Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-08T11:26:35.775Z 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, Illinois 60637, USA E-mail: rdr@chicagobooth.edu
Sheldon M. Ross
Affiliation:
Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California 90089, USA E-mail: smross@usc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

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

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