On the Classification of Universal Rotor-Routers

TL;DR

This paper investigates the classification of rotor-router networks, introducing new rotor classes and an algorithm to determine universality, significantly advancing understanding of deterministic walks on directed graphs.

Contribution

It proposes a conjecture classifying universal rotor types, introduces the compressor algorithm, and proves universality for many rotor classes, expanding the theoretical framework of rotor-router networks.

Findings

The compressor algorithm determines the universality of almost all two-state rotors.

Most rotor types up to length 17 are non-universal, with only 272 exceptions.

New rotor classes like boppy, balanced, and BURD rotors are shown to be universal.

Abstract

The combinatorial theory of rotor-routers has connections with problems of statistical mechanics, graph theory, chaos theory, and computer science. A rotor-router network defines a deterministic walk on a digraph G in which a particle walks from a source vertex until it reaches one of several target vertices. Motivated by recent results due to Giacaglia et al., we study rotor-router networks in which all non-target vertices have the same type. A rotor type r is universal if every hitting sequence can be achieved by a homogeneous rotor-router network consisting entirely of rotors of type r. We give a conjecture that completely classifies universal rotor types. Then, this problem is simplified by a theorem we call the Reduction Theorem that allows us to consider only two-state rotors. A rotor-router network called the compressor, because it tends to shorten rotor periods, is introduced along with an associated algorithm that determines the universality of almost all rotors. New rotor classes, including boppy rotors, balanced rotors, and BURD rotors, are defined to study this algorithm rigorously. Using the compressor the universality of new rotor classes is proved, and empirical computer results are presented to support our conclusions. Prior to these results, less than 100 of the roughly 260,000 possible two-state rotor types of length up to 17 were known to be universal, while the compressor algorithm proves the universality of all but 272 of these rotor types.