Sum-networks from undirected graphs: construction and capacity analysis

TL;DR

This paper introduces a method to construct sum-networks from undirected graphs, providing capacity bounds and conditions for optimal linear coding, thus generalizing previous approaches and answering open questions.

Contribution

It presents a novel procedure for constructing sum-networks with arbitrary capacities from undirected graphs and establishes conditions for achieving maximum computation rates.

Findings

Constructed sum-networks with any arbitrary capacity $rac{p}{q}$

Provided upper bounds on computation rates for these networks

Demonstrated linear codes achieving the bounds under certain conditions

Abstract

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.