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

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050098
Author(s):  
JIAQI FAN ◽  
JIANGWEN GU ◽  
LIFENG XI ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Self Similar

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.

AVERAGE GEODESIC DISTANCE OF NODE-WEIGHTED SIERPINSKI NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950110
Author(s):  
LIFENG XI ◽  
QIANQIAN YE ◽  
JIANGWEN GU
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Sierpinski Gasket ◽  
Weight Vector ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar

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.

WEIGHTED AVERAGE GEODESIC DISTANCE OF PENTADENDRITE NETWORKS

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050075
Author(s):  
YUANYUAN LI ◽  
XIAOMIN REN ◽  
KAN JIANG
Keyword(s):  
Complex Networks ◽  
Geodesic Distance ◽  
Average Distance ◽  
Weighted Average ◽  
Similar Measure ◽  
Important Index ◽  
Self Similar

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

AVERAGE GEODESIC DISTANCE ON SIERPINSKI HEXAGON AND SIERPINSKI HEXAGON NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950077 ◽  
Author(s):  
JIALI ZHU ◽  
LI TIAN ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance

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.

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.

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.

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.

AVERAGE GEODESIC DISTANCE OF SIERPINSKI CARPET

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750061 ◽  
Author(s):  
LUMING ZHAO ◽  
SONGJING WANG ◽  
LIFENG Xi
Keyword(s):  
Finite Type ◽  
Geodesic Distance ◽  
Sierpinski Carpet ◽  
Similar Measure ◽  
Self Similar ◽  
Sierpiński Carpet

We obtain the average geodesic distance on the Sierpinski carpet in terms of the integral of geodesic distance on self-similar measure. We find out the finite pattern phenomenon of integral inspired by the notion of finite type on self-similar sets with overlaps.

NODE-WEIGHTED AVERAGE FERMAT DISTANCES OF FRACTAL TREE NETWORKS

Fractals ◽  
2021 ◽  
Author(s):  
CHEN CHEN ◽  
YING MA ◽  
LIFENG XI
Keyword(s):  
Asymptotic Formula ◽  
Weighted Average ◽  
The Self ◽  
Similar Measure ◽  
Tree Networks ◽  
Self Similar

In this paper, we investigate a class of self-similar networks modeled on a self-similar fractal tree, and use the self-similar measure and the method of finite pattern to obtain the asymptotic formula of node-weighted average Fermat distances on fractal tree networks.

AVERAGE DISTANCE OF SELF-SIMILAR FRACTAL TREES

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850016 ◽  
Author(s):  
TINGTING LI ◽  
KAN JIANG ◽  
LIFENG XI
Keyword(s):  
Exact Solution ◽  
Geodesic Distance ◽  
Average Distance ◽  
Tree Networks ◽  
Initial Graph ◽  
Self Similar ◽  
The Mean ◽  
Reduced Scale

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.

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