AVERAGE GEODESIC DISTANCE OF SIERPINSKI GASKET AND SIERPINSKI NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750044 ◽  
Author(s):  
SONGJING WANG ◽  
ZHOUYU YU ◽  
LIFENG XI

The average geodesic distance is concerned with complex networks. To obtain the limit of average geodesic distances on growing Sierpinski networks, we obtain the accurate value of integral in terms of average geodesic distance and self-similar measure on the Sierpinski gasket. To provide the value of integral, we find the phenomenon of finite pattern on integral inspired by the concept of finite type on self-similar sets with overlaps.

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950110
Author(s):  
LIFENG XI ◽  
QIANQIAN YE ◽  
JIANGWEN GU

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.


Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG

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.


Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050075
Author(s):  
YUANYUAN LI ◽  
XIAOMIN REN ◽  
KAN JIANG

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.


Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750061 ◽  
Author(s):  
LUMING ZHAO ◽  
SONGJING WANG ◽  
LIFENG Xi

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.


2019 ◽  
Vol 7 (1) ◽  
pp. 1-62
Author(s):  
Sizhen Fang ◽  
Dylan King ◽  
Eun Bi Lee ◽  
Robert Strichartz

2007 ◽  
Vol 27 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Mihai Cucuringu ◽  
Robert S. Strichartz

1994 ◽  
Vol 115 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Jun Kigami

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].


2020 ◽  
Vol 7 (4) ◽  
pp. 387-444
Author(s):  
Christian Loring ◽  
W. Jacob Ogden ◽  
Ely Sandine ◽  
Robert Strichartz

2019 ◽  
Vol 3 (1) ◽  
pp. 13
Author(s):  
Melis Güneri ◽  
Mustafa Saltan

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.


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