  The scale-free (SF) networks that have been studied so far contained quenched
disorder generated by random dilution which does not vary with the time. In
practice, if a SF network is to represent, for example, the worldwide web, then
the links between its various nodes may temporarily be lost, and re-established
again later on. This gives rise to SF networks with annealed disorder. Even if
the disorder is quenched, it may be more realistic to generate it by a
dynamical process that is happening in the network. In this paper, we study
diffusion in SF networks with annealed disorder generated by various scenarios,
as well as in SF networks with quenched disorder which, however, is generated
by the diffusion process itself. Several quantities of the diffusion process
are computed, including the mean number of distinct sites visited, the mean
number of returns to the origin, and the mean number of connected nodes that
are accessible to the random walkers at any given time. The results including,
(1) greatly reduced growth with the time of the mean number of distinct sites
visited; (2) blocking of the random walkers; (3) the existence of a phase
diagram that separates the region in which diffusion is possible from one in
which diffusion is impossible, and (4) a transition in the structure of the
networks at which the mean number of distinct sites visited vanishes, indicate
completely different behavior for the computed quantities than those in SF
networks with quenched disorder generated by simple random dilution.
