Numéro spécial : Special Issue on Networks and Statistics
Estimation of Stochastic actor-oriented models for the evolution of networks by generalized method of moments
[Estimation des modèles stochastiques orienté par l’acteur pour l’évolution des réseaux par la méthode des moments généralisées]
Journal de la société française de statistique, Tome 156 (2015) no. 3, pp. 140-165.

Le modèle stochastique orienté par l’acteur (Snijders, Sociological Methodology, 2001) modèle l’évolution temporelle des réseaux, étant donné un panel dans un ensemble fixe d’acteurs, où à chaque vague de panel le réseau entre ces acteurs (une structure de graphe orienté) ainsi que les attributs des acteurs sont observés. Les paramètres de ce modèle sont, d’habitude, estimés par une version d’approximation stochastique de la méthode des moments. Des statistiques qui correspondent aux paramètres d’une manière naturelle sont utilisés pour l’ajustement du modèle. Nous présentons ici un estimateur basé sur la méthode généralisée des moments, utilisant plus de statistiques que de paramètres, pour minimiser la distance entre les statistiques observées et leurs espérances mathématiques. Ici encore, l’équation résultante est résolue par approximation stochastique. Plusieurs questions algorithmiques surviennent qui doivent être résolues afin d’obtenir une procédure stable. Pour quelques exemples, nous étudions le gain résultant de l’efficience statistique.

The stochastic actor-oriented model (Snijders, Sociological Methodology, 2001) models the evolution of networks over time, given panel data in a fixed group of actors, where at each panel wave the network between these actors (a digraph structure) as well as attribute variables for these actors are observed. The parameters of this model usually are estimated by a stochastic approximation version of the method of moments. Statistics that correspond to the parameters in a natural way are used for fitting the model. Here we present an estimator based on the generalized method of moments, i.e., using more statistics than parameters, for minimizing the distance between observed statistics and their expected values. Again, the resulting equation is solved by stochastic approximation. Several algorithmic issues arise that have to be solved in order to obtain a stable procedure. For some examples we study the resulting gain in statistical efficiency.

Keywords: Stochastic actor-oriented models, social networks, generalized method of moments, stochastic approximation algorithm
Mot clés : Modèles stochastiques orienté par l’acteur, réseaux sociaux, méthode des moments généralisées, algorithmes d’approximation stochastique
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     title = {Estimation of {Stochastic} actor-oriented models  for the evolution of networks by generalized method of moments},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {140--165},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
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Amati, Viviana; Schönenberger, Felix; Snijders, Tom A. B. Estimation of Stochastic actor-oriented models  for the evolution of networks by generalized method of moments. Journal de la société française de statistique, Tome 156 (2015) no. 3, pp. 140-165. http://archive.numdam.org/item/JSFS_2015__156_3_140_0/

[1] Burguete, Jose F; Ronald Gallant, A; Souza, Geraldo On unification of the asymptotic theory of nonlinear econometric models, Econometric Reviews, Volume 1 (1982) no. 2, pp. 151-190 | MR | Zbl

[2] Bowman, K. O.; Shenton, L. R. Method of Moments, Encyclopedia of Statistical Sciences (Kotz, S.; Johnson, N. L., eds.), Volume 5, Wiley, New York, 1985, pp. 467-473

[3] Carrington, Peter J; Scott, John; Wasserman, Stanley Models and Methods in Social Network Analysis, Cambridge University Press, New York, 2005

[4] Doreian, Patrick; Stokman, Frans N The dynamics and evolution of social networks, Evolution of Social Networks (Doreian, Patrick; Stokman, Frans, eds.), Routledge, Amsterdam, 1997, pp. 1-18

[5] Doreian, Patrick; Teuter, Klaus; Wang, Chi-Hsein Network Autocorrelation Models: Some Monte Carlo Results, Sociological Methods & Research, Volume 13 (1984) no. 2, pp. 155-200

[6] Freeman, Linton C The impact of computer based communication on the social structure of an emerging scientific specialty, Social Networks, Volume 6 (1984) no. 3, pp. 201-221

[7] Greenan, Charlotte C Diffusion of innovations in dynamic networks, Journal of the Royal Statistical Society: Series A (Statistics in Society), Volume 178 (2015), pp. 147-166 | MR

[8] Hall, A. R. Generalized Method of Moments, Oxford University Press, Oxford, 2005 | MR | Zbl

[9] Hansen, L.P. Large sample properties of generalized method of moments estimators, Econometrica, Volume 50 (1982), pp. 1029-1054 | MR | Zbl

[10] Holland, Paul W; Leinhardt, Samuel A dynamic model for social networks, Journal of Mathematical Sociology, Volume 5 (1977) no. 1, pp. 5-20 | MR | Zbl

[11] Koskinen, Johan; Edling, Christofer Modelling the evolution of a bipartite network-Peer referral in interlocking directorates, Social Networks, Volume 34 (2012) no. 3, pp. 309-322

[12] Koskinen, Johan H.; Snijders, Tom A. B. Bayesian Inference for Dynamic Social Network Data, Journal of Statistical Planning and Inference, Volume 13 (2007), pp. 3930-3938 | MR | Zbl

[13] Leenders, Roger Th AJ Modeling social influence through network autocorrelation: constructing the weight matrix, Social Networks, Volume 24 (2002) no. 1, pp. 21-47

[14] Luce, R.D.; Suppes, P. Preference, utility, and subjective probability, Handbook of Mathematical Psychology, Volume 3 (1965), pp. 249-410

[15] Lospinoso, Joshua A; Schweinberger, Michael; Snijders, Tom A. B.; Ripley, Ruth M Assessing and accounting for time heterogeneity in stochastic actor oriented models, Advances in Data Analysis and Classification, Volume 5 (2011) no. 2, pp. 147-176 | MR | Zbl

[16] McFadden, Daniel Conditional Logit Analysis of Qualitative Choice Behavior, Frontiers in Econometrics (Zarembka, Paul, ed.), Academic Press, New York, 1974, pp. 105-142

[17] Pflug, Georg Ch Non-asymptotic confidence bounds for stochastic approximation algorithms with constant step size, Monatshefte für Mathematik, Volume 110 (1990) no. 3-4, pp. 297-314 | MR | Zbl

[18] Polyak, Boris Teodorovich A new method of stochastic approximation type, Automation and Remote Control, Volume 51 (1990), pp. 937-946 | MR | Zbl

[19] Robins, Garry; Elliott, Peter; Pattison, Philippa Network models for social selection processes, Social Networks, Volume 23 (2001) no. 1, pp. 1-30 | MR | Zbl

[20] Robbins, Herbert; Monro, Sutton A stochastic approximation method, The Annals of Mathematical Statistics, Volume 22 (1951), pp. 400-407 | MR | Zbl

[21] Robins, Garry; Pattison, Philippa; Elliott, Peter Network models for social influence processes, Psychometrika, Volume 66 (2001) no. 2, pp. 161-189 | MR | Zbl

[22] Ripley, Ruth M.; Snijders, Tom A. B.; Bóda, Zsofia; Vörös, András; Preciado, Paulina Manual for RSiena (2015) http://www.stats.ox.ac.uk/siena/ (Technical report)

[23] Ruppert, David Efficient estimations from a slowly convergent Robbins-Monro process (1988) (Technical report)

[24] Scott, John; Carrington, Peter J The SAGE Handbook of Social Network Analysis, SAGE publications, London, 2011

[25] Snijders, Tom A. B.; Koskinen, Johan; Schweinberger, Michael Maximum likelihood estimation for social network dynamics, The Annals of Applied Statistics, Volume 4 (2010) no. 2, pp. 567-588 | MR | Zbl

[26] Snijders, Tom A. B.; Lomi, Alessandro; Torlò, Vanina A model for the multiplex dynamics of two-mode and one-mode networks, with an application to employment preference, friendship, and advice, Social Networks, Volume 35 (2013), pp. 265-276

[27] Snijders, Tom A. B. The statistical evaluation of social network dynamics, Sociological Methodology, Volume 31 (2001) no. 1, pp. 361-395

[28] Snijders, Tom A. B. Stochastic actor-oriented models for network change, Journal of Mathematical Sociology, Volume 21 (1996) no. 1-2, pp. 149-172 | Zbl

[29] Schweinberger, Michael; Snijders, Tom A. B. Markov models for digraph panel data: Monte Carlo-based derivative estimation, Computational Statistics & Data Analysis, Volume 51 (2007) no. 9, pp. 4465-4483 | MR | Zbl

[30] Steglich, Christian E. G.; Snijders, Tom A. B.; Pearson, Michael Dynamic networks and behavior: Separating selection from influence, Sociological Methodology, Volume 40 (2010), pp. 329-393

[31] Snijders, Tom A B; van Duijn, Marijtje A J Simulation for statistical inference in dynamic network models, Simulating Social Phenomena (Conte, R.; Hegselmann, R.; Terna, P., eds.), Springer, Berlin, 1997, pp. 493-512

[32] Snijders, Tom A. B.; Van de Bunt, Gerhard G; Steglich, Christian E G Introduction to stochastic actor-based models for network dynamics, Social Networks, Volume 32 (2010) no. 1, pp. 44-60

[33] Van de Bunt, Gerhard G Friends by choice: An actor-oriented statistical network model for friendship networks through time, Interuniversity Center for Social Science Theory and Methodology, Groningen/Amsterdam, 1999

[34] Van de Bunt, Gerhard G; Van Duijn, Marijtje A. J.; Snijders, Tom A. B. Friendship networks through time: An actor-oriented dynamic statistical network model, Computational & Mathematical Organization Theory, Volume 5 (1999) no. 2, pp. 167-192 | Zbl

[35] Wasserman, Stanley A stochastic model for directed graphs with transition rates determined by reciprocity, Sociological Methodology, Volume 11 (1980), pp. 392-412

[36] Wasserman, Stanley Analyzing social networks as stochastic processes, Journal of the American Statistical Association, Volume 75 (1980) no. 370, pp. 280-294 | Zbl

[37] Wasserman, Stanley; Faust, Katherine Social Network Analysis: Methods and Applications, Cambridge University Press, 1994 | Zbl

[38] Wasserman, Stanley; Iacobucci, Dawn Sequential social network data, Psychometrika, Volume 53 (1988) no. 2, pp. 261-282 | Zbl