Computation in Multicast Networks: Function Alignment and Converse Theorems

The classical problem in a network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers that decode the same set of messages. In this paper, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of two-transmitter two-receiver single-hop linear deterministic networks, the computation capacity is characterized for a linear function (modulo-2 sum) of Bernoulli sources. A new upper bound is derived that is tighter than cut-set-based and genie-aided bounds. A matching inner bound is established via the development of a network decomposition theorem, which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptually simple proof of achievability that generalizes to L-transmitter L-receiver networks.