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

In this paper we explore the features of a graph generated by random walkers with nodes that have evolutionary attractiveness and Boltzmann-like transition probabilities that depend both on the euclidean distance between the nodes and on the ratio ($\beta $) of the attractiveness between them. We show that persistent nodes, i.e., nodes that never been reached by random walker in asymptotic times are possible in the stationary case differently from the case where the attractiveness is fixed and equal to one for all nodes ($\beta =1$). Simultaneously, we also investigate the spectral properties and statistics related to the attractiveness and degree distribution of the evolutionary network. Finally, we study a crossover between persistent phase and no persistent phase and we also show the existence of a special type of transition probability that leads to a power law behaviour for the time evolution of the persistence.