  We study the independence structure of finitely exchangeable distributions
over random vectors and random networks. In particular, we provide necessary
and sufficient conditions for an exchangeable vector so that its elements are
completely independent or completely dependent. We also provide a sufficient
condition for an exchangeable vector so that its elements are marginally
independent. We then generalize these results and conditions for exchangeable
random networks. In this case, it is demonstrated that the situation is more
complex. We show that the independence structure of exchangeable random
networks lies in one of six regimes that are two-fold dual to one another,
represented by undirected and bidirected independence graphs in graphical model
sense with graphs that are complement of each other. In addition, under certain
additional assumptions, we provide necessary and sufficient conditions for the
exchangeable network distributions to be faithful to each of these graphs.
