Fractality and the small-world effect in Sierpinski graphs

Abstract

Although some real networks exhibit self-similarity, there is no standard definition of fractality in graphs. On the other hand, the small-world phenomenon is one of the most important common properties of real interconnection networks. In this paper we relate these two properties. In order to do so, we focus on the family of Sierpinski networks. For the Sierpinski gasket, the Sierpinski carpet and the Sierpinski tetra, we give the basic properties and we calculate the box-counting dimension as a measure of their fractality. We also define a deterministic family of graphs, which we call small-world Sierpinski graphs. We show that our construction preserves the structure of Sierpinski graphs, including its box-counting dimension, while the small-world phenomenon arises. Thus, in this family of graphs, fractality and small-world effect are simultaneously present.