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 s_i each holding independent unit-entropy information X_i wish to communicate the sum sum X_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 sum X_i. We then present an efficient encoding scheme which enables the communication of \sum X_i for the three sources, three terminals case, given that each source terminal pair is connected by em two edge disjoint paths.