Two-Unicast Wireless Networks: Characterizing the Degrees-of-Freedom

TL;DR

This paper characterizes the degrees-of-freedom for two-unicast layered wireless networks, identifying conditions for achievable sum degrees-of-freedom and proposing strategies based on network condensation and interference management.

Contribution

It provides a complete characterization of the sum degrees-of-freedom in layered two-unicast wireless networks, including necessary and sufficient conditions and a novel network condensation approach.

Findings

Sum degrees-of-freedom are 1, 3/2, or 2 with probability 1.

A new network condensation technique simplifies analysis.

Achievability and converse proofs establish the full degrees-of-freedom region.

Abstract

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.