%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 }