Towards a better understanding of the characteristics of fractal networks

Applied Network Science

Research

Enikő Zakar-Polyák¹  Marcell Nagy¹  Roland Molontay² 

[1] Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., 1111, Budapest, Hungary;Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., 1111, Budapest, Hungary;ELKH-BME Stochastics Research Group, Műegyetem rkp. 3., 1111, Budapest, Hungary;

关键词: Fractal networks;    Network models;    Structural properties;    Small world;    Assortativity;    Betweenness centrality;   

DOI  :  10.1007/s41109-023-00537-8

 received in 2023-01-04, accepted in 2023-01-27,  发布年份 2023

来源: Springer

【 摘 要 】

The fractal nature of complex networks has received a great deal of research interest in the last two decades. Similarly to geometric fractals, the fractality of networks can also be defined with the so-called box-covering method. A network is called fractal if the minimum number of boxes needed to cover the entire network follows a power-law relation with the size of the boxes. The fractality of networks has been associated with various network properties throughout the years, for example, disassortativity, repulsion between hubs, long-range-repulsive correlation, and small edge betweenness centralities. However, these assertions are usually based on tailor-made network models and on a small number of real networks, hence their ubiquity is often disputed. Since fractal networks have been shown to have important properties, such as robustness against intentional attacks, it is in dire need to uncover the underlying mechanisms causing fractality. Hence, the main goal of this work is to get a better understanding of the origins of fractality in complex networks. To this end, we systematically review the previous results on the relationship between various network characteristics and fractality. Moreover, we perform a comprehensive analysis of these relations on five network models and a large number of real-world networks originating from six domains. We clarify which characteristics are universally present in fractal networks and which features are just artifacts or coincidences.

【 授权许可】

CC BY   
© The Author(s) 2023. corrected publication 2023

【 图 表 】

40517_2023_258_Article_IEq133.gif

40517_2023_258_Article_IEq132.gif

40517_2023_258_Article_IEq131.gif

40517_2023_258_Article_IEq130.gif

40517_2023_258_Article_IEq129.gif

40517_2023_258_Article_IEq128.gif

40517_2023_258_Article_IEq127.gif

40517_2023_258_Article_IEq126.gif

40517_2023_258_Article_IEq125.gif

Fig. 2

Fig. 1

40517_2023_258_Article_IEq122.gif

Fig. 1

40517_2023_258_Article_IEq115.gif

40517_2023_258_Article_IEq114.gif

40517_2023_258_Article_IEq113.gif

Fig. 1

Fig. 1

40517_2023_256_Article_IEq14.gif

Fig. 8

Fig. 6

Fig. 7

40517_2023_256_Article_IEq11.gif

Fig. 5

40517_2023_256_Article_IEq5.gif

40517_2023_256_Article_IEq4.gif

41116_2023_36_Article_IEq117.gif

Fig. 2

41116_2023_36_Article_IEq116.gif

41116_2023_36_Article_IEq115.gif

41116_2023_36_Article_IEq818.gif

41116_2023_36_Article_IEq817.gif

41116_2023_36_Article_IEq816.gif

41116_2023_36_Article_IEq815.gif

41116_2023_36_Article_IEq813.gif

41116_2023_36_Article_IEq811.gif

41116_2023_36_Article_IEq809.gif

40517_2023_258_Article_IEq102.gif

41116_2023_36_Article_IEq806.gif

41116_2023_36_Article_IEq805.gif

41116_2023_36_Article_IEq803.gif

41116_2023_36_Article_IEq801.gif

41116_2023_36_Article_IEq799.gif

41116_2023_36_Article_IEq797.gif

41116_2023_36_Article_IEq795.gif

Fig. 1

41116_2023_36_Article_IEq793.gif

41116_2023_36_Article_IEq791.gif

41116_2023_36_Article_IEq789.gif

41116_2023_36_Article_IEq786.gif

41116_2023_36_Article_IEq785.gif

41116_2023_36_Article_IEq783.gif

41116_2023_36_Article_IEq781.gif

41116_2023_36_Article_IEq778.gif

41116_2023_36_Article_IEq777.gif

41116_2023_36_Article_IEq774.gif

Fig. 2

Fig. 2

41116_2023_36_Article_IEq768.gif

41116_2023_36_Article_IEq767.gif

41116_2023_36_Article_IEq765.gif

41116_2023_36_Article_IEq763.gif

41116_2023_36_Article_IEq761.gif

41116_2023_36_Article_IEq757.gif

41116_2023_36_Article_IEq753.gif

41116_2023_36_Article_IEq749.gif

41116_2023_36_Article_IEq748.gif

41116_2023_36_Article_IEq747.gif

41116_2023_36_Article_IEq744.gif

41116_2023_36_Article_IEq737.gif

Fig. 1

41116_2023_36_Article_IEq734.gif

41116_2023_36_Article_IEq733.gif

41116_2023_36_Article_IEq731.gif

40517_2023_258_Article_IEq58.gif

Fig. 3

Fig. 1

41116_2023_36_Article_IEq726.gif

Fig. 6

41116_2023_36_Article_IEq722.gif

41116_2023_36_Article_IEq720.gif

40517_2023_258_Article_IEq51.gif

41116_2023_36_Article_IEq718.gif

41116_2023_36_Article_IEq716.gif

41116_2023_36_Article_IEq714.gif

Fig. 2

40517_2023_258_Article_IEq46.gif

41116_2023_36_Article_IEq710.gif

41116_2023_36_Article_IEq708.gif

41116_2023_36_Article_IEq706.gif

41116_2023_36_Article_IEq702.gif

41116_2023_36_Article_IEq701.gif

40517_2023_258_Article_IEq40.gif

41116_2023_36_Article_IEq700.gif

41116_2023_36_Article_IEq699.gif

41116_2023_36_Article_IEq698.gif

40517_2023_258_Article_IEq34.gif

41116_2023_36_Article_IEq692.gif

41116_2023_36_Article_IEq689.gif

41116_2023_36_Article_IEq688.gif

41116_2023_36_Article_IEq686.gif

41116_2023_36_Article_IEq680.gif

Fig. 1

Fig. 1

40517_2023_258_Article_IEq25.gif

41116_2023_36_Article_IEq675.gif

41116_2023_36_Article_IEq674.gif

41116_2023_36_Article_IEq672.gif

41116_2023_36_Article_IEq670.gif

41116_2023_36_Article_IEq667.gif

41116_2023_36_Article_IEq663.gif

41116_2023_36_Article_IEq657.gif

41116_2023_36_Article_IEq656.gif

41116_2023_36_Article_IEq655.gif

41116_2023_36_Article_IEq653.gif

41116_2023_36_Article_IEq649.gif

41116_2023_36_Article_IEq644.gif

Fig. 2

41116_2023_36_Article_IEq639.gif

Fig. 1

41116_2023_36_Article_IEq633.gif

41116_2023_36_Article_IEq631.gif

41116_2023_36_Article_IEq627.gif

41116_2023_36_Article_IEq619.gif

41116_2023_36_Article_IEq616.gif

Fig. 1

41116_2023_36_Article_IEq615.gif

40798_2023_579_Article_IEq3.gif

41116_2023_36_Article_IEq504.gif

41116_2023_36_Article_IEq503.gif

41116_2023_36_Article_IEq502.gif

41116_2023_36_Article_IEq501.gif

Fig. 7

Fig. 2

Fig. 1

Fig. 2

40517_2023_258_Article_IEq8.gif

40517_2023_258_Article_IEq7.gif

Fig. 1

41116_2023_36_Article_IEq500.gif

41116_2023_36_Article_IEq499.gif

41116_2023_36_Article_IEq498.gif

41116_2023_36_Article_IEq497.gif

41116_2023_36_Article_IEq496.gif

Fig. 1

41116_2023_36_Article_IEq493.gif

12888_2023_4776_Article_IEq1.gif

41116_2023_36_Article_IEq490.gif

41116_2023_36_Article_IEq489.gif

41116_2023_36_Article_IEq487.gif

Fig. 4

41116_2023_36_Article_IEq483.gif

Fig. 2

Fig. 2

Fig. 1

41116_2023_36_Article_IEq468.gif

41116_2023_36_Article_IEq466.gif

41116_2023_36_Article_IEq465.gif

Fig. 1

Fig. 1

41116_2023_36_Article_IEq458.gif

41116_2023_36_Article_IEq456.gif

41116_2023_36_Article_IEq454.gif

41116_2023_36_Article_IEq452.gif

41116_2023_36_Article_IEq414.gif

41116_2023_36_Article_IEq412.gif

41116_2023_36_Article_IEq410.gif

41116_2023_36_Article_IEq408.gif

41116_2023_36_Article_IEq354.gif

41116_2023_36_Article_IEq352.gif

40517_2023_256_Article_IEq199.gif

41116_2023_36_Article_IEq278.gif

41116_2023_36_Article_IEq270.gif

41116_2023_36_Article_IEq239.gif