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Convegno della Società Italiana di Statistica

SIS 2017: New Statistical Developments in Data Science pp 105–117Cite as

Bayesian Analysis of ERG Models for Multilevel, Multiplex, and Multilayered Networks with Sampled or Missing Data

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 288)

Abstract

Social network analysis has typically concerned analysis of one type of tie connecting nodes of the same type. It has however been recognised that people are connected through multiple types of ties and that people in addition are affiliated with multiple types of non-people nodes. Exponential random graph models (ERGM) is a family of statistical models for social networks that at this point allows for a number of different types of network data, including one-mode networks, bipartite networks, multiplex data, as well as multilevel network data. Multilevel networks have been proposed as a joint representation of associations between multiple types of entities or nodes, such as people and organization, where two types of nodes gives rise to three distinct types of ties. The typical roster data collection method may be impractical or infeasible when the node sets are hard to detect or define or because of the cognitive demands on respondents. Multilevel multilayered networks allow us to consider a multitude of different sources of data and to sample on different types of nodes and relations. We consider modelling multilevel multilayered networks using exponential random graph models and extend a recently developed Bayesian data-augmentation scheme to allow for partially missing data. We illustrate the proposed inference procedures for the case of multilevel snowball sampling and sampling with error based on the Noordin Top network.

Keywords

  • ERGM
  • Exchange algorithm
  • Missing data
  • Multilevel networks
  • Snowball sampling
  • Social networks

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References

  1. Agneessens, F., Roose, H.: Local structural properties and attribute characteristics in 2-mode networks: \(p^*\) models to map choices of theater events. Journal of Mathematical Sociology 32(3), 204–237 (2008)

    Google Scholar 

  2. Breiger, R.L.: The duality of persons and groups. Soc. Forces 53, 181–190. http://www.jstor.org/stable/10.2307/2576011 (1974)

    CrossRef  Google Scholar 

  3. Broccatelli, C., Everett, M., Koskinen, J.: Temporal dynamics in covert networks. Methodol. Innov. 9, 1–14 (2016)

    Google Scholar 

  4. Butts, C.T.: A perfect sampling method for exponential family random graph models. J. Math. Sociol. 1–20 (2017)

    Google Scholar 

  5. Caimo, A., Friel, N.: Bayesian inference for exponential random graph models. Soc. Netw. 33, 41–55 (2011)

    CrossRef  Google Scholar 

  6. Everton, S.F.: Disrupting Dark Networks. Cambridge University Press, New York (2012)

    CrossRef  Google Scholar 

  7. Frank, O., Strauss, D.: Markov graphs. J. Am. Stat. Assoc. 81, 832–842 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Group, I.C.: Terrorism in indonesia: Noordin’s networks (2006)

    Google Scholar 

  9. Handcock, M.S., Gile, K.: Modeling social networks from sampled data. Ann. Appl. Stat. 4, 5–25 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)

    CrossRef  Google Scholar 

  11. Koskinen, J.: Bayesian analysis of exponential random graphs-estimation of parameters and model selection. Technical Report, Research Report 2004: 2, Department of Statistics, Stockholm University (2004)

    Google Scholar 

  12. Koskinen, J.: The linked importance sampler auxiliary variable Metropolis Hastings algorithm for distributions with intractable normalising constants. Technical Report 1, Department of Psychology, University of Melbourne (2008). (Working paper)

    Google Scholar 

  13. Koskinen, J., Edling, C.: Modelling the evolution of a bipartite network - peer referral in interlocking directorates. Soc. Netw. 34, 309–322 (2012). https://doi.org/10.1016/j.socnet.2010.03.001

    CrossRef  Google Scholar 

  14. Koskinen, J., Snijders, T.: Simulation, estimation and goodness of fit. In: Lusher, D., Koskinen, J., Robins, G. (eds.) Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications, pp. 141–166. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  15. Koskinen, J.H.: Exponential random graph models. In: Wiley StatsRef: Statistics Reference Online, stat08136. Wiley (in press)

    Google Scholar 

  16. Koskinen, J.H., Caimo, A., Lomi, A.: Simultaneous modeling of initial conditions and time heterogeneity in dynamic networks: an application to foreign direct investments. Netw. Sci. 3(1), 58–77 (2015)

    CrossRef  Google Scholar 

  17. Koskinen, J.H., Robins, G.L., Pattison, P.E.: Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation. Stat. Methodol. 7, 366–384 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Koskinen, J.H., Robins, G.L., Wang, P., Pattison, P.E.: Bayesian analysis for partially observed network data, missing ties, attributes and actors. Soc. Netw. 35(4), 514–527 (2013)

    CrossRef  Google Scholar 

  19. Koskinen, J.H., Wang, P., Robins, G.L., Pattison, P.E.: Outliers and influential observations in exponential random graph models. Psychometrika 83(4), 809–830 (2018)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Krivitsky, P.N.: Exponential-family random graph models for valued networks. Electron. J. Stat. 6, 1100 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Lazega, E., Jourda, M.T., Mounier, L., Stofer, R.: Catching up with big fish in the big pond? Multi-level network analysis through linked design. Soc. Netw. 30(2), 159–176 (2008)

    CrossRef  Google Scholar 

  22. Lazega, E., Pattison, P.E.: Multiplexity, generalized exchange and cooperation in organizations: a case study. Soc. Netw. 21(1), 67–90 (1999)

    CrossRef  Google Scholar 

  23. Little, R., Rubin, D.: Statistical Analysis with Missing Data. Wiley, New York (1987)

    MATH  Google Scholar 

  24. Lusher, D., Koskinen, J., Robins, G.: Exponential Random Graph Models. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  25. Møller, J., Pettitt, A.N., Reeves, R., Berthelsen, K.K.: An efficient markov chain monte carlo method for distributions with intractable normalising constants. Biometrika 93(2), 451–458 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Moreno, J., Jennings, M.: Statistics of social configurations. Sociometry 1(3), 342–374 (1938)

    CrossRef  Google Scholar 

  27. Murray, I., Ghahramani, Z., MacKay, D.J.C.: MCMC for doubly-intractable distributions. In: Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence, pp. 359–366. AUAI Press, Corvallis (2006)

    Google Scholar 

  28. Pattison, P., Wasserman, S.: Logit models and logistic regressions for social networks: II. Multivariate relations. Br. J. Math. Stat. Psychol. 52, 169–193 (1999)

    CrossRef  MATH  Google Scholar 

  29. Pattison, P.L., Snijders, T.: Modelling social networks: next steps. In: Lusher, D., Koskinen, J., Robins, G. (eds.) Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications, pp. 287–301. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  30. Propp, J., Wilson, D.: Exact sampling with coupled markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 232–252 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. Roberts, N., Everton, S.F.: Strategies for combating dark networks. J. Soc. Struct. 12(2), (2011)

    Google Scholar 

  32. Robins, G., Pattison, P., Wasserman, S.: Logit models and logistic regressions for social networks, III. Valued relations. Psychometrika 64, 371–394 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Robins, G.L., Alexander, M.: Small worlds among interlocking directors: network structure and distance in bipartite graphs. Comput. Math. Organ. Theory 10, 69–94 (2004)

    CrossRef  MATH  Google Scholar 

  34. Robins, G.L., Pattison, P.E., Wang, P.: Closure, connectivity and degree distributions: Exponential random graph (\(p^*\)) models for directed social networks. Soc. Netw. 31, 105–117 (2009)

    Google Scholar 

  35. Schweinberger, M.: Instability, sensitivity, and degeneracy of discrete exponential families. J. Am. Stat. Assoc. 106(496), 1361–1370 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Schweinberger, M., Krivitsky, P.N., Butts, C.T., Stewart, J.: Exponential-family models of random graphs: Inference in finite-, super-, and infinite-population scenarios. Available at https://arxiv.org/abs/1707.04800 (2018)

  37. Skvoretz, J., Faust, K.: Logit models for affiliation networks. Sociol. Methodol. 29(1), 253–280 (1999)

    CrossRef  Google Scholar 

  38. Snijders, T.A.: Statistical models for social networks. Annu. Rev. Sociol. 37, (2011)

    CrossRef  Google Scholar 

  39. Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S.: New specifications for exponential random graph models. Sociol. Methodol. 36, 99–153 (2006)

    CrossRef  Google Scholar 

  40. Wang, P., Pattison, P., Robins, G.: Exponential random graph model specifications for bipartite networks - a dependence hierarchy. Soc. Netw. 35(2), 211–222 (2013)

    CrossRef  Google Scholar 

  41. Wang, P., Robins, G., Pattison, P., Koskinen, J.H.: MPNet: Program for the simulation and estimation of (p*) exponential random graph models for multilevel networks. Melbourne School of Psychological Sciences, The University of Melbourne, Melbourne (2014)

    Google Scholar 

  42. Wang, P., Robins, G., Pattison, P., Lazega, E.: Exponential random graph models for multilevel networks. Soc. Netw. 35, 96–115 (2013)

    CrossRef  Google Scholar 

  43. Wang, P., Sharpe, K., Robins, G.L., Pattison, P.E.: Exponential random graph (p*) models for affiliation networks. Soc. Netw. 31(1), 12–25 (2009)

    CrossRef  Google Scholar 

  44. Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)

    CrossRef  MATH  Google Scholar 

  45. Wasserman, S., Iacobucci, D.: Statistical modelling of one-mode and two-mode networks: simultaneous analysis of graphs and bipartite graphs. Br. J. Math. Stat. Psychol. 44(1), 13–43 (1991)

    CrossRef  Google Scholar 

  46. White, H.C., Boorman, S.A., Breiger, R.L.: Social Structure from Multiple Networks. I. Blockmodels of Roles and Positions. Am. J. Sociol. 81(4), 730–780 (1976)

    CrossRef  Google Scholar 

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Acknowledgements

The work of Koskinen and Broccatelli is funded by the Leverhulme Trust Grant RPG-2013-140.

Author information

Authors and Affiliations

  1. Melbourne School of Psychological Sciences, The University of Melbourne, Melbourne, Vic, 3010, Australia

    Johan Koskinen & Garry Robins

  2. The Institute of Analytical Sociology, Linköping University, Linköping, Sweden

    Johan Koskinen

  3. The Alan Turing Institute, British Library, 96 Euston Road, London, NW1 2DB, UK

    Johan Koskinen

  4. MRC/CSO Social and Public Health Sciences Unit, University of Glasgow, 200 Renfield Street, Glasgow, G2 3QB 9PL, Scotland

    Chiara Broccatelli

  5. Faculty of Business and Law, Centre for Transformative Innovation, Swinburne University of Technology, Melbourne, Vic, Australia

    Peng Wang

Authors
  1. Johan Koskinen
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  2. Chiara Broccatelli
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  3. Peng Wang
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  4. Garry Robins
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Corresponding author

Correspondence to Johan Koskinen .

Editor information

Editors and Affiliations

  1. Dipartimento di Statistica, Informatica, Applicazioni ‘G. Parenti’ (DiSIA), Università degli Studi di Firenze, Florence, Italy

    Prof. Alessandra Petrucci

  2. Dipartimento Scienze Statistiche, Sapienza Università di Roma, Rome, Italy

    Prof. Filomena Racioppi

  3. Dipartimento di Matematica e Fisica, Università della Campania ‘Luigi Vanvitelli’, Caserta, Italy

    Prof. Rosanna Verde

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Koskinen, J., Broccatelli, C., Wang, P., Robins, G. (2019). Bayesian Analysis of ERG Models for Multilevel, Multiplex, and Multilayered Networks with Sampled or Missing Data. In: Petrucci, A., Racioppi, F., Verde, R. (eds) New Statistical Developments in Data Science. SIS 2017. Springer Proceedings in Mathematics & Statistics, vol 288. Springer, Cham. https://doi.org/10.1007/978-3-030-21158-5_9

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