Self-similarity of complex networks

By Chaoming Song, Shlomo Havlin, Hernán A. Makse. Nature, 2005

1,396 citations recorded by the source corpus.

Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as ‘scale-free’ because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. 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, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given ‘size’. 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 help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

Subfields: Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c2c7ef74173502f07666961dd8426ce1f36e28d3

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Citations in the corpus

A new approach for rescaling land cover data (2008, methodology): “The reverse process of increasing resolution (downscaling) might be achieved using principles of fractal geometry for self-similar objects (Mandelbrot 1967, 1983) by the estimation of powerlaw relationships (e.g., Grayson et al. 1997; Plotnick and Sepkoski 2001; Sole ´ et al. 1997; e.g., Song et al”
Uncovering the overlapping community structure of complex networks in nature and society (2005, neutral): “In the present case the principle of organisation (scaling) is preserved (with some speciﬁc modiﬁcations) when going to the next level in good agreement with the recent ﬁnding of the self-similarity of many complex networks [30].”
Random walks and diffusion on networks (2016, background): “For fractals without a 568 heavy-tailed degree distribution, dB = df [172].”
Extracting the multiscale backbone of complex weighted networks (2009, background): “In the ﬁrst case, nodes sharing a common attribute could be gathered together in the same class –group, community, etc.– and then substituted by a single new unit which represents the whole class in a new network representation of the system [7, 8, 9, 10].”
Network physiology reveals relations between network topology and physiological function (2012, background): “Recent studies have identified networks with complex topologie”
Self-similarity in ultrafast nonlinear optics (2007, background): “similarity are also widespread and can be found in many different settings ranging from natural branching patterns and coastlines [5], to the nodal properties of complex networks such as the world-wide web [6].”
Origins of fractality in the growth of complex networks (2005, background): “While the origin of the scale-free property can be reduced to two basic mechanisms: growth and preferential attachment, as exemplified by the Barabási-Albert model (BA model [9]), the empirical result of fractality cannot be explained only in those terms.”