February  2022, 27(2): 1149-1162. doi: 10.3934/dcdsb.2021084

Mixed Hegselmann-Krause dynamics

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

Received  October 2020 Revised  December 2020 Published  February 2022 Early access  March 2021

The original Hegselmann-Krause (HK) model consists of a set of $ n $ agents that are characterized by their opinion, a number in $ [0, 1] $. Each agent, say agent $ i $, updates its opinion $ x_i $ by taking the average opinion of all its neighbors, the agents whose opinion differs from $ x_i $ by at most $ \epsilon $. There are two types of HK models: the synchronous HK model and the asynchronous HK model. For the synchronous model, all the agents update their opinion simultaneously at each time step, whereas for the asynchronous HK model, only one agent chosen uniformly at random updates its opinion at each time step. This paper is concerned with a variant of the HK opinion dynamics, called the mixed HK model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at each update. The degree of the stubbornness of agents can be different and/or vary over time. An agent is not stubborn or absolutely open-minded if its new opinion at each update is the average opinion of its neighbors, and absolutely stubborn if its opinion does not change at the time of the update. The particular case where, at each time step, all the agents are absolutely open-minded is the synchronous HK model. In contrast, the asynchronous model corresponds to the particular case where, at each time step, all the agents are absolutely stubborn except for one agent chosen uniformly at random who is absolutely open-minded. We first show that some of the common properties of the synchronous HK model, such as finite-time convergence, do not hold for the mixed model. We then investigate conditions under which the asymptotic stability holds, or a consensus can be achieved for the mixed model.

Citation: Hsin-Lun Li. Mixed Hegselmann-Krause dynamics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1149-1162. doi: 10.3934/dcdsb.2021084
References:
[1] L. W. BeinekeP. J. Cameron and R. J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 2004. 
[2]

T. Biyikoglu, J. Leydold and P. F. Stadler, Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems, , Springer-Verlag, Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-73510-6.

[3]

S. R. Etesami and T. Başar, Game-theoretic analysis of the Hegselmann-Krause model for Opinion dynamics in finite dimensions, IEEE Transactions on Automatic Control, 60 (2015), 1886-1897.  doi: 10.1109/TAC.2015.2394954.

[4]

W. Han, C. Huang and J. Yang, Opinion clusters in a modified Hegselmann-Krause model with heterogeneous bounded confidences and stubbornness, Physica A: Statistical Mechanics and its Applications, 531 (2019), Article 121791. doi: 10.1016/j.physa.2019.121791.

[5] R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, Cambridge, 2013. 

show all references

References:
[1] L. W. BeinekeP. J. Cameron and R. J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 2004. 
[2]

T. Biyikoglu, J. Leydold and P. F. Stadler, Laplacian Eigenvectors of Graphs: Perron-Frobenius and Faber-Krahn Type Theorems, , Springer-Verlag, Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-73510-6.

[3]

S. R. Etesami and T. Başar, Game-theoretic analysis of the Hegselmann-Krause model for Opinion dynamics in finite dimensions, IEEE Transactions on Automatic Control, 60 (2015), 1886-1897.  doi: 10.1109/TAC.2015.2394954.

[4]

W. Han, C. Huang and J. Yang, Opinion clusters in a modified Hegselmann-Krause model with heterogeneous bounded confidences and stubbornness, Physica A: Statistical Mechanics and its Applications, 531 (2019), Article 121791. doi: 10.1016/j.physa.2019.121791.

[5] R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, Cambridge, 2013. 
[1]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003

[2]

Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015

[3]

Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457

[4]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009

[5]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control and Related Fields, 2022, 12 (1) : 115-146. doi: 10.3934/mcrf.2021004

[6]

Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235

[7]

Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007

[8]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016

[9]

Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267

[10]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19

[11]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545

[12]

Marina Dolfin, Mirosław Lachowicz. Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10 (4) : 877-896. doi: 10.3934/nhm.2015.10.877

[13]

GuanLin Li, Sebastien Motsch, Dylan Weber. Bounded confidence dynamics and graph control: Enforcing consensus. Networks and Heterogeneous Media, 2020, 15 (3) : 489-517. doi: 10.3934/nhm.2020028

[14]

Mohammad A. Safi, Abba B. Gumel. Global asymptotic dynamics of a model for quarantine and isolation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 209-231. doi: 10.3934/dcdsb.2010.14.209

[15]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206

[16]

Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116

[17]

Pengyan Wang, Wenxiong Chen. Hopf's lemmas for parabolic fractional p-Laplacians. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022089

[18]

Sun-Ho Choi, Hyowon Seo. Rumor spreading dynamics with an online reservoir and its asymptotic stability. Networks and Heterogeneous Media, 2021, 16 (4) : 535-552. doi: 10.3934/nhm.2021016

[19]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

[20]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (265)
  • HTML views (400)
  • Cited by (0)

Other articles
by authors

[Back to Top]