  The deterministic wireless relay network model, introduced by Avestimehr et
al., has been proposed for approximating Gaussian relay networks. This model,
known as the ADT network model, takes into account the broadcast nature of
wireless medium and interference. Avestimehr et al. showed that the Min-cut
Max-flow theorem holds in the ADT network.
  In this paper, we show that the ADT network model can be described within the
algebraic network coding framework introduced by Koetter and Medard. We prove
that the ADT network problem can be captured by a single matrix, called the
"system matrix". We show that the min-cut of an ADT network is the rank of the
system matrix; thus, eliminating the need to optimize over exponential number
of cuts between two nodes to compute the min-cut of an ADT network.
  We extend the capacity characterization for ADT networks to a more general
set of connections. Our algebraic approach not only provides the Min-cut
Max-flow theorem for a single unicast/multicast connection, but also extends to
non-multicast connections such as multiple multicast, disjoint multicast, and
two-level multicast. We also provide sufficiency conditions for achievability
in ADT networks for any general connection set. In addition, we show that the
random linear network coding, a randomized distributed algorithm for network
code construction, achieves capacity for the connections listed above.
  Finally, we extend the ADT networks to those with random erasures and cycles
(thus, allowing bi-directional links). Note that ADT network was proposed for
approximating the wireless networks; however, ADT network is acyclic.
Furthermore, ADT network does not model the stochastic nature of the wireless
links. With our algebraic framework, we incorporate both cycles as well as
random failures into ADT network model.
