Computing linear functions by linear coding over networks

TL;DR

This paper investigates the conditions under which linear functions can be computed over networks using linear coding, providing algebraic tests and identifying classes of functions with guaranteed or non-guaranteed computability.

Contribution

It introduces an algebraic test for network computability of linear functions and characterizes classes of functions that can or cannot be computed with linear codes.

Findings

Algebraic test for linear function computability in networks

Identification of a class of functions computable under a cut-based condition

Demonstration that some functions are not computable despite the cut condition

Abstract

We consider the scenario in which a set of sources generate messages in a network and a receiver node demands an arbitrary linear function of these messages. We formulate an algebraic test to determine whether an arbitrary network can compute linear functions using linear codes. We identify a class of linear functions that can be computed using linear codes in every network that satisfies a natural cut-based condition. Conversely, for another class of linear functions, we show that the cut-based condition does not guarantee the existence of a linear coding solution. For linear functions over the binary field, the two classes are complements of each other.