# American Institute of Mathematical Sciences

## Mixed Hegselmann-Krause dynamics

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

Received  October 2020 Revised  December 2020 Published  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021084
##### References:
 [1] L. W. Beineke, P. J. Cameron and R. J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, Cambridge, 2013.   Google Scholar

show all references

##### References:
 [1] L. W. Beineke, P. J. Cameron and R. J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, Cambridge, 2013.   Google Scholar
 [1] Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004 [2] Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021035 [3] Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015 [4] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [5] Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 [6] Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021049 [7] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [8] Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021027 [9] Kun Hu, Yuanshi Wang. Dynamics of consumer-resource systems with consumer's dispersal between patches. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021077 [10] Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3357-3380. doi: 10.3934/dcdsb.2020235 [11] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [12] Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 [13] Yu Jin, Xiao-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1991-2010. doi: 10.3934/dcdsb.2020362 [14] Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623 [15] Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021064 [16] Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier. Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks & Heterogeneous Media, 2021, 16 (2) : 257-281. doi: 10.3934/nhm.2021006 [17] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [18] Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021054 [19] Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 [20] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

2019 Impact Factor: 1.27