  Algorithms and dynamics over networks often involve randomization, and
randomization may result in oscillating dynamics which fail to converge in a
deterministic sense. In this paper, we observe this undesired feature in three
applications, in which the dynamics is the randomized asynchronous counterpart
of a well-behaved synchronous one. These three applications are network
localization, PageRank computation, and opinion dynamics. Motivated by their
formal similarity, we show the following general fact, under the assumptions of
independence across time and linearities of the updates: if the expected
dynamics is stable and converges to the same limit of the original synchronous
dynamics, then the oscillations are ergodic and the desired limit can be
locally recovered via time-averaging.
