  We consider two-source two-destination (i.e., two-unicast) multi-hop wireless
networks that have a layered structure with arbitrary connectivity. We show
that, if the channel gains are chosen independently according to continuous
distributions, then, with probability 1, two-unicast layered Gaussian networks
can only have 1, 3/2 or 2 sum degrees-of-freedom (unless both
source-destination pairs are disconnected, in which case no degrees-of-freedom
can be achieved). We provide sufficient and necessary conditions for each case
based on network connectivity and a new notion of source-destination paths with
manageable interference. Our achievability scheme is based on forwarding the
received signals at all nodes, except for a small fraction of them in at most
two key layers. Hence, we effectively create a "condensed network" that has at
most four layers (including the sources layer and the destinations layer). We
design the transmission strategies based on the structure of this condensed
network. The converse results are obtained by developing information-theoretic
inequalities that capture the structures of the network connectivity. Finally,
we extend this result and characterize the full degrees-of-freedom region of
two-unicast layered wireless networks.
