论文标题
顶点加强随机步行，并在完整图上进行指数互动
Vertex reinforced random walks with exponential interaction on complete graphs
论文作者
论文摘要
我们描述了一个$ M $顶点增强的模型，并在完整的图表上使用$ d \ geq 2 $顶点。随机步行到给定顶点的过渡概率指数取决于所有步行到该顶点的访问的比例。访问的单个比例是由可以将其设置为等于任何实际数字的强度参数调节的。该模型涵盖了各种各样的相互作用，包括任何两个随机步行以及自我增强的相互作用之间的不同顶点排斥和吸引力强度。我们表明，通过相互作用的随机步道收敛（A.S。）定义的经验顶点职业的过程量表到光滑矢量场诱导的流量的极限集。此外，如果场的平衡集由孤立的点形成，则顶点占用度量将（A.S.）汇聚到田间的平衡。这些事实是通过构建严格的Lyapunov函数来显示的。我们表明，如果相互作用强度参数的绝对值小于某个上限，那么，对于任何图（$ d \ geq 2 $）的任何随机步行（$ M \ geq 2 $），则顶点占用度量将收敛于独特的平衡。我们提供了两个其他示例的驱动随机步行，$ m = d = 2 $和$ m = 3 $，$ d = 2 $。后者用于研究$ \ mathbb {z} $的三个指数排斥随机步行的一些属性。
We describe a model for $m$ vertex reinforced interacting random walks on complete graphs with $d\geq 2$ vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks ($m\geq 2$) on any graph ($d \geq 2$), the vertex occupation measure converges toward a unique equilibrium. We provide two additional examples of repelling random walks for the cases $m=d=2$ and $m=3$, $d=2$. The latter is used to study some properties of three exponentially repelling random walks on $\mathbb{Z}$.
