Fractal networks with Sturmian structure

Abstract

Many real complex networks behave similarly with the scale-free and the small-world properties. In this paper, we create a special hierarchical network with the Fibonacci word. This network is self-similar on some scales due to Fibonacci word’s recurrence property. Based on the construction of the graph, we study the clustering coefficient, the average path length and the cumulative degree distribution of our network. These results show the scale-free and the small-world effects of the evolving network. Moreover, the monotony of the standardized average path length and the average clustering coefficient is coincided with that of the Fibonacci word by applying the balance property of the Sturmian word. We think the Sturmian word can be an useful tool to describe some complex networks.