  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.
