SCALE-FREE AND SMALL-WORLD PROPERTIES OF VAF FRACTAL NETWORKS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650033 ◽  
Author(s):  
HAO LI ◽  
JIAN HUANG ◽  
ANBO LE ◽  
QIN WANG ◽  
LIFENG XI
Keyword(s):  
Small World ◽  
The Self ◽  
Sierpinski Carpet ◽  
Scale Free ◽  
Evolving Networks ◽  
Self Similar ◽  
Sierpiński Carpet

In this paper, we investigate the vertical-affiliation-free (VAF) evolving networks whose node set is the basic squares in the process of generating the Sierpinski carpet and edge exists between any two nodes if and only if the corresponding basic squares intersect just on their boundary. Although the VAF networks gets rid of the hierarchial organizations produced naturally by the self-similar structures of fractals, we still prove that they are scale-free and have the small-world effect.

SMALL-WORLD AND SCALE-FREE PROPERTIES OF FRACTAL NETWORKS MODELED ON n-DIMENSIONAL SIERPINSKI PYRAMID

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750057 ◽  
Author(s):  
CHENG ZENG ◽  
MENG ZHOU
Keyword(s):  
Small World ◽  
The Self ◽  
Scale Free ◽  
Evolving Networks ◽  
Self Similar

In this paper, we construct evolving networks based on the construction of the [Formula: see text]-dimensional Sierpinski pyramid by the self-similar structure. We show that such networks have scale-free and small-world effects.

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.

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.

FRACTAL NETWORKS ON SIERPINSKI-TYPE POLYGON

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050087
Author(s):  
CHENG ZENG ◽  
YUMEI XUE ◽  
MENG ZHOU
Keyword(s):  
Degree Distribution ◽  
Path Length ◽  
Small World ◽  
Average Path Length ◽  
Clustering Coefficient ◽  
Scale Free ◽  
Evolving Networks ◽  
Power Law Exponent ◽  
Self Similar ◽  
Encoding Method

In this paper, the evolving networks are created from a series of Sierpinski-type polygon by applying the encoding method in fractal and symbolic dynamical system. Based on the self-similar structures of our networks, we study the cumulative degree distribution, the clustering coefficient and the standardized average path length. The power-law exponent of the cumulative degree distribution is deduced to be [Formula: see text] and the average clustering coefficients have a uniform lower bound [Formula: see text]. Moreover, we find the asymptotic formula of the average path length of our proposed networks. These results show the scale-free and the small-world effects of these networks.

The Shortest Path to Network Geometry

2021 ◽  
Author(s):  
M. Ángeles Serrano ◽  
Marián Boguñá
Keyword(s):  
Routing Protocols ◽  
Hyperbolic Plane ◽  
Network Science ◽  
Small World ◽  
The Self ◽  
Geometric Description ◽  
Apparent Lack ◽  
Network Geometry ◽  
Connectivity Patterns ◽  
Self Similar

Real networks comprise from hundreds to millions of interacting elements and permeate all contexts, from technology to biology to society. All of them display non-trivial connectivity patterns, including the small-world phenomenon, making nodes to be separated by a small number of intermediate links. As a consequence, networks present an apparent lack of metric structure and are difficult to map. Yet, many networks have a hidden geometry that enables meaningful maps in the two-dimensional hyperbolic plane. The discovery of such hidden geometry and the understanding of its role have become fundamental questions in network science giving rise to the field of network geometry. This Element reviews fundamental models and methods for the geometric description of real networks with a focus on applications of real network maps, including decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks by geometric renormalization.

Epidemic spreading on evolving networks

2019 ◽  
Vol 33 (23) ◽  
pp. 1950266 ◽  
Author(s):  
Jin-Xuan Yang
Keyword(s):  
Preferential Attachment ◽  
Small World ◽  
Network Evolution ◽  
Epidemic Spreading ◽  
Epidemic Threshold ◽  
Scale Free ◽  
Evolving Networks ◽  
Power Law Exponent ◽  
Free Network ◽  
Over Time

Network structure will evolve over time, which will lead to changes in the spread of the epidemic. In this work, a network evolution model based on the principle of preferential attachment is proposed. The network will evolve into a scale-free network with a power-law exponent between 2 and 3 by our model, where the exponent is determined by the evolution parameters. We analyze the epidemic spreading process as the network evolves from a small-world one to a scale-free one, including the changes in epidemic threshold over time. The condition of epidemic threshold to increase is given with the evolution processes. The simulated results of real-world networks and synthetic networks show that as the network evolves at a low evolution rate, it is more conducive to preventing epidemic spreading.

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.

A SMALL-WORLD AND SCALE-FREE NETWORK GENERATED BY SIERPINSKI TETRAHEDRON

Fractals ◽  
2016 ◽  
Vol 24 (01) ◽  
pp. 1650001 ◽  
Author(s):  
JIN CHEN ◽  
FEI GAO ◽  
ANBO LE ◽  
LIFENG XI ◽  
SHUHUA YIN
Keyword(s):  
Small World ◽  
Scale Free Network ◽  
Fractal Scaling ◽  
Scale Free ◽  
Evolving Networks ◽  
Free Network

The Sierpinski tetrahedron is used to construct evolving networks, whose vertexes are all solid regular tetrahedra in the construction of the Sierpinski tetrahedron up to the stage [Formula: see text] and any two vertexes are neighbors if and only if the corresponding tetrahedra are in contact with each other on boundary. We show that such networks have the small-world and scale-free effects, but are not fractal scaling.

DYNAMICS BEHAVIORS OF WEIGHTED LOCAL-WORLD EVOLVING NETWORKS WITH EXTENDED LINKS

2009 ◽  
Vol 20 (11) ◽  
pp. 1719-1735 ◽  
Author(s):  
GUANGHUI WEN ◽  
ZHISHENG DUAN
Keyword(s):  
Weak Interactions ◽  
Small World ◽  
Local Information ◽  
Scale Free ◽  
Evolving Networks ◽  
Free Behavior ◽  
Small World Property ◽  
Evolving Model ◽  
Analytical Expressions ◽  
Good Agreement

In this paper, we present a local-world evolving model to characterize weighted networks. By introducing the extended links to mimic the weak interactions between the nodes in different local-worlds, the model yields scale-free behavior as well as the small-world property, as confirmed in many real networks. With the increase of the local information, the generated network undergoes a transition from assortative to disassortative, meanwhile the small-world property is preserved. It indicates that the small-world property is a universal characteristic in our model. The numerical simulation results are in good agreement with the analytical expressions.

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