We consider two-source two-destination (i.e., two-unicast) multihop 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.
Later, in a recent independent work REF such a scheme is employed and the authors characterize the degrees-of-freedom region of two unicast Gaussian networks almost surely.
Two-Unicast Wireless Networks: Characterizing the Degrees of Freedom
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Mathematics Computer Science