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Sampling networks from their posterior predictive distribution

Published online by Cambridge University Press:  03 April 2014

RAVI GOYAL ,
JOSEPH BLITZSTEIN  and
VICTOR DE GRUTTOLA
RAVI GOYAL
Affiliation:
Department of Biostatistics, Harvard School of Public Health, Boston, MA, 02115, USA (e-mail: [email protected])
JOSEPH BLITZSTEIN
Affiliation:
Department of Statistics, Harvard University, Cambridge, MA 02138-2901, USA (e-mail: [email protected])
VICTOR DE GRUTTOLA
Affiliation:
Department of Biostatistics, Harvard School of Public Health, Boston, MA, 02115, USA (e-mail: [email protected])
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Abstract

Recent research indicates that knowledge about social networks can be leveraged to increase efficiency of interventions (Valente, 2012). However, in many settings, there exists considerable uncertainty regarding the structure of the network. This can render the estimation of potential effects of network-based interventions difficult, as providing appropriate guidance to select interventions often requires a representation of the whole network. In order to make use of the network property estimates to simulate the effect of interventions, it may be beneficial to sample networks from an estimated posterior predictive distribution, which can be specified using a wide range of models. Sampling networks from a posterior predictive distribution of network properties ensures that the uncertainty about network property parameters is adequately captured. The tendency for relationships among network properties to exhibit sharp thresholds has important implications for understanding global network topology in the presence of uncertainty; therefore, it is essential to account for uncertainty. We provide detail needed to sample networks for the specific network properties of degree distribution, mixing frequency, and clustering. Our methods to generate networks are demonstrated using simulated data and data from the National Longitudinal Study of Adolescent Health.

Keywords

network generationposterior predictive distributionnetwork statistics
Type
Research Article
Information
Network Science , Volume 2 , Issue 1 , April 2014 , pp. 107 - 131
Copyright
Copyright © Cambridge University Press 2014 

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References

Amanatidis, Y., Green, B., & Mihail, M. (2008). Graphic realizations of joint-degree matrices. Unpublished.Google Scholar
Bansal, S., Pourbohloul, B., & Meyers, L. A. (2006). A comparative analysis of influenza vaccination programs. Plos Medicine, 3, 387.CrossRefGoogle ScholarPubMed
Barabasi, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512.Google Scholar
Blitzstein, J., & Diaconis, P. (2010). A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet mathematics, 6 (4), 487–520.Google Scholar
Boily, M.-C., Mâsse, B., Alsallaq, R., Padian, N., Eaton, J., Vesga, J., & Hallett, T. (2012). Hiv treatment as prevention: Considerations in the design, conduct, and analysis of cluster randomized controlled trials of combination hiv prevention. Plos Medicine, 9 (7), e1001250.Google Scholar
Bollobás, B. (2001). Random graphs (2 ed.). New York: Cambridge University Press.Google Scholar
Britton, T., Deijfen, M. & Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics, 124 (6), 1377–1397.Google Scholar
Caimo, A., & Friel, N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33 (1), 41–55.Google Scholar
Christakis, N., & Fowler, J. (2008). The collective dynamics of smoking in a large social network. New England Journal of Medicine, 358 (21), 2249–2258.Google Scholar
Doyle, J. C., Alderson, D., Li, L., Low, S., Roughan, M., Shalunov, S., . . . Willinger, W. (2005). The ‘robust yet fragile’ nature of the internet. Proceedings of the National Academy of Sciences, 102 (41), 14497–14502.CrossRefGoogle ScholarPubMed
Dunbar, R. I. M. (1993). Coevolution of neocortical size, group size, and language in humans. Behavioural and Brain Sciences, 16, 681–735.Google Scholar
Erdős, P., & Rényi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.Google Scholar
Frank, O. (2005). Network sampling and model fitting. In Carrington, P. J., Scott, J., & Wasserman, S. (Eds.), Models and methods in social network analysis (pp. 31–56) Chap. 3. Cambridge: Cambridge University Press.Google Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81, 832–842.Google Scholar
Friedgut, E., & Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proceedings of the American Mathematical Society, 124 (10), 2993–3002.CrossRefGoogle Scholar
Hakimi, S. L. (1962). On realizability of a set of integers as degrees of the vertices of a linear graph. Journal of the Society for Industrial and Applied Mathematics, 10, 496–506.Google Scholar
Handcock, M. S. (2003). Degreenet: Models for skewed count distributions relevant to networks. Seattle, WA. Version 1.2 . Project home. Retrieved from http://statnet.org.Google Scholar
Handcock, M. S., & Gile, K. J. (2010). Modeling social networks from sampled data. Annals of Applied Statistics, 4, 5–25.Google Scholar
Handcock, M. S., Hunter, D. R., Butts, C. T., Goodreau, S. M., Krivitsky, P. N., & Morris, M. (2012). ERGM: A package to fit, simulate and diagnose exponential-family models for networks. Seattle, WA. Version 3.0-1. Project home Retrieved from http://urlstatnet.org.Google Scholar
Harris, K. M., & Udry, J. R. (2012). National longitudinal study of adolescent health (add health), 1994-2008. Ann Arbor, MI: Inter-University Consortium for Political and Social Research, doi:10.3886/icpsr21600-v9edn.Google Scholar
Havel, V. (1955). A remark on the existence of finite graphs. Časopis pro pěstování matematiky, 80, 477–480.CrossRefGoogle Scholar
Kolaczyk, E. (2009). Statistical analysis of network data: Methods and models. New York: Springer Science+Business Media, LLC. Chap. Sampling and Estimation in Network Graphs, pp. 123–152.CrossRefGoogle Scholar
Koskinen, J. H, Robins, G. L, Wang, P., & Pattison, P. E. (2013). Bayesian analysis for partially observed network data, missing ties, attributes and actors. Social Networks, 35 (4), 514–527.Google Scholar
Mahadevan, P., Krioukov, D., Fall, K., & Vahdat, A. (2006). Systematic topology analysis and generation using degree correlations. In ACM SIGCOMM Computer Communication Review, 36(4), 135–146. New York: ACM.Google Scholar
Maslov, S., & Sneppen, K. (2002). Specificiy and stability in topology of protein networks. Science, 296 (5569), 910–913.Google Scholar
McNeely, C. A., Nonnemaker, J. M., & Blum, R. W. (2002). Promoting school connectedness: Evidence from the national longitudinal study of adolescent health. Journal of School Health, 72, 138–146.Google Scholar
McPherson, M., Smith-Lovin, L., & Brashears, M. E. (2006). Social isolation in america: Changes in core discussion networks over two decades. American Sociological Review, 71 (3), 353–375.Google Scholar
Meyers, L. A., Newman, M. E. J., Martin, M., & Schrag, S. (2003). Applying network theory to epidemics: Control measures for mycoplasma pneumoniae outbreaks. Emerging Infectious Diseases, 9 (2), 204.Google Scholar
Meyers, L. A., Pourbohloul, B., Newman, M. E. J., Skowronski, D. M, & Brunham, R. C. (2005). Network theory and sars: predicting outbreak diversity. Journal of Theoretical Biology, 232 (1), 71–81.CrossRefGoogle ScholarPubMed
Mills, H. L., Cohen, T., & Colijn, C. (2011). Modelling the performance of isoniazid preventive therapy for reducing tuberculosis in hiv endemic settings: The effects of network structure. Journal of the Royal Society Interface, 8 (63), 1510–1520.CrossRefGoogle ScholarPubMed
Morris, M., Goodreau, S., & Moody, J. (2007). Sexual networks, concurrency, and std/hiv. In Holmes, K. K., Sparling, P. F., & Stamm, W. E. (Eds.), Sexually transmitted diseases (pp. 109–126). New York, NY, USA: McGraw-Hill International Book Co.Google Scholar
Morris, M., Kurth, A., Hamilton, D., Moody, J., & Wakefield, S. (2009). Concurrent partnerships and hiv prevalence disparities by race: Linking science and public health practice. American Journal of Public Health, 99 (6), 1023–1031.CrossRefGoogle ScholarPubMed
Murphy, K. P. (2007). Conjugate bayesian analysis of the gaussian distribution.Google Scholar
Newman, M. (2002). Assortative mixing in networks. Physical Review Letters, 89 (20), 208701.Google Scholar
Newman, M. E. (2010). Networks an introduction. New York: Oxford University Press.Google Scholar
Palombi, L., Bernava, G. M., Nucita, A., Giglio, P., Liotta, G., Nielsen-Saines, K., . . . Marazzi, M. C. (2012). Predicting trends in hiv-1 sexual transmission in sub-saharan africa through the drug resource enhancement against aids and malnutrition model: Antiretrovirals for reduction of population infectivity, incidence and prevalence at the district level. Clinical Infectious Diseases, 55 (2), 268–275.Google Scholar
Pattison, P. E., Robins, G. L., Snijders, T. A. B., & Wang, P. (2013). Conditional estimation of exponential random graph models from snowball sampling designs. Journal of Mathematical Psychology, 57 (6), 284–296.CrossRefGoogle Scholar
Robert, C. P., & Casella, G. (2004). Monte carlo statistical methods. New York: Springer.CrossRefGoogle Scholar
Shalizi, C. R., & Rinaldo, A. (2013). Consistency under sampling of exponential random graph models. Annals of Statistics, 41 (2), 508–535.CrossRefGoogle ScholarPubMed
Valente, T. W. (2012). Network interventions. Science, 337, 49–53.Google Scholar
Valente, T. W., Hoffman, B. R., Ritt-Olson, A., Lichtman, K., & Johnson, C. A. (2003). Effects of a social-network method for group assignment strategies on peer-led tobacco prevention programs in schools. American Journal of Public Health, 93 (11), 1837–1843.Google Scholar
Vázquez, A., Pastor-Satorras, R., & Vespignani, A. (2002). Large-scale topological and dynamical properties of the internet. Physical Reviewe E, 65 (6), 66130.Google Scholar
Wang, R., Goyal, R., Lei, Q., Essex, M., & De Gruttola, V. (in press). Sample size considerations in the design of cluster randomized trials of combination hiv prevention. Clinical Trials.Google Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Internet Mathematics, 393 (6684), 397–498.Google Scholar
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