Fractal scaling and self-similar connectivity behaviour of scale-free
(SF) networks are reviewed and investigated in diverse aspects. We
first recall an algorithm of box-covering that is useful and easy to
implement in SF networks, the so-called random sequential
box-covering. Next, to understand the origin of the fractal scaling,
fractal networks are viewed as comprising of a skeleton and shortcuts.
The skeleton, embedded underneath the original network, is a spanning
tree specifically based on the edge-betweenness centrality or load. We
show that the skeleton is a non-causal tree, either critical or
supercritical. We also study the fractal scaling property of the
k-core of a fractal network and find that as k increases, not only
does the fractal dimension of the k-core change but also eventually
the fractality no longer holds for large enough k. Finally, we study
the self-similarity, manifested as the scale-invariance of the degree
distribution under coarse-graining of vertices by the box-covering
method. We obtain the condition for self-similarity, which turns out
to be independent of the fractality, and find that some non-fractal
networks are self-similar. Therefore, fractality and self-similarity
are disparate notions in SF networks.
%0 Journal Article
%1 Kim_2007
%A Kim, J S
%A Goh, K-I
%A Kahng, B
%A Kim, D
%D 2007
%K fractals, networkx, renormalization self_similarity,
%N 177
%T Fractality and self-similarity in scale-free networks
%V 9
%X Fractal scaling and self-similar connectivity behaviour of scale-free
(SF) networks are reviewed and investigated in diverse aspects. We
first recall an algorithm of box-covering that is useful and easy to
implement in SF networks, the so-called random sequential
box-covering. Next, to understand the origin of the fractal scaling,
fractal networks are viewed as comprising of a skeleton and shortcuts.
The skeleton, embedded underneath the original network, is a spanning
tree specifically based on the edge-betweenness centrality or load. We
show that the skeleton is a non-causal tree, either critical or
supercritical. We also study the fractal scaling property of the
k-core of a fractal network and find that as k increases, not only
does the fractal dimension of the k-core change but also eventually
the fractality no longer holds for large enough k. Finally, we study
the self-similarity, manifested as the scale-invariance of the degree
distribution under coarse-graining of vertices by the box-covering
method. We obtain the condition for self-similarity, which turns out
to be independent of the fractality, and find that some non-fractal
networks are self-similar. Therefore, fractality and self-similarity
are disparate notions in SF networks.
@article{Kim_2007,
abstract = {Fractal scaling and self-similar connectivity behaviour of scale-free
(SF) networks are reviewed and investigated in diverse aspects. We
first recall an algorithm of box-covering that is useful and easy to
implement in SF networks, the so-called random sequential
box-covering. Next, to understand the origin of the fractal scaling,
fractal networks are viewed as comprising of a skeleton and shortcuts.
The skeleton, embedded underneath the original network, is a spanning
tree specifically based on the edge-betweenness centrality or load. We
show that the skeleton is a non-causal tree, either critical or
supercritical. We also study the fractal scaling property of the
k-core of a fractal network and find that as k increases, not only
does the fractal dimension of the k-core change but also eventually
the fractality no longer holds for large enough k. Finally, we study
the self-similarity, manifested as the scale-invariance of the degree
distribution under coarse-graining of vertices by the box-covering
method. We obtain the condition for self-similarity, which turns out
to be independent of the fractality, and find that some non-fractal
networks are self-similar. Therefore, fractality and self-similarity
are disparate notions in SF networks.
},
added-at = {2010-05-10T08:12:01.000+0200},
author = {Kim, J S and Goh, K-I and Kahng, B and Kim, D},
biburl = {https://www.bibsonomy.org/bibtex/2d4d141195b0518ca67be21d8e6601437/dhruvbansal},
file = {/home/dhruv/projects/work/papers/papers/Kim_2007.pdf},
interhash = {f71a26df315136d857ad23ec705c30d8},
intrahash = {d4d141195b0518ca67be21d8e6601437},
keywords = {fractals, networkx, renormalization self_similarity,},
month = {June},
number = 177,
read = {nil},
timestamp = {2010-05-10T08:12:04.000+0200},
title = {Fractality and self-similarity in scale-free networks},
volume = 9,
year = 2007
}