  Byzantine consensus is a classical problem in distributed computing. Each
node in a synchronous system starts with a binary input. The goal is to reach
agreement in the presence of Byzantine faulty nodes. We consider the setting
where communication between nodes is modelled via a directed hypergraph. In the
classical point-to-point communication model, the communication between nodes
is modelled as a simple graph where all messages sent on an edge are private
between the two endpoints of the edge. This allows a faulty node to equivocate,
i.e., lie differently to its different neighbors. Different models have been
proposed in the literature that weaken equivocation. In the local broadcast
model, every message transmitted by a node is received identically and
correctly by all of its neighbors. In the hypergraph model, every message
transmitted by a node on a hyperedge is received identically and correctly by
all nodes on the hyperedge. Tight network conditions are known for each of the
three cases for undirected (hyper)graphs. For the directed models, tight
conditions are known for the point-to-point and local broadcast models.
  In this work, we consider the directed hypergraph model that encompasses all
the models above. Each directed hyperedge consists of a single head (sender)
and at least one tail (receiver), This models a local multicast channel where
messages transmitted by the sender are received identically by all the
receivers in the hyperedge. For this model, we identify tight network
conditions for consensus. We observe how the directed hypergraph model reduces
to each of the three models above under specific conditions. In each case, we
relate our network condition to the corresponding known tight conditions. The
directed hypergraph model also encompasses other practical network models of
interest that have not been explored previously, as elaborated in the paper.
