ASYMPTOTIC FORMULA OF AVERAGE DISTANCES ON FRACTAL NETWORKS MODELED BY SIERPINSKI TETRAHEDRON

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950120
Author(s):  
JUAN DENG ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Average Distance ◽  
The Self ◽  
Renewal Theorem ◽  
Self Similarity ◽  
Evolving Networks

This paper concerns the average distances of evolving networks modeled by Sierpinski tetrahedron. We express the limit of average distances on reorganized networks as an integral of geodesic distance on Sierpinski tetrahedron. Based on the self-similarity and renewal theorem, we obtain the asymptotic formula on the average distance of our evolving networks.

ASYMPTOTIC FORMULA ON AVERAGE PATH LENGTH OF A SPECIAL NETWORK BASED ON SIERPINSKI CARPET

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850039 ◽  
Author(s):  
YUMEI XUE ◽  
DONGXUE ZHOU
Keyword(s):  
Asymptotic Formula ◽  
Path Length ◽  
Average Path Length ◽  
The Self ◽  
Renewal Theorem ◽  
Self Similarity ◽  
Sierpinski Carpet ◽  
Sierpiński Carpet

In this paper, we construct a special network based on the construction of the Sierpinski carpet. Using the self-similarity and renewal theorem, we obtain the asymptotic formula for the average path length of our evolving network.

Weighted average geodesic distance of Vicsek network in three-dimensional space

2021 ◽  
pp. 2150077
Author(s):  
Yan Liu ◽  
Meifeng Dai ◽  
Yuanyuan Guo
Keyword(s):  
Dimensional Space ◽  
Geodesic Distance ◽  
Three Dimensional ◽  
Weighted Average ◽  
Weight Vector ◽  
Natural Generalization ◽  
The Self ◽  
Self Similarity ◽  
Similar Measure ◽  
Three Dimensional Space

Fractal generally has self-similarity. Using the self-similarity of fractal, we can obtain some important theories about complex networks. In this paper, we concern the Vicsek fractal in three-dimensional space, which provides a natural generalization of Vicsek fractal. Concretely, the Vicsek fractal in three-dimensional space is obtained by repeatedly removing equilateral cubes from an initial equilateral cube of unit side length, at each stage each remaining cube is divided into [Formula: see text] smaller cubes of which [Formula: see text] are kept and the rest discarded, where [Formula: see text] is odd. In addition, we obtain the skeleton network of the Vicsek fractal in three-dimensional space. Then we focus on weighted average geodesic distance of the Vicsek fractal in three-dimensional space. Take [Formula: see text] as an example, we define a similar measure on the Vicsek fractal in three-dimensional space by weight vector and calculate the weighted average geodesic distance. At the same time, asymptotic formula of weighted average geodesic distance on the skeleton network is also obtained. Finally, the general formula of weighted average geodesic distance should be applicable to the models when [Formula: see text], the base of a power, is odd.

SMALL-WORLD AND SCALE-FREE PROPERTIES OF THE n-DIMENSIONAL SIERPINSKI CUBE NETWORKS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050001
Author(s):  
CHENG ZENG ◽  
MENG ZHOU ◽  
YUMEI XUE
Keyword(s):  
Small World ◽  
The Self ◽  
Self Similarity ◽  
Scale Free ◽  
Evolving Networks

In this paper, we construct evolving networks from [Formula: see text]-dimensional Sierpinski cube. Using the self-similarity of Sierpinski cube, we show the evolving networks have scale-free and small-world properties.

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

AVERAGE DISTANCES OF A FAMILY OF NON-P.C.F. SELF-SIMILAR NETWORKS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050040
Author(s):  
BING ZHAO ◽  
JIANGWEN GU ◽  
LIFENG XI
Keyword(s):  
Geodesic Distance ◽  
Average Distance ◽  
The Self ◽  
Similar Measure ◽  
Geometric Patterns ◽  
Finite Set ◽  
Self Similar

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.

THE SCALE-FREE AND SMALL-WORLD PROPERTIES OF COMPLEX NETWORKS ON SIERPINSKI-TYPE HEXAGON

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050054
Author(s):  
KUN CHENG ◽  
DIRONG CHEN ◽  
YUMEI XUE ◽  
QIAN ZHANG
Keyword(s):  
Power Law ◽  
Path Length ◽  
Small World ◽  
The Self ◽  
Renewal Theorem ◽  
Self Similarity ◽  
Scale Free ◽  
Power Law Exponent ◽  
Self Similar ◽  
Encoding Method

In this paper, a network is generated from a Sierpinski-type hexagon by applying the encoding method in fractal. The criterion of neighbor is established to quantify the relationships among the nodes in the network. Based on the self-similar structures, we verify the scale-free and small-world effects. The power-law exponent on degree distribution is derived to be [Formula: see text] and the average clustering coefficients are shown to be larger than [Formula: see text]. Moreover, we give the bounds of the average path length of our proposed network from the renewal theorem and self-similarity.

AVERAGE DISTANCE OF SUBSTITUTION NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950097 ◽  
Author(s):  
QIANQIAN YE ◽  
LIFENG XI
Keyword(s):  
Linear Equation ◽  
Metric Spaces ◽  
Deterministic Model ◽  
Geodesic Distance ◽  
Average Distance ◽  
The Self ◽  
Similar Measure ◽  
Self Similar

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.

ASYMPTOTIC FORMULA OF ECCENTRIC DISTANCE SUM FOR VICSEK NETWORK

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850027 ◽  
Author(s):  
QIANQIAN YE ◽  
LONG HE ◽  
QIN WANG ◽  
LIFENG XI
Keyword(s):  
Asymptotic Formula ◽  
The Self ◽  
Self Similarity ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

In this work, we study the eccentric distance sum of Vicsek networks. To obtain the eccentric distance sum of networks, we investigate the corresponding integral on self-similar measure for Vicsek fractals. We use the self-similarity of distance and measure to solve the integral.

scholarly journals CAN AGN FEEDBACK BREAK THE SELF-SIMILARITY OF GALAXIES, GROUPS, AND CLUSTERS?

2014 ◽  
Vol 783 (1) ◽  
pp. L10 ◽  
Author(s):  
M. Gaspari ◽  
F. Brighenti ◽  
P. Temi ◽  
S. Ettori
Keyword(s):  
The Self ◽  
Self Similarity
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