Sum-networks from incidence structures: construction and capacity analysis

TL;DR

This paper introduces a method to construct sum-networks from incidence structures, analyzes their capacity, and reveals that their computation capacity strongly depends on the finite field's characteristic, unlike multiple unicast networks.

Contribution

The paper presents an algorithm for constructing sum-networks from incidence structures and characterizes their capacity, highlighting the characteristic dependence of sum-network capacity.

Findings

Sum-network capacity varies with the finite field's characteristic.

Examples show capacity can be high over one characteristic and near zero over another.

Sum-networks can have multiple capacities depending on the alphabet, unlike multiple unicast networks.

Abstract

A sum-network is an instance of a network coding problem over a directed acyclic network in which each terminal node wants to compute the sum over a finite field of the information observed at all the source nodes. Many characteristics of the well-studied multiple unicast network communication problem also hold for sum-networks due to a known reduction between instances of these two problems. In this work, we describe an algorithm to construct families of sum-network instances using incidence structures. The computation capacity of several of these sum-network families is characterized. We demonstrate that unlike the multiple unicast problem, the computation capacity of sum-networks depends on the characteristic of the finite field over which the sum is computed. This dependence is very strong; we show examples of sum-networks that have a rate-1 solution over one characteristic but a rate close to zero over a different characteristic. Additionally, a sum-network can have an arbitrary different number of computation capacities for different alphabets. This is contrast to the multiple unicast problem where it is known that the capacity is independent of the network coding alphabet.