Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T20:48:10.880Z Has data issue: false hasContentIssue false
  • English
  • Français

A PROBABILISTIC FRIENDSHIP NETWORK MODEL

Published online by Cambridge University Press:  03 June 2020

Rebecca Dizon-Ross  and
Sheldon M. Ross Open the ORCID record for Sheldon M. Ross [Opens in a new window]
Rebecca Dizon-Ross
Affiliation:
Booth School of Business, University of Chicago, Chicago, Illinois60637, USA E-mail: rdr@chicagobooth.edu
Sheldon M. Ross
Affiliation:
Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California90089, USA E-mail: smross@usc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a friendship model in which each member of a community has a latent value such that the probability that any two individuals are friends is a function of their latent values. We consider such questions as does information that i and j are both friends with k make it more likely that i and j are themselves friends. Among other things, we show that for fixed sets S and T, the more friends that i has in S, then the stochastically more friends i has in T. We consider how a variation of the friendship paradox applies to our model. We also study the distribution of the number of friendless individuals in the community and derive a bound on the total variation distance between it and a Poisson with the same mean.

Keywords

associated random variablesfriendless individualsfriendship modelfriendship paradoxstochastic networkPoisson approximation
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 1 , January 2022 , pp. 1 - 16
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbe, E. (2018). Community detection and stochastic block models: Recent developments. Journal of Machine Learning Research 18(1): 1–86.Google Scholar
Barbour, A.D., Holst, L., & Janson, S. (1992). Poisson approximations, vol. 1. Oxford: The Clarendon Press.Google Scholar
Barlow, R.E., & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring: To Begin With.Google Scholar
Berenhaut, K., & Jiang, H. (2019). The friendship paradox for weighted and directed networks. Probability in the Engineering and Informational Sciences 33(1): 136–145.CrossRefGoogle Scholar
Cao, Y., & Ross, S.M. (2016). The friendship paradox. The Mathematical Scientist 41(1): 61–64.Google Scholar
Daudin, J.J., Picard, F., & Robin, S. (2008). A mixture model for random graphs. Statistics and Computing 18: 173–183.CrossRefGoogle Scholar
Erdos, P., & Renyi, A. (1959). On random graphs. Publicationes Mathematicae 6: 290–297.Google Scholar
Farago, A. (2010). On the structure of classes of random graphs. Chicago Journal of Theoretical Computer Science: 1–15.10.4086/cjtcs.2010.005CrossRefGoogle Scholar
Feld, S.L. (1991). Why your friends have more friends than you do. American Journal of Sociology 96(6): 1464–1477.CrossRefGoogle Scholar
Hoff, P., Raftery, A., & Handcock, M. (2002). Latent space approaches to social network analysis. Journal of the American Statistical Association 97: 1090–1098.CrossRefGoogle Scholar
Jackson, M.O. (2008). Social and economic networks. Princeton NJ: Princeton University Press.CrossRefGoogle Scholar
Jackson, M.O., & Rogers, B. W. (2017). Meeting strangers and friends of friends: How random are social networks. American Economic Review 97(3): 890–915.CrossRefGoogle Scholar
Lovasz, L., & Szegedy, B. (2006). Limits of dense graph sequences. Journal of Combinatorial Theory, Series B 96(6): 933–957.CrossRefGoogle Scholar
Nowicki, K., & Snijders, T.A.B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association 96(455): 1077–1087.CrossRefGoogle Scholar
Ortega, E.M., Alonso, J., & Ortega, I. (2013). Stochastic comparisons of mixtures of parametric families in stochastic epidemics. Mathematical Biosciences 243(1): 18–27.CrossRefGoogle ScholarPubMed
Rastelli, R., Friel, N., & Raftery, A. (2016). Properties of latent variable network models. Network Science 4(4): 407–432.CrossRefGoogle Scholar
Robins, G., & Pattison, P. (2001). Random graphs models for temporal processes in social networks. The Journal of Mathematical Sociology 25: 5–41.CrossRefGoogle Scholar
Robins, G., Pattison, P., Kalish, Y., & Lusher, D. (2006). An introduction to exponential random praph (p*) models for social networks. Social Networks 29(2): 173–191.CrossRefGoogle Scholar
Ross, S.M. (2016). Improved Chen-Stein bounds on the probability of a union. Journal of Applied Probability 53(4): 1265–1270.CrossRefGoogle Scholar
Snijders, T.A.B. (2011). Statistical models for social networks. Annual Review of Sociology 37: 131–153.10.1146/annurev.soc.012809.102709CrossRefGoogle Scholar
Van Der Hofstad, R. (2013). Random graphs and complex networks. Cambridge: Cambridge University Press.Google Scholar