Organic Structures Emerging From Bio-Inspired Graph-Rewriting Automata

TL;DR

This work extends cellular automata into Graph-Rewriting Automata (GRA) by evolving graph topology via local division rules, and develops linear-algebraic tools to study long-time dynamics. Graphs are encoded by an adjacency matrix $A$ and binary state vector $S$, with a configuration rule combining local state and neighbor activity to produce a new state and potential divisions, formalized through a finite rule set and a Wolfram-like encoding. By implementing a sparse-array approach, the authors compute configuration vectors $C = 4S + AS$, apply rules to obtain $S$, and iteratively perform divisions to grow the graph, enabling scalable exploration of thousands of rules. An exhaustive study of single-division rules reveals diverse growth behaviors—halted, linear, quadratic, exponential, and unclassified—with notable examples such as rule 256 (exponential growth) and rule 2182 (quasi-quadratic growth); some rules also generate organic-looking, algae-like graph structures, illustrating the potential of GRA to model natural complex systems. The work emphasizes a practical, hardware-friendly methodology and points to broad future extensions, including continuous states, non-binary topologies, and directed graphs, with code and implementations shared for reproducibility.

Abstract

Graph-Rewriting Automata (GRA) are an extension of Cellular Automata to a dynamic structure using local graph-rewriting rules. This work introduces linear algebra based tools that allow for a practical investigation of their behavior in deeply extended time scales. A natural subset of GRA is explored in different ways thereby demonstrating the benefits of this method. Some elements of the subset were discovered to create chaotic patterns of growth and others to generate organic-looking graph structures. These phenomena suggest a strong relevance of GRA in the modeling natural complex systems. The approach presented here can be easily adapted to a wide range of GRA beyond the chosen subset.

Figures (14)

• Figure 1: Example of evolution starting from a simple initial graph.
• Figure 2: Example of the graph $G_0$ from Figure \ref{['example']}.
• Figure 3: Eight possible configurations.
• Figure 4: Rule of the example in Figure \ref{['example']} ($n=765=10,1111,1101_2$).
• Figure 5: Division operated on the second vertex of the graph from Figure \ref{['example-graph']}.