This document discusses how to care for graph automata. It begins by introducing graph automata as a type of cellular automata where cells are connected by edges rather than existing on a grid. Graph automata are able to reproduce with fewer cells than traditional cellular automata by reconstructing edges between dividing cells. The document then covers classifications of graph automata behavior analogous to Wolfram's classifications of cellular automata, exploring the "edge of chaos" for graph automata by varying the number of nodes and graphs. It concludes by discussing potential new aspects of graph automata, such as introducing energy conservation and modeling the growth of living things.

1. How to care for graph automata
Kazuya Horibe
1
codes written by AK

2. Self Introduction
Kazuya Horibe (@shikihuton)
Osaka University D1
Research
- Traveling waves on the brain
- Dynamical system analysis of
recurrent neural networks
- Urban planning theory
- Chaotic scattering
- Graph automata (Near future)
2
[Tallinen et al. 2016]
[Bleher et al. 1990]

3. Table of contents
3
Motivation
What’s graph automata?
- Cellular automata (CA)
- Graph automata (GA)
- Self-reproduction
- High-dimensional Nonlinear Dynamics
How does graph automata evolve?

4. Motivation
4
The origin of and evolution of
life
What is the smallest component
of life?
- Self-reproduction
- Sustainability
The origin of intelligence
[Freitas et al. 1983]
Life
Intelligence
Jump

5. Graph automata
5
Briefly say …
Cellular automata having graph
structure
- Cell: division, annihilation
- Edge: reconstruction
[Kataoka 2006]
codes written by AK

6. Cellular automata (CA)
6
Setting
- Many cells matrix
- Each cell have K states
Time evolution (Rule)
- Each cell changes its state
depending on the states of
its neighboring cell．
Game of Life
https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
x
y
Basic rule
Black:Dead, White: Alive

7. Classification of
one-dimensional CA
7
Wolfram’s classification [Wolfram 1983]
Class I Dead
Class II Periodic
Class III Random
Class IV
Others (Periodic & Random)
e.g. Game of Life
time
space
Rule 32
Rule 108
Rule 22
Rule 110

8. Edge of chaos
8
Langton’s lambda parameter [Langton 1990]
- K: Number of cell conditions
- N: Number of neighborhoods of each cell (including
itself)
- nq: The number of states to stop at the next step in the
all neighborhood state
Class I → Class II → Class IV → Class III
Dead Periodic RandomOthers
= 1
nq
KN
= 1

9. GA can write self-
reproduction simple.
9
Self-reproduction
- CA (29 states, 130,622 cells)
[Neumann 1966]
- GA (4 states, 4 cells)
[Tomita et al. 2002]
A new to o l to approach
important scientific problems
s u c h a s e v o l u t i o n o f
morphology
[Neumann 1966] [Tomita et al. 2002]

10. Graph automata (GA)
10
Tomita, et al. (2002)
- Automata on graph
- Edge-connected cells are
neighboring cells.
- Self-reproduction automata
- Reconstruction of edges, cell
division / annihilation (Rule)
codes written by AK

11. High-dimensional
Nonlinear Dynamics
11
The classification of all rules for the case of K = 2.
[Kataoka 2006]
- Factory behavior
- Self-reproduction
- 4096 ({0, 1, 2, 3, 4, 5}→{0, 1, 2, 3}) rules 100 steps Factory behavior
Self-reproduction

12. To higher: Edge of
chaos for GA
12
The analogues classification to Wolfram's. [Kitajima
2011]
Nnode：Number of nodes
Ngraph：Number of graphs
Vgraph：Diversity of graphs
Nnode = 46, Ngraph =
5, Vgraph = 3

13. 13
ClassA ClassB
ClassDClassC

14. To higher: Edge of
chaos for GA
14
The analogues classification to Wolfram's. The exploration
of the analog to Langron's lambda for graph automata
[Kitajima 2011]
Class A Nnodes (No development)
Class B Nnodes (Elongation)
Class C Nnodes，Ngraphs (Self-Reproduction)
Class D Nnodes，Ngraphs, Vnodes (Others)
There are both power / index how to increase.

15. New Aspects of GA
15
Interactions between graphs
Introduce energy conservation law
Ground the body plan of living
things

16. Reference
16
• [Tallinen et al. 2016] Tallinen, Tuomas, et al. "On the growth and form of cortical convolutions." Nature Physics 12.6 (2016): 588.
• [Blender et al. 1990] Bleher, Siegfried, Celso Grebogi, and Edward Ott. "Bifurcation to chaotic scattering." Physica D: Nonlinear Phenomena 46.1
(1990): 87-121.
• [Freitas et al. 1982] Freitas Jr, Robert A., Timothy J. Healy, and James E. Long. "Advanced automation for space missions." Journal of the
Astronautical Sciences 30.1 (1982): 221.
• [Wolfram 1983] Wolfram, Stephen. "Statistical mechanics of cellular automata." Reviews of modern physics 55.3 (1983): 601.
• [Langton 1990] Langton, Chris G. "Computation at the edge of chaos: phase transitions and emergent computation." Physica D: Nonlinear
Phenomena 42.1-3 (1990): 12-37.
• [Kataoka 2006] N. Kataoka, “Graph Automata and Degree of Freedom” Proccedings of the conference on New Aspects of High-dimensional
Nonlinear Dynamics (2006)
• [Tomita et al. 2002] Tomita, Kohji, Haruhisa Kurokawa, and Satoshi Murata. "Graph automata: natural expression of self-reproduction." Physica
D: Nonlinear Phenomena 171.4 (2002): 197-210.
• [Neumann 1966] Von Neumann, John, and Arthur Walter Burks. Theory of self-reproducing automata. Urbana: University of Illinois Press, 1996.
• [Kitajima 2011] A. Kitajima, “グラフオートマトン∼自発的に自由度を変化させる力学系∼”, Young Scientists Meeting on Statistical Physics and Information
Processing (2011)
• [Kitajima, Kikuchi 2011] A. Kitajima, M. Kikuchi, “グラフオートマトンの振る舞いによる分類”, The physical Society of Japan (2011)
• [Lindenmayer 1968] Lindenmayer, Aristid. "Mathematical models for cellular interactions in development I. Filaments with one-sided
inputs." Journal of theoretical biology 18.3 (1968): 280-299.
• https://en.wikipedia.org/wiki/Elementary_cellular_automaton (2018.06.08)
• https://staff.aist.go.jp/k.tomita/ga/ (2018.06.13)
• https://en.wikipedia.org/wiki/Codd%27s_cellular_automaton (2018.06.27)