Fractal networks with hierarchical structure: Mean fermat distance and small-world effect

Abstract

The Fermat point of a triangle is the point with the minimal total distance from the three vertices of the triangle. Meanwhile, the total distance from the three vertices to the Fermat point is called the Fermat distance. In this paper, we discuss the Fermat distance on some networks. We study the mean Fermat distance of an unweighted and undirected hierarchical network [Formula: see text], which is obtained by analytical method and iterative calculation. We then reveal the relation between the mean Fermat distance [Formula: see text] and the mean geodesic distance [Formula: see text] in general networks, namely, [Formula: see text]. The ratio of the mean Fermat distance to the mean geodesic distance of [Formula: see text] tends to [Formula: see text], which is the lower bound of the inequality above. Moreover, the result shows the small-world effect of [Formula: see text]. Finally, we illustrate that the mean Fermat distance is significant in both small-world and scale-free networks.