Fri, 25 Jul 2008 15:26:13 BST CiteULike: bigbossman self-similarity http://www.citeulike.org/user/bigbossman/tag/self-similarity CiteULike.org en-gb Copyright © 2004-2008 citeulike.org http://www.citeulike.org/user/bigbossman/article/2197822 <i>(10 Oct 2007)</i><br /><br />We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization. Self-similarity of complex networks and hidden metric spaces Angeles Serrano Dmitri Krioukov Marian Boguna (10 Oct 2007) 2008-01-05T19:09:07-00:00 2007 complex metrics networks self-similarity http://www.citeulike.org/user/bigbossman/article/977167 <i>(3 Mar 2005)</i><br /><br />Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \citeab-review,mendes,vespignani,newman,amaral. A large number of real networks are called “scale-free” because they show a power-law distribution of the number of links per node \citeab-review,barabasi1999,faloutsos. However, it is widely believed that complex networks are not length-scale invariant or self-similar. This conclusion originates from the “small-world” property of these networks, which implies that the number of nodes increases exponentially with the “diameter” of the network \citeerdos,bollobas,milgram,watts, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena. Self-similarity of complex networks Chaoming Song Shlomo Havlin Hernan Makse (3 Mar 2005) 2006-12-06T22:32:03-00:00 2005 fractal self-similarity