  In Part II [3] we carried out a detailed mean-square-error analysis of the
performance of asynchronous adaptation and learning over networks under a
fairly general model for asynchronous events including random topologies,
random link failures, random data arrival times, and agents turning on and off
randomly. In this Part III, we compare the performance of synchronous and
asynchronous networks. We also compare the performance of decentralized
adaptation against centralized stochastic-gradient (batch) solutions. Two
interesting conclusions stand out. First, the results establish that the
performance of adaptive networks is largely immune to the effect of
asynchronous events: the mean and mean-square convergence rates and the
asymptotic bias values are not degraded relative to synchronous or centralized
implementations. Only the steady-state mean-square-deviation suffers a
degradation in the order of $\nu$, which represents the small step-size
parameters used for adaptation. Second, the results show that the adaptive
distributed network matches the performance of the centralized solution. These
conclusions highlight another critical benefit of cooperation by networked
agents: cooperation does not only enhance performance in comparison to
stand-alone single-agent processing, but it also endows the network with
remarkable resilience to various forms of random failure events and is able to
deliver performance that is as powerful as batch solutions.
