Complex Networks from Simple Rewrite Systems

TL;DR

This paper explores how simple rewrite rules in colored network systems can generate a wide variety of complex networks, some resembling natural objects, through simulation and mathematical analysis.

Contribution

It introduces a new class of simple, rule-based network systems and provides an empirical classification of their diverse dynamic behaviors.

Findings

Generated networks range from simple to highly complex.

Some rules produce structures similar to natural objects.

The study offers an empirical classification of network dynamics.

Abstract

Complex networks are all around us, and they can be generated by simple mechanisms. Understanding what kinds of networks can be produced by following simple rules is therefore of great importance. We investigate this issue by studying the dynamics of extremely simple systems where are `writer' moves around a network, and modifies it in a way that depends upon the writer's surroundings. Each vertex in the network has three edges incident upon it, which are colored red, blue and green. This edge coloring is done to provide a way for the writer to orient its movement. We explore the dynamics of a space of 3888 of these `colored trinet automata' systems. We find a large variety of behaviour, ranging from the very simple to the very complex. We also discover simple rules that generate forms which are remarkably similar to a wide range of natural objects. We study our systems using simulations (with appropriate visualization techniques) and analyze selected rules mathematically. We arrive at an empirical classification scheme which reveals a lot about the kinds of dynamics and networks that can be generated by these systems.