Spatial patterns of random walkers under evolution of the attractiveness: persistent nodes, degree distribution, and spectral properties

摘要

In this paper we explore the features of a graph generated by random walkerswith nodes that have evolutionary attractiveness and Boltzmann-like transitionprobabilities that depend both on the euclidean distance between the nodes andon the ratio ($\beta $) of the attractiveness between them. We show thatpersistent nodes, i.e., nodes that never been reached by random walker inasymptotic times are possible in the stationary case differently from the casewhere the attractiveness is fixed and equal to one for all nodes ($\beta =1$).Simultaneously, we also investigate the spectral properties and statisticsrelated to the attractiveness and degree distribution of the evolutionarynetwork. Finally, we study a crossover between persistent phase and nopersistent phase and we also show the existence of a special type of transitionprobability that leads to a power law behaviour for the time evolution of thepersistence.