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196
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1415 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Evolution of networks
 Adv. Phys
, 2002
"... We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence rece ..."
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Cited by 269 (2 self)
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We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scalefree and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect. We discuss how growing networks selforganize into scalefree structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems
A Brief History of Generative Models for Power Law and Lognormal Distributions
 INTERNET MATHEMATICS
"... Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying ..."
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Cited by 253 (7 self)
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Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying
Stochastic Models for the Web Graph
, 2000
"... The web may be viewed as a directed graph each of whose vertices is a static HTML web page, and each of whose edges corresponds to a hyperlink from one web page to another. In this paper we propose and analyze random graph models inspired by a series of empirical observations on the web. Our graph m ..."
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Cited by 214 (10 self)
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The web may be viewed as a directed graph each of whose vertices is a static HTML web page, and each of whose edges corresponds to a hyperlink from one web page to another. In this paper we propose and analyze random graph models inspired by a series of empirical observations on the web. Our graph models differ from the traditional Gn;p models in two ways: 1. Independently chosen edges do not result in the statistics (degree distributions, clique multitudes) observed on the web. Thus, edges in our model are statistically dependent on each other. 2. Our model introduces new vertices in the graph as time evolves. This captures the fact that the web is changing with time. Our results are two fold: we show that graphs generated using our model exhibit the statistics observed on the web graph, and additionally, that natural graph models proposed earlier do not exhibit them. This remains true even when these earlier models are generalized to account for the arrival of vertices over time. In particular, the sparse random graphs in our models exhibit properties that do not arise in far denser random graphs generated by ErdosR'enyi models.
Power laws, Pareto distributions and Zipf’s law
 Contemporary Physics
, 2005
"... When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, econ ..."
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Cited by 176 (0 self)
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When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people’s personal fortunes all appear to follow power laws. The origin of powerlaw behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of powerlaw forms and the theories proposed to explain them. I.
The LargeScale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth
 Cognitive Science
"... We present statistical analyses of the largescale structure of three types of semantic networks: word associations, WordNet, and Roget's thesaurus. We show that they have a smallworld structure, characterized by sparse connectivity, short average pathlengths between words, and strong local clu ..."
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Cited by 122 (1 self)
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We present statistical analyses of the largescale structure of three types of semantic networks: word associations, WordNet, and Roget's thesaurus. We show that they have a smallworld structure, characterized by sparse connectivity, short average pathlengths between words, and strong local clustering. In addition, the distributions of the number of connections follow power laws that indicate a scalefree pattern of connectivity, with most nodes having relatively few connections joined together through a small number of hubs with many connections. These regularities have also been found in certain other complex natural networks, such as the world wide web, but they are not consistent with many conventional models of semantic organization, based on inheritance hierarchies, arbitrarily structured networks, or highdimensional vector spaces. We propose that these structures reflect the mechanisms by which semantic networks grow. We describe a simple model for semantic growth, in which each new word or concept is connected to an existing network by differentiating the connectivity pattern of an existing node. This model generates appropriate smallworld statistics and powerlaw connectivity distributions, and also suggests one possible mechanistic basis for the effects of learning history variables (ageofacquisition, usage frequency) on behavioral performance in semantic processing tasks.
The Small World of Human Language
 Proceedings of The Royal Society of London. Series B, Biological Sciences
, 2001
"... this paper, we show that the cooccurrence of words in sentences relies on the network structure of the lexicon, the properties of which are analysed in depth. As we will show in this paper, human language can be described in terms of a graph of word interactions. This graph has some unexpected prop ..."
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Cited by 81 (5 self)
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this paper, we show that the cooccurrence of words in sentences relies on the network structure of the lexicon, the properties of which are analysed in depth. As we will show in this paper, human language can be described in terms of a graph of word interactions. This graph has some unexpected properties (shared by other biological and technological networks (Amaral et al. 2000; Strogatz 2001)) that might underlie its diversity and exibility, and create new questions about its origins and organization
Interpolating between types and tokens by estimating powerlaw generators
 In Advances in Neural Information Processing Systems 18
, 2006
"... Standard statistical models of language fail to capture one of the most striking properties of natural languages: the powerlaw distribution in the frequencies of word tokens. We present a framework for developing statistical models that generically produce powerlaws, augmenting standard generative ..."
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Cited by 76 (13 self)
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Standard statistical models of language fail to capture one of the most striking properties of natural languages: the powerlaw distribution in the frequencies of word tokens. We present a framework for developing statistical models that generically produce powerlaws, augmenting standard generative models with an adaptor that produces the appropriate pattern of token frequencies. We show that taking a particular stochastic process – the PitmanYor process – as an adaptor justifies the appearance of type frequencies in formal analyses of natural language, and improves the performance of a model for unsupervised learning of morphology. 1
Graph mining: Laws, generators, and algorithms
 ACM COMPUTING SURVEYS
, 2006
"... How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation i ..."
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Cited by 72 (7 self)
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How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: "How can we generate synthetic but realistic graphs?" To answer this, we must first understand what patterns are common in realworld graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.