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Sum-networks from undirected graphs: construction and capacity analysis

We consider a directed acyclic network with multiple sources and multiple terminals where each terminal is interested in decoding the sum of independent sources generated at the source nodes. We describe a procedure whereby a simple undirected graph can be used to construct such a sum-network and demonstrate an upper bound on its computation rate. Furthermore, we show sufficient conditions for the construction of a linear network code that achieves this upper bound. Our procedure allows us to construct sum-networks that have any arbitrary computation rate $\frac{p}{q}$ (where $p,q$ are non-negative integers). Our work significantly generalizes a previous approach for constructing sum-networks with arbitrary capacities. Specifically, we answer an open question in prior work by demonstrating sum-networks with significantly fewer number of sources and terminals.