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Mixed HegselmannKrause dynamics
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA 
The original HegselmannKrause (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 openminded 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 openminded 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 openminded. We first show that some of the common properties of the synchronous HK model, such as finitetime 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.
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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: PerronFrobenius and FaberKrahn Type Theorems, , SpringerVerlag, Berlin Heidelberg, 2007. doi: 10.1007/9783540735106. Google Scholar 
[3] 
S. R. Etesami and T. Başar, Gametheoretic analysis of the HegselmannKrause model for Opinion dynamics in finite dimensions, IEEE Transactions on Automatic Control, 60 (2015), 18861897. doi: 10.1109/TAC.2015.2394954. Google Scholar 
[4] 
W. Han, C. Huang and J. Yang, Opinion clusters in a modified HegselmannKrause 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 
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