Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom.
Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom.
Phys Rev E. 2017 Sep;96(3-1):032313. doi: 10.1103/PhysRevE.96.032313. Epub 2017 Sep 25.
We study the dynamics of opinion formation in a heterogeneous voter model on a complete graph, in which each agent is endowed with an integer fitness parameter k≥0, in addition to its + or - opinion state. The evolution of the distribution of k-values and the opinion dynamics are coupled together, so as to allow the system to dynamically develop heterogeneity and memory in a simple way. When two agents with different opinions interact, their k-values are compared, and with probability p the agent with the lower value adopts the opinion of the one with the higher value, while with probability 1-p the opposite happens. The agent that keeps its opinion (winning agent) increments its k-value by one. We study the dynamics of the system in the entire 0≤p≤1 range and compare with the case p=1/2, in which opinions are decoupled from the k-values and the dynamics is equivalent to that of the standard voter model. When 0≤p<1/2, agents with higher k-values are less persuasive, and the system approaches exponentially fast to the consensus state of the initial majority opinion. The mean consensus time τ appears to grow logarithmically with the number of agents N, and it is greatly decreased relative to the linear behavior τ∼N found in the standard voter model. When 1/2<p≤1, agents with higher k-values are more persuasive, and the system initially relaxes to a state with an even coexistence of opinions, but eventually reaches consensus by finite-size fluctuations. The approach to the coexistence state is monotonic for 1/2<p<p_{o}≃0.8, while for p_{o}≤p≤1 there are damped oscillations around the coexistence value. The final approach to coexistence is approximately a power law t^{-b(p)} in both regimes, where the exponent b increases with p. Also, τ increases respect to the standard voter model, although it still scales linearly with N. The p=1 case is special, with a relaxation to coexistence that scales as t^{-2.73} and a consensus time that scales as τ∼N^{β}, with β≃1.45.
我们研究了完全图上异质投票者模型中意见形成的动力学，其中每个个体除了具有+或-意见状态外，还具有一个整数适应值参数 k≥0。k 值分布的演化和意见动态是耦合在一起的，因此可以以简单的方式使系统动态地发展出异质性和记忆。当具有不同意见的两个个体相互作用时，比较它们的 k 值，并且以概率 p 具有较低值的个体采用具有较高值的个体的意见，而以概率 1-p 则相反。保持意见（获胜个体）的个体将其 k 值增加 1。我们在整个 0≤p≤1 范围内研究系统的动力学，并与 p=1/2 的情况进行比较，在这种情况下，意见与 k 值解耦，动力学与标准投票者模型等效。当 0≤p<1/2 时，具有较高 k 值的个体的说服力较小，系统以指数速度快速接近初始多数意见的共识状态。共识时间 τ似乎与个体数量 N 呈对数增长，并且相对于标准投票者模型中发现的线性行为 τ∼N 大大减少。当 1/2<p≤1 时，具有较高 k 值的个体的说服力较大，系统最初松弛到意见共存的状态，但最终通过有限大小的波动达到共识。对于 1/2<p<p_{o}≃0.8，接近共存状态的过程是单调的，而对于 p_{o}≤p≤1，围绕共存值存在阻尼振荡。在这两种情况下，最终接近共存的过程大约是 t^{-b(p)}的幂律，其中指数 b 随 p 增加。此外，τ相对于标准投票者模型增加，尽管它仍然与 N 线性缩放。p=1 的情况很特殊，它具有与 t^{-2.73}的标度关系的松弛到共存的过程和与 τ∼N^{β}的标度关系的共识时间，其中β≃1.45。
