Structural Properties and Complexity of a New Network Class: Collatz Step Graphs

In this paper, we introduce a biologically inspired model to generate complex networks. In contrast to many other construction procedures for growing networks introduced so far, our method generates networks from one-dimensional symbol sequences that are related to the so called Collatz problem from number theory. The major purpose of the present paper is, first, to derive a symbol sequence from the Collatz problem, we call the step sequence, and investigate its structural properties. Second, we introduce a construction procedure for growing networks that is based on these step sequences. Third, we investigate the structural properties of this new network class including their finite scaling and asymptotic behavior of their complexity, average shortest path lengths and clustering coefficients. Interestingly, in contrast to many other network models including the small-world network from Watts & Strogatz, we find that CS graphs become ‘smaller’ with an increasing size.