From Global Linear Computations to Local Interaction Rules

TL;DR

This paper characterizes which global linear transformations can be computed by networks of locally interacting agents, showing that only those with positive determinants are feasible, and provides methods to find optimal local rules.

Contribution

It establishes a necessary and sufficient condition for local computability of linear transformations and develops an optimal control framework for designing local interaction rules.

Findings

Linear transformations with positive determinant are computable locally.

Optimal local rules can be derived through control problem formulations.

Simulations demonstrate the effectiveness of the proposed methods.

Abstract

A network of locally interacting agents can be thought of as performing a distributed computation. But not all computations can be faithfully distributed. This paper investigates which global, linear transformations can be computed using local rules, i.e., rules which rely solely on information from adjacent nodes in a network. The main result states that a linear transformation is computable in finite time using local rules if and only if the transformation has positive determinant. An optimal control problem is solved for finding the local interaction rules, and simulations are performed to elucidate how optimal solutions can be obtained.