We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, which are inspired by the DeGroot–Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper, we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the $\ell _1$-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear—both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

中文翻译：

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意见动力学中随机模型的稳定性理论

我们考虑了一类保留概率单纯形的非线性映射，即随机映射，其灵感来自影响网络上的信念/意见传播的 DeGroot-Friedkin 模型。相应的动力学模型描述了相互作用物种的概率分布的演变。这种概率转移机制非线性地依赖于当前状态的模型通常被称为非线性马尔可夫链。在本文中，我们开发了稳定性结果并研究了代表性意见模型的行为。稳定性证书基于 $\ell_1$-metric 中非线性演化的收缩性。我们将该理论应用于两种类型的意见模型，其中转移概率对当前状态的适应是指数和线性的——这两种模型都可以表现出广泛的行为。我们讨论连续时间和其他概括。