Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T13:20:17.207Z Has data issue: false hasContentIssue false
  • English
  • Français

On the steady state of continuous-time stochastic opinion dynamics with power-law confidence

Part of: Miscellaneous applications of operator theory

Published online by Cambridge University Press:  16 September 2021

François Baccelli Open the ORCID record for François Baccelli [Opens in a new window] ,
Sriram Vishwanath  and
Jae Oh Woo Open the ORCID record for Jae Oh Woo [Opens in a new window]
François Baccelli*
Affiliation:
The University of Texas at Austin and INRIA Paris
Sriram Vishwanath*
Affiliation:
The University of Texas at Austin
Jae Oh Woo*
Affiliation:
The University of Texas at Austin
*
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA.
***Postal address: Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, Austin, TX 78712, USA.
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces a non-linear and continuous-time opinion dynamics model with additive noise and state-dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of the opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed-form expressions for this class of dynamics are obtained for the two-agent case. However, the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

Keywords

Opinion dynamicscontinuous-time Markov processMellin transformbounded confidencestochastic intensityadditive noisenon-local PDE

MSC classification

Primary: 47N30: Applications in probability theory and statistics
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 746 - 772
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Acemoğlu, D., Como, G., Fagnani, F. and Ozdaglar, A. (2013). Opinion fluctuations and disagreement in social networks. Math. Operat. Res. 38, 1–27.CrossRefGoogle Scholar
Acemoğlu, D., Mostagir, M. and Ozdaglar, A. (2014). State-dependent opinion dynamics. In Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP), pp. 4773–4777.10.1109/ICASSP.2014.6854508CrossRefGoogle Scholar
Acemoğlu, D. and Ozdaglar, A. (2011). Opinion dynamics and learning in social networks. Dynamic Games Appl. 1, 3–49.CrossRefGoogle Scholar
Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Baccelli, F., Chatterjee, A. and Vishwanath, S. (2015). Pairwise stochastic bounded confidence opinion dynamics: Heavy tails and stability. In Proc. IEEE Conf. Computer Communications (INFOCOM), pp. 1831–1839.10.1109/INFOCOM.2015.7218565CrossRefGoogle Scholar
Baccelli, F., Kim, K. B. and De Vleeschauwer, D. (2005). Analysis of the competition between wired, DSL and wireless users in an access network. In Proc. 24th Ann. Joint Conf. IEEE Comp. Commun. Soc., Vol. 1, pp. 362–373.CrossRefGoogle Scholar
Baccelli, F., Kim, K. B. and McDonald, D. R. (2007). Equilibria of a class of transport equations arising in congestion control. Queueing Systems 55, 1–8.CrossRefGoogle Scholar
Baccelli, F., McDonald, D. R. and Reynier, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49, 77–97.10.1016/S0166-5316(02)00136-0CrossRefGoogle Scholar
Baccelli, F. and Woo, J. O. (2016). On the entropy and mutual information of point processes. In Proc. 2016 IEEE Int. Symp. Information Theory (ISIT), pp. 695–699.CrossRefGoogle Scholar
Bhati, D., Kgosi, P. and Rattihalli, R. N. (2011). Distribution of geometrically weighted sum of Bernoulli random variables. Appl. Math. 2, 1382–1386.CrossRefGoogle Scholar
BjÖrk, T. (2011). An introduction to point processes from a martingale point of view.Google Scholar
Blondel, V. D., Hendrickx, J. M. and Tsitsiklis, J. N. (2010). Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM J. Control Optim. 48, 5214–5240.10.1137/090766188CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2012). Handbook of Brownian Motion: Facts and Formulae. BirkhÄuser, Basel.Google Scholar
Brugna, C. and Toscani, G. (2015). Kinetic models of opinion formation in the presence of personal conviction. Phys. Rev. E 92, 052818.CrossRefGoogle ScholarPubMed
Como, G. and Fagnani, F. (2011). Scaling limits for continuous opinion dynamics systems. Ann. Appl. Prob. 21, 1537–1567.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes, Vol. II. Springer, New York.Google Scholar
Deffuant, G. Neau, D., Amblard, F. and Weisbuch, G. (2000). Mixing beliefs among interacting agents. Adv. Complex Syst. 3, 87–98.CrossRefGoogle Scholar
DeGroot, M. H. (1974). Reaching a consensus. J. Am. Statist. Assoc. 69, 118–121.10.1080/01621459.1974.10480137CrossRefGoogle Scholar
Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. Appl. Prob. 34, 85–111.CrossRefGoogle Scholar
Fagnani, F. and Zampieri, S. (2008). Randomized consensus algorithms over large scale networks. IEEE J. Sel. Areas Commun. 26, 634–649.CrossRefGoogle Scholar
Ghaderi, J. and Srikant, R. (2013). Opinion dynamics in social networks: A local interaction game with stubborn agents. In Proc. IEEE American Control Conf. (ACC), pp. 1982–1987.CrossRefGoogle Scholar
Hegselmann, R. and Krause, U. (2002). Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Social Simul. 5.Google Scholar
Holley, R. A. and Liggett. T. M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Prob. 3, pp. 643–663.CrossRefGoogle Scholar
Hollot, C. V., Misra, V., Towsley, V. and Gong, W.-B. (2001). A control theoretic analysis of RED. In Proc. 20th Ann. Joint Conf. IEEE Comp. Commun. Soc., Vol. 3, pp. 1510–1519.Google Scholar
Kallenberg, O. (2006). Foundations of Modern Probability. Springer, New York.Google Scholar
Karatzas, I. and Shreve, S. (2012). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Ken-Iti, S. (1999). LÉvy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Krishnamoorthy, K. (2016). Handbook of Statistical Distributions with Applications. CRC Press, Boca Raton.CrossRefGoogle Scholar
Lobel, I., Ozdaglar, A. and Feijer, D. (2011). Distributed multi-agent optimization with state-dependent communication. Math. Program. 129, 255–284.CrossRefGoogle Scholar
Melamed, B. and Whitt, W. (1990). On arrivals that see time averages. Operat. Res. 38, 156–172.CrossRefGoogle Scholar
Melamed, B. and Whitt, W. (1990). On arrivals that see time averages: A martingale approach. J. Appl. Prob. 27, 376–384.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (2012). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Mirtabatabaei, A. and Bullo, F. (2012). Opinion dynamics in heterogeneous networks: Convergence conjectures and theorems. SIAM J. Control Optim. 50, 2763–2785.10.1137/11082751XCrossRefGoogle Scholar
Mobilia, M., Petersen, A. and Redner, S. (2007). On the role of zealotry in the voter model. J. Statist. Mech. Theory Exp. 2007, P08029.CrossRefGoogle Scholar
MÖrters, P. and Peres, Y. (2010). Brownian Motion. Cambridge University Press.Google Scholar
Olshevsky, A. and Tsitsiklis, J. N. (2009). Convergence speed in distributed consensus and averaging. SIAM J. Control Optim. 48, 33–55.CrossRefGoogle Scholar
Ramanan, K. (2006). Reflected diffusions defined via the extended Skorokhod map. Electron. J. Prob. 11, 934–992.10.1214/EJP.v11-360CrossRefGoogle Scholar
Toscani, G. Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496.CrossRefGoogle Scholar
Wolff, R. W. (1982). Poisson arrivals see time averages. Operat. Res. 30, 223–231.CrossRefGoogle Scholar
Yen, J.-Y. and Yor, M. (2014). Local Times and Excursion Theory for Brownian Motion. Springer, New York.Google Scholar
Yildiz, E., Ozdaglar, A., Acemoglu, D., Saberi, A. and Scaglione, A. (2013). Binary opinion dynamics with stubborn agents. ACM Trans. Econom. Comput. 1, 19.Google Scholar