Fractal dimension of complex network is an open issue. A large number of real networks are referred to as. However, it is widely believed that complex networks are not invariant or self similar under a lengthscale transformation. 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. 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. The influence of three statistical variables on self. 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. Self similarity of complex networks monday december 12, 2005 starts at 12. We demonstrate that the self similarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural.
Pdf selfsimilarity of complex networks chaoming song. The original preprint appeared as research report no. A surface such as a plane or the surface of a cylinder. Complex networks are widely used to model the structure of many complex systems in nature and society. 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.
In order to confirm the self similarity of complex networks, song et al. 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. Self similarity of complex networks december 12, 2005. 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. After analyzing a variety of real complex networks, song et al. 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. Makse 1 1 levich institute and physics department, city college of new york, new york, ny 10031, us. To unravel ubiquitous characteristics that complex networks may have in common, we adopt the clustering coefficient as the probability measure, and present a systematic analysis of various types of complex networks from the perspective of statistical selfsimilarity. Testing statistical selfsimilarity in the topology of river. Pdf on the capacity of fractal d2d social networks with. Fractals and self similarity indiana university math. Fractals and self similarity indiana university math ematics.
The reason for the self similarity property of complex network is still an open issue. Under certain conditions on the generators, the rsn model has been shown to obey mean self. Representation by complex networks of complex systems has been proved to be generally successful to describe their various features. In researching self similarity, for example, chaoming song and shlowo havlin measure the self similarity of complex networks using renormalization procedure17. Chaoming song advanced the selfsimilarity of complex networks, it used different edges of box covered the network, which obey the power law25. Uncovering the overlapping community structure of complex. Complex networks are researched by several disparate disciplines. Selfsimilarity of complex networks and hidden metric. Fractal networks are those where the exponent d b in eq. Chaoming song northeastern university, center for complex network research wednesday, march 2, 2011 4. Measures of selfsimilarity of complex networks aldo. The self similarity research of complex is just use interaction of nodes to study the micro evolving of networks. In this paper, we focus on the influence of degree, betweenness, and coreness on selfsimilarity of complex network.
Selfsimilarity of human protein interaction networks. Complex networks have been studied extensively owing to their relevance to many real systems such as the. Target control of complex networks nature communications. Selfsimilarity of complex networks and hidden metric spaces. Pdf selfsimilarity of complex networks researchgate. The possibility of a unique growth mechanism promises to uncover universal origins of collective behaviour.
Many objects in the real world, such as coastlines, are statistically selfsimilar. The selfsimilarity in the distribution of the clustering coef. In mathematics, a selfsimilar object is exactly or approximately similar to a part of itself i. Selfsimilarity and scale invariance are traditionally known as characteristics of certain geometric objects, such. Selfsimilarity and scaling theory of complex networks dr. In respect of self similarity of the general complex systems to which biological networks belong, the work of song et al 6 is seminal. 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.
For instance, a side of the koch snowflake is both symmetrical and scaleinvariant. Pdf selfsimilarity of complex networks hernan makse. 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. Nsh 202 network has become a key approach to understanding systems of interacting objects, unifying diverse phenomena including. In particular, the emergence of selfsimilarity in complex networks raises the fundamental question of. Specifically, the fractal networks are characterized by the direct social connection and the selfsimilarity. Modeling the selfsimilarity in complex networks based on.
Newly won evidence shows that many realworld network systems obey a powerlaw scaling, just as if they were fractal shapes. They analysed a variety of real complex networks and found that these systems consist of self repeating patterns. Although the traditional fractal theory does not distinguish between fractality and selfsimilarity, in complex networks these two properties can be considered to be distinct. Self similarity and scaling theory of complex networks dr. Makse1 1 levich institute and physics department, city. In the next section, some complex networks are investigated and the results show that the minimum number of the boxes used to cover the networks and the box size follows a power law rule, which reflects the self similarity of the complex networks.
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. Abstract complex networks are widely used to model the structure of many complex systems in nature and society. Ltww94 lan traffic is consistent with self similarity as well as wan traffic pf95 and web traffic cb96. Statistical selfsimilar properties of complex networks. 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. Specifically, the fractal networks are characterized by the direct social connection and the self similarity. Pdf selfsimilarity of complex networks chaoming song academia. Recently, complex networks have attracted much attention in diverse areas. 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. Modelling the selfsimilarity in complex networks based on.
Although proposed the complex network has selfsimilarity, they havent. Some nodes are removed from the original network based on the definitions of degree, betweenness, and coreness in the ascending and descending order. Fractal networks are those where the exponentdb in eq. Target control of complex networks pubmed central pmc. Selfsimilarity of complex networks article pdf available in nature 4337024. It has been shown that the smallworld property and the scalefree property are the two. Selfsimilarity networks and selfsimilarity network group. Surprisingly, this selfsimilarity under different length scales seems to be a more general feature that also applies in nonfractal networks such as the internet.
This result comes as a surprise, because the exponential. Selfsimilarity of complex networks chaoming song1, shlomo havlin2, and hern. Selfsimilarity of complex networks university of miami. Origins of fractality in the growth of complex networks. 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. Testing statistical selfsimilarity in the topology of. 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. In this paper, we focus on the influence of degree, betweenness, and coreness on self similarity of complex network. Selfsimilarity of carbon market from complex network. The selfsimilarity research of complex is just use interaction of nodes to study the micro evolving of networks. The maximum capacity of fractal d2d devicetodevice social networks with both direct and hierarchical communications is studied in this paper.
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. 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. 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. The reason is that many complex networks are not explicitly embedded in any physical space. Hub repulsion plays an important role in fractal topologies. Boxcovering algorithm for fractal dimension of complex. Network science is widely used in many academic research. Recently, fractal and selfsimilarity of complex networks have attracted much attention. We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural. Could this be the harbinger of a new architectural law for complex systems. In researching selfsimilarity, for example, chaoming song and shlowo havlin measure the selfsimilarity of complex networks using renormalization procedure17. 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.
Selfsimilarity of carbon market from complex network analysis. 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. The reason for the selfsimilarity property of complex network is still an open issue. 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. It is observed that hub repulsion is the key principle that leads to the fractal structure of networks. 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. 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. Request pdf selfsimilarity of complex networks and hidden metric spaces we demonstrate that the selfsimilarity of some scalefree networks with respect to a. The reason is that many complex networks are not explicitly. Here we introduce a boxcovering algorithm that outperforms previous ones in most cases.
Recently, fractal and self similarity of complex networks have attracted much attention. 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. 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. Selfsimilarity of complex networks university of miamis. 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. 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. Recently, selfsimilarity of complex networks have attracted much attention.
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