  Smoothed analysis is a framework suggested for mediating gaps between
worst-case and average-case complexities. In a recent work, Dinitz et
al.~[Distributed Computing, 2018] suggested to use smoothed analysis in order
to study dynamic networks. Their aim was to explain the gaps between real-world
networks that function well despite being dynamic, and the strong theoretical
lower bounds for arbitrary networks. To this end, they introduced a basic model
of smoothing in dynamic networks, where an adversary picks a sequence of
graphs, representing the topology of the network over time, and then each of
these graphs is slightly perturbed in a random manner. The model suggested
above is based on a per-round noise, and our aim in this work is to extend it
to models of noise more suited for multiple rounds. This is motivated by
long-lived networks, where the amount and location of noise may vary over time.
To this end, we present several different models of noise. First, we extend the
previous model to cases where the amount of noise is very small. Then, we move
to more refined models, where the amount of noise can change between different
rounds, e.g., as a function of the number of changes the network undergoes. We
also study a model where the noise is not arbitrarily spread among the network,
but focuses in each round in the areas where changes have occurred. Finally, we
study the power of an adaptive adversary, who can choose its actions in
accordance with the changes that have occurred so far. We use the flooding
problem as a running case-study, presenting very different behaviors under the
different models of noise, and analyze the flooding time in different models.
