List of figures:
One of the basins of attraction of a random Boolean network.
The network size, n=13. The maximum connections (wires) for any cell,
k=3. The complete basin of attraction field of 15 basins is shown
below. The basin links 604 states, of which 523 are garden-of-Eden
states. The attractor cycle has a period of 7. The basin can be
regenerated in DDLab by setting any state belonging to it as the seed,
for example the attractor state shown in detail. The significance of
these "state transition diagrams" is to show how states (the nodes) are
linked by directed arcs (lines). The actual position of nodes and the
length of arcs is set automaticaly in DDLab according to a graphic
convention, where the direction of time is inwards from garden-of-Eden
states to the attractor, then clock-wise.
The network's "memory".
The network architecture results in a specific basin of attraction field,
which shows how the network dynamics hierarchically catagorises state-space
into seperate basins, trees and sub-trees, the network's "memory".
Small mutations
to network architecture, for example moving one wire, of flipping
one bit in a rule, will usually have a small effect on the resulting
basin of attraction field, though hitting a sensitive spot can have
drastic consequences. There are also sets of neutral architectures
that produce equivelent basin of attraction fields.
A detail of the 2nd basin in the basin of
attraction field of the CA above.
The states are shown as a 4x4 bit patterns.
Various predefined neighbourhoods on either a square or hex lattice.
Max-k=13
unless totalistic-rules-only was selected.
Network wiring can also be represented between cells
arranged in a circle.
For small networks cells are shown a discs, the examples
above are for an RBN, n=30, firstly highlighting just
one cell, then showing the wiring in the whole network.
For larger networks as in the example right for n=500, cells are represented by short radial lines. In both cases the "out-degree" is represented by the radial red lines outside the circle. |
Look-up frequency histograms relating to figure above.
Suppressing neighborhoods are indicated with a dot.
Unfiltered and partly filtered space-time patterns of
k=3 rule 18. (transformed to k=5 rule 030c030c).
n=150, about 130 time-steps from the same random initial
state, showing discontinuities within the chaotic domain.
Examples of glider-guns. 127 time-steps.
The k=5 rule numbers are shown in hex.
(a) A compound glider, (b) a glider with a period of
106 time-steps, (c) a compound glider-gun. 168 time-steps.
The k=5 rule numbers are shown in hex.
Classifying a
random sample of k=5 rules by plotting
mean entropy against standard deviation of the entropy,
with the frequency of rules within a 128x128 grid shown
on the vertical axis.
Classifying
random samples of k=6 k=7 rules by plotting
mean entropy against standard deviation of the entropy,
with the frequency of rules within a 128x128 grid shown
on the vertical axis.
(a) CA - wiring randomised in stages | (b) CA wiring, mixed rules | (c) RBN - random wiring, mixed rules |
Fozen elements that have stabilized
for 20 time-steps are shown, 0s-green, 1s red, otherwise white,
for left: C=25% and right: 52%.
The log-log ``damage
spread'' histogram for C=52%, sample size about 1000.
The Derrida plot
for C=0%, 25%, 52%, and 75%, for
1 time-step, H_{t}=0-0.3, interval = 5,
sample for each H_{t} =25.
Ordered dynamics. Rule 01dc3610, n=40, Z=0.5625, lambda_{ratio}=0.668. below: The complete sub-tree 7 levels deep, with 58153 nodes, G-density=0.931.
Complex dynamics. Rule 6c1e53a8, n=50, Z=0.727, lambda_{ratio}=0.938. below: The sub-tree, stopped after 12 levels, with 144876 nodes, G-density=0.692.
Chaotic dynamics. Rule 994a6a65, n=50, Z=0.938, lambda_{ratio}=0.938. below: The sub-tree, stopped after about 75 levels, with 9446 nodes, G-density=0.487.
The basin of attraction fields of two RBN (n=6, k=3) with just a 1 bit difference in one of their rules, to illustrate the effect mutation, and perturbation of a state. Some differences in the fields are evident. The result of a 1 bit perturbation to a reference state of all 1s (rs) is indicated by the location of its its 1 bit mutants (m).