Search

In this paper, we establish the theory of finite geodesic patterns on limit space to calculate the average distances of touching networks. [ABSTRACT FROM AUTHOR]

The geodesic structures on self-similar fractals are interesting. One feature of geodesic structures is average distance on fractal. We investigate the Sierpinski-like carpet and obtain the average distance in terms of self-similar measure. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a family of non-p.c.f. self-similar networks. Although the boundary of each fractal piece is not a finite set, we obtain the finite geometric patterns for the integral of geodesic distance on the self-similar measure, and then calculate its average distance. [ABSTRACT FROM AUTHOR]

This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector. [ABSTRACT FROM AUTHOR]

The substitution network is a deterministic model of evolving self-similar networks. For normalized substitution networks, the limit of metric spaces with respect to networks is a self-similar fractal and the limit of average distances on networks is the integral of geodesic distance of the fractal on the self-similar measure. After some technical handles, we establish the finiteness of integrals and obtain a linear equation set to solve the average distance on the fractal. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the average geodesic distance on the Sierpinski hexagon in terms of finite patterns on integrals. Applying this result, we also obtain the asymptotic formula for average distances of Sierpinski hexagon networks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method which can generate a family of growing symmetrical tree networks. The networks are constructed by replacing each edge with a reduced-scale of the initial graph. Repeating this procedure, we obtain the fractal networks. In this paper, we define the average geodesic distance of fractal tree in terms of some integral, and calculate its accurate value. We find that the limit of the average geodesic distance of the finite networks tends to the average geodesic distance of the fractal tree. This result generalizes the paper [Z. Zhang, S. Zhou, L. Chen, M. Yin and J. Guan, Exact solution of mean geodesic distance for Vicsek fractals, J. Phys. A: Math. Gen. 41(48) (2008) 7199-7200] for which the mean geodesic distance of Vicsek fractals was considered. [ABSTRACT FROM AUTHOR]

The average distance between points of a fractal is proposed as a natural measure of the way in which the points of a fractal are distributed. The average distance between points of the Cantor Set is found to be $\frac{2}{5}$, the average distance between points of the Cantor p-set is $\frac{p+1}{p+3}$, and the average distance between points of the Fat Cantor p-set is $\frac{2}{3(2-p)}$. A general formula for computing the average distance between points of a self-similar set satisfying the open set condition is found. [ABSTRACT FROM AUTHOR]