Steve E. Harris, Bruce Sawhill, Andrew Wuensche, Stuart Kauffman.

Derrida plots made with *DDLab*

These plots were made for random Boolean networks, n=1000, for k=3, 4, and 5, using
the old+new data. The C values are higher in all cases.

k=3, C=0.795

k=4, C=0.708

k=5, C=0.649

There are two diagrams for each k.

**left**: an overall plot showing rules at random, and with canalizing set.
The normalized Hamming range is 0 to 0.5. Each Hamming distance is
plotted for a sample of 25 initial pairs of states.

**right**: details of the initial Hamming range (0 to 0.01) to calculate
the Hamming coefficient based on the first 5 steps.
The average slope between these points and the
origin is taken as the initial slope.
If the initial slope = x degrees, then the Derrida coefficient,
Dc=log2(tan(x)).
A slope of 45 degrees (Dc=0) indicates the dynamics is balanced between
order and chaos. A slope above the 45 degree diagonal (positive Dc)
is in the chaotic regime, below (negative Dc) is in the ordered regime.
If Derrida plot is based on more than 1 iteration, the slope
above or below the diagonal is amplified, giving a clear indication
where it lies relative to the 45 degree diagonal.

The plots how 4 curves. The top curve is for rules set at random.
The 3 lower curves are for canalizing set, for 1, 2 and 3 iterations.
Each Hamming distance is
plotted for a sample of 1000 initial pairs of states. These are typical plots,
and the variance for other networks with the same parameters was found to
be low.

Comments: the curves no longer lie on each other, but become shallower with greater k. The k=3 curve in the old paper looks too shallow compared with new k=3 curve, which is steeper despite having greater C.

Click on each image to see it full size.

k=3, n=1000. Steep curve: random rules. Shallow curve: C=0.795. | Derrida coefficient. Steep curve: random rules, Dc=+0.58 Shallow curve, 1 itteration, Dc=-0.9 |

k=4 n=1000. Steep curve: random rules. Shallow curve: C=0.708. | Derrida coefficient. Steep slope: random rules, Dc=+1.31 Shallow curve, 1 itteration, Dc=-0.25 |

k=5 n=1000. Steep curve: random rules. Shallow curve: C=0.649. | Derrida coefficient. Steep curve: random rules, Dc=+1.33 Shallow curve, 1 itteration, Dc=-0.39 |

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Last modified: June 2001