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

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 G(n), which is obtained by analytical method and iterative calculation. We then reveal the relation between the mean Fermat distance (F) over bar and the mean geodesic distance (d) over bar in general networks, namely, 3/2 (d) over bar < <(F)over bar> <= 2 (d) over bar. The ratio of the mean Fermat distance to the mean geodesic distance of G(n), tends to 3/2, which is the lower bound of the inequality above. Moreover, the result shows the small-world effect of G(n). Finally, we illustrate that the mean Fermat distance is significant in both small-world and scale-free networks.

Automated summary

The Fermat point is the point in a triangle that has the minimum total distance from the three vertices. This paper explores the Fermat distance on various networks and reveals the relationship between the mean Fermat distance and the mean geodesic distance. It also demonstrates the small-world effect of the networks studied.