Asymptotic properties of distributed social sampling algorithm

Abstract

Social sampling is a novel randomized message passing protocol inspired by social communication for opinion formation in social networks. In a typical social sampling algorithm, each agent holds a sample from the empirical distribution of social opinions at initial time, and it collaborates with other agents in a distributed manner to estimate the initial empirical distribution by randomly sampling a message from current distribution estimate. In this paper, we focus on analyzing the theoretical properties of the distributed social sampling algorithm over random networks. First, we provide a framework based on stochastic approximation to study the asymptotic properties of the algorithm. Then, under mild conditions, we prove that the estimates of all agents converge to a common random distribution, which is composed of the initial empirical distribution and the accumulation of quantized error. Besides, by tuning algorithm parameters, we prove the strong consistency, namely, the distribution estimates of agents almost surely converge to the initial empirical distribution. Furthermore, the asymptotic normality of estimation error generated by distributed social sample algorithm is addressed. Finally, we provide a numerical simulation to validate the theoretical results of this paper.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Anderson B D O, Ye M. Recent advances in the modelling and analysis of opinion dynamics on influence networks. Int J Autom Comput, 2019, 16: 129–149

    Article  Google Scholar 

  2. 2

    Flache A, Mäs M, Feliciani T, et al. Models of social influence: towards the next frontiers. J Artif Soc Social Simulat, 2017, 20: 2

    Article  Google Scholar 

  3. 3

    Proskurnikov A V, Tempo R. A tutorial on modeling and analysis of dynamic social networks. Part I. Annu Rev Control, 2017, 43: 65–79

    Article  Google Scholar 

  4. 4

    Proskurnikov A V, Tempo R. A tutorial on modeling and analysis of dynamic social networks. Part II. Annu Rev Control, 2018, 45: 166–190

    MathSciNet  Article  Google Scholar 

  5. 5

    Holley R A, Liggett T M. Ergodic theorems for weakly interacting infinite systems and the voter model. Ann Probab, 1975, 3: 643–663

    MathSciNet  Article  Google Scholar 

  6. 6

    Acemoglu D, Dahleh M A, Lobel I, et al. Bayesian learning in social networks. Rev Economic Studies, 2011, 78: 1201–1236

    MathSciNet  Article  Google Scholar 

  7. 7

    Narayanan H, Niyogi P. Language evolution, coalescent processes, and the consensus problem on a social network. J Math Psychol, 2014, 61: 19–24

    MathSciNet  Article  Google Scholar 

  8. 8

    Xiao Y P, Li X X, Liu Y N, et al. Correlations multiplexing for link prediction in multidimensional network spaces. Sci China Inf Sci, 2018, 61: 112103

    Article  Google Scholar 

  9. 9

    Friedkin N E, Proskurnikov A V, Tempo R, et al. Network science on belief system dynamics under logic constraints. Science, 2016, 354: 321–326

    MathSciNet  Article  Google Scholar 

  10. 10

    Hegselmann R, Krause U. Opinion dynamics and bounded confidence models, analysis, and simulation. J Artif Soc Social Simulat, 2002, 5: 1–33

    Google Scholar 

  11. 11

    Zhang J, Hong Y. Opinion evolution analysis for short-range and long-range Deffuant-Weisbuch models. Physica A-Stat Mech Appl, 2013, 392: 5289–5297

    MathSciNet  Article  Google Scholar 

  12. 12

    Pineda M, Toral R, Hernández-García E. Noisy continuous-opinion dynamics. J Stat Mech, 2009, 2009: P08001

    Article  Google Scholar 

  13. 13

    Boyd S, Ghosh A, Prabhakar B, et al. Randomized gossip algorithms. IEEE Trans Inform Theor, 2006, 52: 2508–2530

    MathSciNet  Article  Google Scholar 

  14. 14

    Lou Y C, Strub M, Li D, et al. Reference point formation in social networks, wealth growth, and inequality. SSRN J, 2017. doi: https://doi.org/10.2139/ssrn.3013124

  15. 15

    Frasca P, Ishii H, Ravazzi C, et al. Distributed randomized algorithms for opinion formation, centrality computation and power systems estimation: a tutorial overview. Eur J Control, 2015, 24: 2–13

    MathSciNet  Article  Google Scholar 

  16. 16

    Friedkin N E, Johnsen E C. Social influence networks and opinion change. Adv Group Process, 1999, 16: 1–29

    Google Scholar 

  17. 17

    Ravazzi C, Frasca P, Tempo R, et al. Ergodic randomized algorithms and dynamics over networks. IEEE Trans Control Netw Syst, 2015, 2: 78–87

    MathSciNet  Article  Google Scholar 

  18. 18

    Acemoglu D, Bimpikis K, Ozdaglar A. Dynamics of information exchange in endogenous social networks. Theor Economics, 2014, 9: 41–97

    MathSciNet  Article  Google Scholar 

  19. 19

    Acemoğlu D, Como G, Fagnani F, et al. Opinion fluctuations and disagreement in social networks. Math Ope Res, 2013, 38: 1–27

    MathSciNet  Article  Google Scholar 

  20. 20

    Ceragioli F, Frasca P. Consensus and disagreement: the role of quantized behaviors in opinion dynamics. SIAM J Control Opt, 2018, 56: 1058–1080

    MathSciNet  Article  Google Scholar 

  21. 21

    Sarwate A D, Javidi T. Distributed learning of distributions via social sampling. IEEE Trans Automat Contr, 2015, 60: 34–45

    MathSciNet  Article  Google Scholar 

  22. 22

    Degroot M H. Reaching a consensus. J Am Stat Assoc, 1974, 69: 118–121

    Article  Google Scholar 

  23. 23

    Borkar V, Varaiya P P. Asymptotic agreement in distributed estimation. IEEE Trans Automat Contr, 1982, 27: 650–655

    MathSciNet  Article  Google Scholar 

  24. 24

    Tsitsiklis J N, Athans M. Convergence and asymptotic agreement in distributed decision problems. IEEE Trans Automat Contr, 1984, 29: 42–50

    Article  Google Scholar 

  25. 25

    Olfati-Saber R, Murray R M. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Automat Contr, 2004, 49: 1520–1533

    MathSciNet  Article  Google Scholar 

  26. 26

    Kar S, Moura J M F. Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise. IEEE Trans Signal Process, 2009, 57: 355–369

    MathSciNet  Article  Google Scholar 

  27. 27

    Huang M, Manton J H. Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior. SIAM J Control Opt, 2009, 48: 134–161

    MathSciNet  Article  Google Scholar 

  28. 28

    Fang H, Chen H F, Wen L. On control of strong consensus for networked agents with noisy observations. J Syst Sci Complex, 2012, 25: 1–12

    MathSciNet  Article  Google Scholar 

  29. 29

    Leblanc H J, Zhang H, Koutsoukos X, et al. Resilient asymptotic consensus in robust networks. IEEE J Sel Areas Commun, 2013, 31: 766–781

    Article  Google Scholar 

  30. 30

    Zong X F, Li T, Zhang J F. Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises. Automatica, 2019, 99: 412–419

    MathSciNet  Article  Google Scholar 

  31. 31

    Zong X F, Li T, Zhang J F. Consensus conditions of continuous-time multi-agent systems with additive and multiplicative measurement noises. SIAM J Control Opt, 2018, 56: 19–52

    MathSciNet  Article  Google Scholar 

  32. 32

    Wang Y H, Lin P, Hong Y G. Distributed regression estimation with incomplete data in multi-agent networks. Sci China Inf Sci, 2018, 61: 092202

    MathSciNet  Article  Google Scholar 

  33. 33

    Rajagopal R, Wainwright M J. Network-based consensus averaging with general noisy channels. IEEE Trans Signal Process, 2011, 59: 373–385

    MathSciNet  Article  Google Scholar 

  34. 34

    Meyer C D. Matrix Analysis and Applied Linear Algebra. Philadelphia: SIAM, 2000

    Google Scholar 

  35. 35

    Durrett R. Probability Theory and Examples. Camberidge: Camberidge Press, 2010

    Google Scholar 

  36. 36

    Chen H F. Stochastic Approximation and Its Applications. New York: Kluwer Academic Publishers, 2003

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0901900) and National Natural Science Foundation of China (Grant No. 61573345).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xingkang He.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, Q., He, X. & Fang, H. Asymptotic properties of distributed social sampling algorithm. Sci. China Inf. Sci. 63, 112202 (2020). https://doi.org/10.1007/s11432-019-9890-5

Download citation

Keywords

  • social networks
  • opinion formation
  • social sampling
  • stochastic approximation
  • random networks
  • asymptotic normality