Often, a complex system consists of network elements, or nodes, that remains fixed in number over the course of the evolution of the system. However, this isn't a requirement, and the number of elements in the system can change as the system evolves. Perhaps the simplest example of this scenario is the substitution system, which consists of an alphabet that forms a string, and a set of rules that state how the string changes over time through substitutions of its elements.
Although simple to describe, substitution systems can give rise to unpredictable and surprisingly complex behaviour.
The examples of the substitution systems described here are derived from Stephen Wolfram's
A New Kind of Science. Netomaton supports the creation
of such substitution systems, through the SubstitutionSystem class.
An example of a substitution system is given below. It is a reproduction of the system on page 82 (left) of NKS:
import netomaton as ntm
system = ntm.SubstitutionSystem(rules={
"2": "21",
"1": "12"
}, axiom=[2])
trajectory = ntm.evolve(network=system.network,
initial_conditions=system.initial_conditions,
activity_rule=system.activity_rule, timesteps=6)
padded = system.pad(trajectory)
ntm.plot_grid(padded, show_grid=True)
The full source code for this example can be found here. See also the demos here and here.
A more complex example is given below, and reproduces the system on page 87 (b) of NKS:
import netomaton as ntm
system = ntm.SubstitutionSystem(rules={
"33": "3",
"32": "12",
"31": "1",
"23": "",
"22": "",
"21": "3",
"13": "1",
"12": "12",
"11": "3"
}, axiom=[1, 2, 3, 2])
trajectory = ntm.evolve(network=system.network,
initial_conditions=system.initial_conditions,
activity_rule=system.activity_rule, timesteps=101)
padded = system.pad(trajectory)
ntm.plot_grid(padded, show_grid=True)
The full source code for this example can be found here. See also the demo here.
For more information, see:
https://www.wolframscience.com/nks/p82--substitution-systems/