL, Landon BE. Physician patient-sharing networks and the cost and intensity of care in US hospitals. Med Care. 2012a;50:152–60.PubMedPubMedCentralCrossRefGoogle Scholar
  7. Barnett ML, Keating NL, Christakis NA, O’Malley AJ, Landon BE. Reasons for referral among primary care and specialist physicians. J Gen Intern Med. 2012b;27:506–12.PubMedCrossRefGoogle Scholar
  8. Berkman L, Glass T. Social integration, social methods, social support, and health. In: Social epidemiology. New York: Oxford University Press; 2000. p. 137–73.Google Scholar
  9. Boguñá M, Pastor-Satorras R, Díaz-Guilera A, Arenas A. Models of social networks based on social distance attachment. Phys Rev E. 2004;70:056122.  https://doi.org/10.1103/PhysRevE.70.056122.CrossRefGoogle Scholar
  10. Bonacich P. Power and centrality: a family of measures. Am J Sociol. 1987;92:1170–82.CrossRefGoogle Scholar
  11. Borgatti S, Everett M. Network analysis of 2-mode data. Soc Networks. 1997;19:243–69.CrossRefGoogle Scholar
  12. Breiger R. The duality of persons and groups. Soc Forces. 1974;53:181–90.CrossRefGoogle Scholar
  13. Cartwright D, Harrary F. A generalization of Heider’s theory. Psychol Rev. 1956;63:277–92.PubMedCrossRefGoogle Scholar
  14. Centola D. Failure in complex social networks. Math Sociol. 2009;33:64–8.CrossRefGoogle Scholar
  15. Choi D, Wolfe P, Airoldi E. Stochastic blockmodels with growing number of classes. Arxiv preprint. 2010;arXiv:1011.4644.Google Scholar
  16. Christakis N, Fowler J. The spread of obesity in a large social network over 32 years. N Engl J Med. 2007;357:370–9.PubMedPubMedCentralCrossRefGoogle Scholar
  17. Christakis NA, Fowler JH. Social contagion theory: examining dynamic social networks and human behavior. Stat Med. 2013;32:556–77.PubMedCrossRefGoogle Scholar
  18. Coleman J, Katz E, Menzel H. The diffusion of innovations among physicians. Sociometry. 1957;20:253–70.CrossRefGoogle Scholar
  19. Coleman J, Katz E, et al. Medical innovation: a diffusion study. Indianapolis: Bobbs-Merrill; 1966.Google Scholar
  20. Davidsen J, Ebel H, Bornholdt S. Emergence of a small world from local interactions: modeling acquaintance networks. Phys Rev Lett. 2002;88:128701.  https://doi.org/10.1103/PhysRevLett.88.128701.CrossRefPubMedGoogle Scholar
  21. Dorogovtsev SN, Mendes JFF, Samukhin AN. Structure of growing networks with preferential linking. Phys Rev Lett. 2000;85:4633–6.  https://doi.org/10.1103/PhysRevLett.85.4633.CrossRefPubMedGoogle Scholar
  22. Duijn MV, Snijders TAB, Zijlstra B. P2: a random effects model with covariates for directed graphs. Statistica Neerlandica. 2004;58:234–54.CrossRefGoogle Scholar
  23. Erdős P, Rényi A. Random graphs. Publ Math. 1959;6:290–7.Google Scholar
  24. Faust K. Centrality in affliation networks. Soc Networks. 1997;19:157–91.CrossRefGoogle Scholar
  25. Feller W. An introduction to probability theory and its applications, vol. 2. New York: Wiley; 1966.Google Scholar
  26. Festinger L. The analysis of sociograms using matrix algebra. Hum Relat. 1949;2:153–8.CrossRefGoogle Scholar
  27. Fineberg S, Wasserman S. Categorical data analysis of single sociometric relations. In: Sociological methodology. New Jersey: Jossey-Bass; 1981. p. 156–92.Google Scholar
  28. Fletcher JM. Social interactions and smoking: evidence using multiple student cohorts, instrumental variables, and school fixed effects. Health Econ. 2008;19:466–84.CrossRefGoogle Scholar
  29. Fletcher JM, Lehrer SF. The effect of adolescent health on educational outcomes: causal evidence using genetic lotteries between siblings. Canadian labor market and skills researcher network, working paper no. 32. 2009.Google Scholar
  30. Fortunato S. Community detection in graphs. Phys Reports. 2010;486:75–174.CrossRefGoogle Scholar
  31. Frank O, Strauss D. Markov graphs. J Am Stat Assoc. 1986;81:832–42.CrossRefGoogle Scholar
  32. Freeman L. Centrality in social networks, I. Conceptual clarification. Soc Networks. 1979;1:215–39.CrossRefGoogle Scholar
  33. Freeman L. The development of social network analysis: a study in the sociology of science. Vancouver: Empirical Press; 2004.Google Scholar
  34. Goh K-I, Cusick ME, Valle D, Childs B, Vidal M, Barabasi A-L. The human disease network. Proc Natl Acad Sci. 2007;104:8685–90. http://www.pnas.org/content/104/21/8685.abstractPubMedCrossRefGoogle Scholar
  35. Goldenberg A, Zheng AX, Fineberg SE, Airoldi EM. A survey of statistical network models. Found Trends Mach Learn. 2009;2:129–233.CrossRefGoogle Scholar
  36. Goodreau S. Advances in exponential random graph (p*) models applied to a large social network. Soc Networks. 2007;29:231–48.PubMedPubMedCentralCrossRefGoogle Scholar
  37. Granovetter MS. The strength of weak ties. Am J Sociol. 1973;78:1360–80.CrossRefGoogle Scholar
  38. Guimera R, Nunes Amaral LA. Functional cartography of complex metabolic networks. Nature. 2005;433:895–900.PubMedPubMedCentralCrossRefGoogle Scholar
  39. Haines V, Hurlbert J. Network range and health. J Health Soc Behav. 1992;33:254–66.PubMedCrossRefPubMedCentralGoogle Scholar
  40. Handcock MS, Robins GL, Snijders TAB, Moody J, Besag J. Assessing degeneracy in statistical models of social networks. J Am Stat Assoc. 2003;76:33–50.Google Scholar
  41. Handcock M, Raftery A, Tantrum J. Model-based clustering for social networks. J Roy Stat Soc A. 2007;170:301–54.CrossRefGoogle Scholar
  42. Handcock MS, Hunter DR, Butts CT, Goodreau SM, Krivitsky PN, Morris M. ergm: A package to fit, simulate and diagnose exponential-family models for networks, http://CRAN.R-project.org/package=ergm. Version 2.2-6. 2010. Project home page at http://statnetproject.org
  43. Hanneke S, Fu W, Xing EP. Discrete temporal models of social networks. Electron J Stat. 2010;4:585–605.CrossRefGoogle Scholar
  44. Harary F. On the notion of balance of a signed graph. Mich Math J. 1953;2:143–6.CrossRefGoogle Scholar
  45. Harary F. The number of linear, directed rooted and connected graphs. Trans Am Math Soc. 1955;78:445–63.CrossRefGoogle Scholar
  46. Heider F. Attitudes and cognitive orientation. J Psychol. 1946;21:107–12.PubMedCrossRefGoogle Scholar
  47. Hidalgo CA, Blumm N, Barabasi A-L, Christakis NA. A dynamic network approach for the study of human phenotypes. PLoS Comput Biol. 2009;5:e1000353.  https://doi.org/10.1371/journal.pcbi.1000353.CrossRefPubMedPubMedCentralGoogle Scholar
  48. Hoff PD. Bilinear mixed effects models for dyadic data. J Am Stat Assoc. 2005;100:286–95.CrossRefGoogle Scholar
  49. Hoff P. Modeling homophily and stochastic equivalence in symmetric relational data. In: Advances in neural information processing systems, vol. 20. Cambridge, MA: MIT Press; 2008. p. 657–64.Google Scholar
  50. Hoff PD, Raftery AE, Handcock MS. Latent space models for social networks analysis. J Am Stat Assoc. 2002;97:1090–8.CrossRefGoogle Scholar
  51. Holland P, Leinhardt S. An exponential family of probability-distributions for directed-graph. J Am Stat Assoc. 1981;76:33–50.CrossRefGoogle Scholar
  52. Holland P, Laskey K, Leinhardt S. Stochastic blockmodels: some first steps. Soc Networks. 1983;5:109–37.CrossRefGoogle Scholar
  53. House J, Kahn R. Measures and concepts of social support. In: Social support and health. Orlando: Academic; 1985. p. 83–108.Google Scholar
  54. Huisman M, Van Duijn M. Software for statistical analysis of social networks. In: The Sixth International Conference on Logic and Methodology; Amsterdam: 2004.Google Scholar
  55. Huisman M, Van Duijn M. Software for social networks analysis. In: Models and methods in social network analysis. Cambridge: Cambridge University Press; 2005.Google Scholar
  56. Hunter D. Curved exponential family models for social networks. Soc Networks. 2007;29:216–30.PubMedPubMedCentralCrossRefGoogle Scholar
  57. Hunter DR, Handcock MS. Inference in curved exponential family models for networks. J Comput Graph Stat. 2006;15:565–83.CrossRefGoogle Scholar
  58. Iwashyna TJ, Chang VW, Zhang JX, Christakis AN. Physician social networks and variation in prostate cancer treatment in three cities. Health Serv Res. 2002;37:1531–51.PubMedPubMedCentralCrossRefGoogle Scholar
  59. Karrer B, Newman MEJ. Stochastic blockmodels and community structure in networks. Phys Rev E. 2011;83:016107.  https://doi.org/10.1103/PhysRevE.83.016107.CrossRefGoogle Scholar
  60. Katz L. On the matrix analysis of Sociometric data. Sociometry. 1947;10:233–41.CrossRefGoogle Scholar
  61. Katz L. A new status index derived from sociometric analysis. Psychometrika. 1953;18:39–43.CrossRefGoogle Scholar
  62. Katz L, Powell JH. Measurement of the tendency toward reciprocation of choice. Sociometry. 1955;18:659–65.CrossRefGoogle Scholar
  63. Keating NL, Ayanian JZ, Cleary PD, et al. Factors affecting influential discussions among physicians: a social network analysis of a primary care practice. J Gen Intern Med. 2007;22:794–8.PubMedPubMedCentralCrossRefGoogle Scholar
  64. Klovdahl A. Social networks and the spread of infectious diseases. Soc Sci Med. 1985;21:1203–16.PubMedCrossRefGoogle Scholar
  65. Kossinets G, Watts DJ. Empirical analysis of an evolving social network. Science. 2006;311:88–90. http://www.sciencemag.org/content/311/5757/88.abstractPubMedCrossRefGoogle Scholar
  66. Krapivsky PL, Redner S, Leyvraz F. Connectivity of growing random networks. Phys Rev Lett. 2000;85:4629–32.  https://doi.org/10.1103/PhysRevLett.85.4629.CrossRefPubMedGoogle Scholar
  67. Krivitsky PN. Exponential-family random graph models for valued networks. 2012. arXiv preprint, 1101.1359v2 [stat.ME] 19 Jan 2012.Google Scholar
  68. Krivitsky PN, Handcock MS. Fitting position latent cluster models for social networks with latentnet. J Stat Softw. 2008;24. http://statnetproject.org
  69. Krivitsky PN, Handcock MS. A separable model for dynamic networks. 2010. arXiv preprint, 1011.1937v1[stat.ME].Google Scholar
  70. Kumpula JM, Onnela J-P, Saramäki J, Kaski K, Kertész J. Emergence of communities in weighted networks. Phys Rev Lett. 2007;99:228701.  https://doi.org/10.1103/PhysRevLett.99.228701.CrossRefPubMedGoogle Scholar
  71. Landon BE, Keating NL, Barnett ML, Onnela JP, Paul S, OˆaMalley AJ, Keegan T, Christakis NA. Variation in patient-sharing networks of physicians across the United States. JAMA. 2012;308:265–73.PubMedPubMedCentralGoogle Scholar
  72. Laumann E, Marsden P, Prensky D. The boundary specification problem in network analysis. In: Burt R, Minor M, editors. Applied network analysis: a methodological introduction. Beverly Hills: Sage; 1983. p. 18–34.Google Scholar
  73. Lorrain F, White H. Structural equivalence of individuals in social networks. J Math Sociol. 1971;1:49–80.CrossRefGoogle Scholar
  74. Lyons R. The spread of evidence-poor medicine via flawed social-network analyses. Stat Polit Policy. 2011;2:1–26.Google Scholar
  75. Manski CA. Identification of endogenous social effects: the reflection problem. Rev Econ Stud. 1993;60:531–42.CrossRefGoogle Scholar
  76. Marsden P. Network methods in social epidemiology. In: Methods in social epidemiology. New York: Jossey-Bass; 2006. p. 267–86.Google Scholar
  77. Marsden PV, Friedkin NE. Network studies of social influence. Sociol Methods Res. 1993;22:127–51.CrossRefGoogle Scholar
  78. Marsili M, Vega-Redondo F, Slanina F. The rise and fall of a networked society: a formal model. Proc Natl Acad Sci USA. 2004;101:1439–42.PubMedCrossRefGoogle Scholar
  79. McPherson ML, Smith-Lovin C, et al. Birds of a feather: homophily in social networks. Annu Rev Sociol. 2001;27:415–44.CrossRefGoogle Scholar
  80. Moreno JL. Who shall survive? Nervous and mental disease processing. The University of Michigan, Ann Arbor; 1934.Google Scholar
  81. Mucha PJ, Richardson T, Macon K, Porter MA, Onnela J-P. Community structure in time-dependent, multiscale, and multiplex networks. Science. 2010;328:876–8. http://www.sciencemag.org/content/328/5980/876.abstractPubMedCrossRefGoogle Scholar
  82. Newcomb TM. An approach to the study of communicative acts. Psychol Rev. 1953;60:393–404.PubMedCrossRefGoogle Scholar
  83. Newman ME. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys Rev. 2001;64:016132.Google Scholar
  84. Newman MEJ. Modularity and community structure in networks. Proc Natl Acad Sci. 2006;103:8577–82.PubMedCrossRefGoogle Scholar
  85. Newman M. Networks: an introduction. New York: Oxford University Press; 2010.CrossRefGoogle Scholar
  86. Newman MEJ. Communities, modules and large-scale structure in networks. Nat Phys. 2012;8:25–31.CrossRefGoogle Scholar
  87. Newman MEJ, Girvan M. Mixing patterns and community structure in networks. In: Pastor-Satorras R, Rubi J, Diaz-Guilera A, editors. Statistical mechanics of complex networks. Berlin: Springer; 2003.Google Scholar
  88. Newman MEJ, Girvan M. Finding and evaluating community structure in networks. Phys Rev E. 2004;69:026113.  https://doi.org/10.1103/PhysRevE.69.026113.CrossRefGoogle Scholar
  89. Nowicki K, Snijders TAB. Estimation and prediction for stochastic blockstructures. J Am Stat Assoc. 2001;96:1077–87.CrossRefGoogle Scholar
  90. O’Malley AJ. The analysis of social network data: an exciting frontier for statisticians. Stat Med. 2013;32:539–55.PubMedCrossRefGoogle Scholar
  91. O’Malley AJ, Christakis NA. Longitudinal analysis of large social networks: estimating the effect of health traits on changes in friendship ties. Stat Med. 2011;30:950–64.PubMedPubMedCentralCrossRefGoogle Scholar
  92. O’Malley AJ, Marsden PV. The analysis of social networks. Health Serv Outcome Res Methodol. 2008;8:222–69.CrossRefGoogle Scholar
  93. O’Malley AJ, Arbesman S, Steiger DM, Fowler JH, Christakis NA. Egocentric social network structure, health, and pro-social behaviors in a National Panel Study of Americans. PLoS One. 2012;7:e36250.  https://doi.org/10.1371/journal.pone.0036250.CrossRefPubMedPubMedCentralGoogle Scholar
  94. Opsahl T. Triadic closure in two-mode networks: redefining the global and local clustering coefficients. Soc Networks. 2011; 34.  https://doi.org/10.1016/j.socnet.2011.07.001.CrossRefGoogle Scholar
  95. Opsahl T, Agneessens F, Skvoretz J. Node centrality in weighted networks: generalizing degree and shortest paths. Soc Networks. 2010;32:245–51.CrossRefGoogle Scholar
  96. Palla G, Derenyi I, Farkas I, Vicsek T. Uncovering the overlapping community structure of complex networks in nature and society. Nature. 2005;435:814–8.  https://doi.org/10.1038/nature03607.CrossRefPubMedGoogle Scholar
  97. Paul S, O’Malley AJ. Hierarchical longitudinal models of relationships in social networks. J R Stat Soc Ser C Appl Stat. 2013;62:705–22.PubMedPubMedCentralGoogle Scholar
  98. Pham HH, O’Malley AS, Bach PB, Saiontz-Martinez C, Schrag D. Primary care physicians’ links to other physicians through Medicare patients: the scope of care coordination. Ann Intern Med. 2009;150:236–42.PubMedPubMedCentralCrossRefGoogle Scholar
  99. Piraveenan M, Prokopenko M, Zomaya AY. Assortative mixing in directed biological networks. IEEE Trans Comput Biol Bioinform. 2010;9:66–78. To appear.PubMedCrossRefGoogle Scholar
  100. Pollack CE, Weissman G, Bekelman J, Liao K, Armstrong K. Physician social networks and variation in prostate cancer treatment in three cities. Health Serv Res. 2012;47:380–403.PubMedPubMedCentralCrossRefGoogle Scholar
  101. Porter MA, Onnela J-P, Mucha PJ. Communities in networks. Not Am Math Soc. 2009;56(1082–1097):1164–6.Google Scholar
  102. Price DDS. A general theory of bibliometric and other cumulative advantage processes. J Am Soc Inf Sci. 1976;27:292–306.  https://doi.org/10.1002/asi.4630270505.CrossRefGoogle Scholar
  103. Robins G, Pattison P, Woolcock J. Small and other worlds: global network structures from local processes. Am J Sociol. 2005;110:894–936.CrossRefGoogle Scholar
  104. Robins GL, Snijders TAB, Wang P, Handcock MS, Pattison PE. Recent developments in exponential random graph (p) models for social networks. Soc Networks. 2007;29:192–215.CrossRefGoogle Scholar
  105. Robins GL, Pattison PE, Wang P. Closure, connectivity and degree distributions: exponential random graph (p*) models for directed social networks. Soc Networks. 2009;31:105–7.CrossRefGoogle Scholar
  106. Rubin D. Bayesian inference for causal effects: the role of randomization. Ann Stat. 1978;6:34–58.CrossRefGoogle Scholar
  107. Seidman SB. Network structure and minimum degree. Soc Networks. 1983;5:269–87.CrossRefGoogle Scholar
  108. Shalizi RR, Rinaldo A. Consistency under sampling of exponential random graph models. 2012. arXiv preprint. arXiv:1111.3054v3Google Scholar
  109. Shalizi CR, Thomas AC. Homophily and contagion are generically confounded in observational social network studies. Sociol Methods Res. 2011;40:211–39.PubMedPubMedCentralCrossRefGoogle Scholar
  110. Simmel G. The sociology of Georg Simmel. New York: The Free Press; 1908.Google Scholar
  111. Snijders T. The degree variance: an index of graph heterogeneity. Soc Networks. 1981;3:163–74.CrossRefGoogle Scholar
  112. Snijders T. Stochastic actor-oriented models for network change. J Math Sociol. 1996;21:149–72.CrossRefGoogle Scholar
  113. Snijders TAB. The statistical evaluation of social network dynamics. In: Sociological methodology. Oxford, UK: Basil Blackwell; 2001. p. 361–95.Google Scholar
  114. Snijders TAB. Models for longitudinal social network data. In: Models and methods in social network analysis. Cambridge: Cambridge University Press; 2005. p. 215–47.CrossRefGoogle Scholar
  115. Snijders TAB. Statistical methods for network dynamics. In: Luchini SR et al., editors. Proceedings of the XLIII Scientific Meeting, Italian Statistical Society, Basil Blackwell, Ltd; 2006. p. 281–96Google Scholar
  116. de Solla Price DJ. Networks of scientific papers. Science. 1965;149:510–5. http://www.sciencemag.org/content/149/3683/510.short.CrossRefGoogle Scholar
  117. Steglich C, Snijders TAB, Pearson M. Dynamic networks and behavior: separating selection from influence. Sociol Methodol. 2010;40:329–93.CrossRefGoogle Scholar
  118. Szabo G, Barabasi AL. Network effects in service usage. 2007. Arxiv preprint. http://lanl.arxiv.org/abs/physics/0611177
  119. Thompson S. Adaptive web sampling. Biometrics. 2006;62:1224–34.PubMedCrossRefGoogle Scholar
  120. Thompson S, Frank O. Mode-based estimation with link-tracing sampling designs. Survey Methodol. 2000;26:87–98.Google Scholar
  121. Thompson S, Seber GAF. Adaptive sampling. New York: Wiley; 1996.Google Scholar
  122. Toivonen R, Onnela J-P, Saramäki J, Hyvönen J, Kaski K. A model for social networks. Phys A Stat Mech Appl. 2006;371:851–60. http://www.sciencedirect.com/science/article/pii/S0378437106003931CrossRefGoogle Scholar
  123. Traud AL, Mucha PJ, Porter MA. Social structure of Facebook networks. Phys A Stat Mech Appl. 2012;391:4165–80. http://www.sciencedirect.com/science/article/pii/S0378437111009186CrossRefGoogle Scholar
  124. VanderWeele TJ. Sensitivity analysis for contagion effects in social networks. Sociol Methods Res. 2011;40:240–55.PubMedPubMedCentralCrossRefGoogle Scholar
  125. VanderWeele TJ, Ogburn EL, Tchetgen Tchetgen EJ. Why and when “Flawed” social network analyses still yield valid tests of no contagion. Stat Polit Policy. 2012;3:1050.  https://doi.org/10.1515/2151-7509.1050.
  126. Vázquez A. Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. Phys Rev E. 2003;67:056104.  https://doi.org/10.1103/PhysRevE.67.056104.CrossRefGoogle Scholar
  127. Wang W, Wong G. Stochastic Blockmodels for directed graphs. J Am Stat Assoc. 1987;82:8–19.CrossRefGoogle Scholar
  128. Wang P, Sharpe K, Robins GL, Pattison PE. Exponential random graph (p*) models for affiliation networks. Soc Networks. 2009;31:12–25.CrossRefGoogle Scholar
  129. Wasserman SS, Faust K. Social network analysis: methods and applications. Cambridge: Cambridge University Press; 1994.CrossRefGoogle Scholar
  130. Wasserman S, Pattison P. Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p. Psychometrika. 1996;61:401–25.CrossRefGoogle Scholar
  131. Westveld AH, Hoff PD. A mixed effect model for longitudinal relational and network data, with applications to international trade and conflict. Ann Appl Stat. 2011;5:843–72.CrossRefGoogle Scholar
  132. White D, Harary F. The cohesiveness of blocks in social networks: node connectivity and conditional density. Sociol Methodol. 2001;31:305–59.CrossRefGoogle Scholar
  133. Wong LH, Pattison P, Robins G. A spatial model for social networks. Phys A Stat Mech Appl. 2006;360:99–120. http://www.sciencedirect.com/science/article/pii/S0378437105004334CrossRefGoogle Scholar
  134. Zijlstra BJH, Duijn MV, Snijders TAB. The multilevel P2 model: a random effects model for the analysis of multiple social networks. Methodology. 2006;2:42–7.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Alistair James O’Malley
    • 1
    • 3
    Email author
  • Jukka-Pekka Onnela
    • 2
  1. 1.The Dartmouth Institute for Health Policy and Clinical Practice, Department of Biomedical Data ScienceGeisel School of Medicine at DartmouthLebanonUSA
  2. 2.Department of BiostatisticsHarvard School of Public HealthBostonUSA
  3. 3.Department of Health Care PolicyHarvard Medical SchoolBostonUSA

Personalised recommendations