Communicating the Sum of Sources Over a Network

We consider the network communication scenario, over directed acyclic networks with unit capacity edges in which a number of sources si each holding independent unit-entropy information X
_i
wish to communicate the sum ΣX
_i
to a set of terminals t
_j
. We show that in the case in which there are only two sources or only two terminals, communication is possible if and only if each source terminal pair s
_i
/t
_j
is connected by at least a single path. For the more general communication problem in which there are three sources and three terminals, we prove that a single path connecting the source terminal pairs does not suffice to communicate ΣX
_i
. We then present an efficient encoding scheme which enables the communication of ΣX
_i
for the three sources, three terminals case, given that each source terminal pair is connected by two edge disjoint paths.