Self-similarity of complex networks pdf

Selfsimilarity of human protein interaction networks. Pdf selfsimilarity of complex networks hernan makse. Some scientists interpret network science as a new paradigm shift 32, however, complex networks did not only in uence the research community, but also appeared in popular literature 3, 67 and mass media 7. Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of 1,2,3,4. Nsh 202 network has become a key approach to understanding systems of interacting objects, unifying diverse phenomena including. Selfsimilarity of complex networks and hidden metric spaces. The possibility of a unique growth mechanism promises to uncover universal origins of collective behaviour. Recently, we introduced an analytical framework to study the controllability of complex networks via a combination of tools from network science, control theory and statistical physics12. Many objects in the real world, such as coastlines, are statistically selfsimilar. This result comes as a surprise, because the exponential increase in equation 1 has led to the general understanding that complex networks are not self similar, since self similarity requires a powerlaw relation between n and l. Recently, complex networks have attracted much attention in diverse areas.

Request pdf selfsimilarity of complex networks and hidden metric spaces we demonstrate that the selfsimilarity of some scalefree networks with respect to a. Abstract complex networks are widely used to model the structure of many complex systems in nature and society. By mapping the structural controllability problem to a maximum matching problem 14, we were able to identify a minimum set of driver nodes of size n d. Pdf selfsimilarity of complex networks chaoming song academia. Self similarity and scaling theory of complex networks dr. Makse 1 1 levich institute and physics department, city college of new york, new york, ny 10031, us. Uncovering the overlapping community structure of complex.

Fractal dimension of complex network is an open issue. Complex networks have been studied extensively owing to their relevance to many real systems such as the. Scale invariance is an exact form of self similarity where at any magnification there is a smaller piece of the object that is similar to the whole. Modeling the selfsimilarity in complex networks based on. Origins of fractality in the growth of complex networks. Hutchinson this is a retyped texd version of the article from indiana university mathematics journal 30 1981, 7747 with some minor formatting changes, a few old \typos corrected, and hopefully few new ones introduced. In respect of self similarity of the general complex systems to which biological networks belong, the work of song et al 6 is seminal. Recently, fractal and self similarity of complex networks have attracted much attention. For instance, a side of the koch snowflake is both symmetrical and scaleinvariant. In mathematics, a selfsimilar object is exactly or approximately similar to a part of itself i. This result comes as a surprise, because the exponential increase in equation 1 has led to the general understanding that complex networks are not selfsimilar, since selfsimilarity requires a powerlaw relation between n and l. Selfsimilarity of complex networks university of miami. Although the traditional fractal theory does not distinguish between fractality and selfsimilarity, in complex networks these two properties can be considered to be distinct.

Recently, fractal and selfsimilarity of complex networks have attracted much attention. The selfsimilarity in the distribution of the clustering coef. Here we introduce a boxcovering algorithm that outperforms previous ones in most cases. Selfsimilarity networks and selfsimilarity network group. Target control of complex networks pubmed central pmc. Jan 27, 2005 complex networks have been studied extensively owing to their relevance to many real systems such as the worldwide web, the internet, energy landscapes and biological and social networks 1,2,3,4. The selfsimilarity of complex networks is typically investigated through computational algorithms, the primary task of which is to cover the structure with a minimal number of boxes.

We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural. Some nodes are removed from the original network based on the definitions of degree, betweenness, and coreness in the ascending and descending order. Thus a line has a dimension of one 1d because only one coordinate is needed to specify a point on it for example, the point at 5 on a number line. Complex networks are researched by several disparate disciplines. We demonstrate that the self similarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Complex networks are widely used to model the structure of many complex systems in nature and society. This result comes as a surprise, because the exponential. It has been shown that the smallworld property and the scalefree property are the two. Newly won evidence shows that many realworld network systems obey a powerlaw scaling, just as if they were fractal shapes. Pdf on the capacity of fractal d2d social networks with. The goal of this research work is to propose two measures of selfsimilarity of complex networks based on the geometric measure of fractal dimension named dsummability. Makse1 1 levich institute and physics department, city college of new york, new york, ny 10031, us 2 minerva center and department of physics, barilan university, ramat gan 52900, israel dated. Testing statistical selfsimilarity in the topology of.

In this paper, we focus on the influence of degree, betweenness, and coreness on self similarity of complex network. Under certain conditions on the generators, the rsn model has been shown to obey mean self. In researching selfsimilarity, for example, chaoming song and shlowo havlin measure the selfsimilarity of complex networks using renormalization procedure17. We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. 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. Jan 27, 2005 a large number of real networks are referred to as scalefree because they show a powerlaw distribution of the number of links per node. Selfsimilarity of carbon market from complex network analysis. Measures of selfsimilarity of complex networks aldo. Pdf selfsimilarity of complex networks chaoming song. Representation by complex networks of complex systems has been proved to be generally successful to describe their various features.

The reason is that many complex networks are not explicitly. In physics and mathematics, the dimension of a mathematical space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Could this be the harbinger of a new architectural law for complex systems. Network science is widely used in many academic research. After analyzing a variety of real complex networks, song et al. In researching self similarity, for example, chaoming song and shlowo havlin measure the self similarity of complex networks using renormalization procedure17. Specifically, the fractal networks are characterized by the direct social connection and the selfsimilarity. Ltww94 lan traffic is consistent with self similarity as well as wan traffic pf95 and web traffic cb96. The reason for the self similarity property of complex network is still an open issue. A surface such as a plane or the surface of a cylinder. Selfsimilarity of complex networks and hidden metric.

Surprisingly, this selfsimilarity under different length scales seems to be a more general feature that also applies in nonfractal networks such as the internet. The original preprint appeared as research report no. The self similarity research of complex is just use interaction of nodes to study the micro evolving of networks. Surprisingly, this self similarity under different length scales seems to be a more general feature that also applies in nonfractal networks such as the internet. Selfsimilarity and scaling theory of complex networks dr. Selfsimilarity and scale invariance are traditionally known as characteristics of certain geometric objects, such. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the selfsimilarity properties that we measured in the real networks. Boxcovering algorithm for fractal dimension of complex. In order to confirm the self similarity of complex networks, song et al.

It is observed that hub repulsion is the key principle that leads to the fractal structure of networks. We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden. Fractals and self similarity indiana university math ematics. Jan 27, 2005 this result comes as a surprise, because the exponential increase in equation 1 has led to the general understanding that complex networks are not self similar, since self similarity requires a. In particular, the emergence of selfsimilarity in complex networks raises the fundamental question of. Pdf selfsimilarity of complex networks researchgate.

In the past decade network science offered deep insights into the structure and dynamics of complex networked systems 1,2,3,4,5,6,7. A large number of real networks are referred to as. The reason for the selfsimilarity property of complex network is still an open issue. Statistical selfsimilar properties of complex networks. Makse1 1 levich institute and physics department, city. Chaoming song advanced the selfsimilarity of complex networks, it used different edges of box covered the network, which obey the power law25. Specifically, the fractal networks are characterized by the direct social connection and the self similarity. Modelling the selfsimilarity in complex networks based on. Although proposed the complex network has selfsimilarity, they havent.

Selfsimilarity of carbon market from complex network. Complex networks have been studied extensively owing to their relevance to many real systems such as the worldwide web, the internet, energy landscapes and biological and social networks 1,2,3,4. The maximum capacity of fractal d2d devicetodevice social networks with both direct and hierarchical communications is studied in this paper. The reason is that many complex networks are not explicitly embedded in any physical space. Self similarity of complex networks monday december 12, 2005 starts at 12. Selfsimilarity of complex networks university of miamis.

The influence of three statistical variables on self. N2 complex networks have been studied extensively owing to their relevance to many real systems such as the worldwide web, the internet, energy landscapes and biological and social networks. The selfsimilarity research of complex is just use interaction of nodes to study the micro evolving of networks. We demonstrate that the self similarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural. Fractal networks are those where the exponent d b in eq. Although the traditional fractal theory does not distinguish between fractality and self similarity, in complex networks these two properties can be considered to be distinct. Self similarity of complex networks december 12, 2005. Fractal networks are those where the exponentdb in eq. However, it is widely believed that complex networks are not invariant or self similar under a lengthscale transformation. Fractals and self similarity indiana university math. Selfsimilarity of complex networks article pdf available in nature 4337024.