  A classic and fundamental result about the decomposition of random sequences
into a mixture of simpler ones is de Finetti's Theorem. In its original form it
applies to infinite 0-1 valued exchangeable sequences. Later it was extended
and generalized in numerous directions. After reviewing this line of
development, we present our new decomposition theorem, covering cases that have
not been previously considered. We also introduce a novel way of applying these
types of results in the analysis of random networks. For self-containment, we
provide the introductory exposition in more details than usual, with the intent
of making it also accessible to readers who may not be closely familiar with
the subject.
