Figure 1.
Collatz sequences and Collatz graph. Left: Examples of Collatz sequences, Cn, for the first 20 natural numbers.
Right: A network representaion of these sequences is called the Collatz graph.
Figure 2.
Collatz step graph. Left: The results of the step function as a function of n. Bottom: The step sequence that is based on the results of the step function.
Right: From a CS graph can be constructed.
Figure 9.
Left: CS graph, , with
nodes,
edges and a diameter of
. Right: CS graph,
, with
nodes,
edges and a diameter of
.
Figure 3.
A: Autocorrelation function, , of the Collatz step sequence. B: Histogram for even and odd sequence elements of
.
Figure 4.
Power law scaling of the edge weights in CS graphs.
The color corresponds to different sizes of with
(green, red, blue).
Figure 5.
A: Examples of lossless transformations underlying the construction of power graphs and the edge reduction measure.
B: The edge reduction of CS graphs as a function of the length of the step sequence. C: Scaling of the size, , of CS graphs as a function of the length,
, of the step sequence.
Figure 6.
The edge reduction for random networks (A) and regular trees (B).
The x-axis gives the number of nodes in these networks. Figure C and D show the in- and out-degree distribution of a CS graph for .
Figure 7.
CS graph with nodes obtained for
.
A: Adjacency matrix . B: Binary adjacency matrix
. C: Non-equal elements
. D: Non-symmetric elements
.
Figure 8.
Average shortest path lengths.
A: Scaling of the average path length in dependence on the size of the step sequence . B: Histograms of the shortest path lengths for four different values of
. The colors correspond to the vertical lines in Fig. A.
Table 1.
The parameters of a logistic function obtained from a nonlinear regression.
Table 2.
The parameters of the logistic decay function obtained from a nonlinear regression.
Figure 10.
Average shortest path lengths and clustering coefficients.
CS graphs (blue), random (red), small-world (green), scale-free (purple) and biological (yellow) networks. The blue cross indicates the limiting value for CS graphs.