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272
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 1407 (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 268 (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
The Average Distance in a Random Graph with Given Expected Degrees
"... Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d wher ..."
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Cited by 191 (13 self)
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Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d where ˜ d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1/k β for some fixed exponent β. For the case of β> 3, we prove that the average distance of the power law graphs is almost surely of order log n / log ˜ d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having n c / log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.
Random graph models of social networks
"... We describe some new exactly solvable models of the structure of social networks, based on random graphs with arbitrary degree distributions. We give models both for simple unipartite networks, such as acquaintance networks, and bipartite networks, such as affiliation networks. We compare the predic ..."
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Cited by 141 (1 self)
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We describe some new exactly solvable models of the structure of social networks, based on random graphs with arbitrary degree distributions. We give models both for simple unipartite networks, such as acquaintance networks, and bipartite networks, such as affiliation networks. We compare the predictions of our models to data for a number of realworld social networks and find that in some cases the models are in remarkable agreement with the data, while in others the agreement is poorer, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.
Analysis of Topological Characteristics of Huge Online Social Networking Services
 In Proceedings of the 16th international conference on World Wide Web (WWW’07
, 2007
"... Abstract — Social networking services are a fastgrowing business in the Internet. However, it is unknown if online relationships and their growth patterns are the same as in reallife social networks. In this paper, we compare the structures of three online social networking services: Cyworld, MySp ..."
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Cited by 122 (5 self)
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Abstract — Social networking services are a fastgrowing business in the Internet. However, it is unknown if online relationships and their growth patterns are the same as in reallife social networks. In this paper, we compare the structures of three online social networking services: Cyworld, MySpace, and orkut, each with more than 10 million users, respectively. We have access to complete data of Cyworld’s ilchon (friend) relationships and analyze its degree distribution, clustering property, degree correlation, and evolution over time. We also use Cyworld data to evaluate the validity of snowball sampling method, which we use to crawl and obtain partial network topologies of MySpace and orkut. Cyworld, the oldest of the three, demonstrates a changing scaling behavior over time in degree distribution. The latest Cyworld data’s degree distribution exhibits a multiscaling behavior, while those of MySpace and orkut have simple scaling behaviors with different exponents. Very interestingly, each of the two exponents corresponds to the different segments in Cyworld’s degree distribution. Certain online social networking services encourage online activities that cannot be easily copied in real life; we show that they deviate from closeknit online social networks which show a similar degree correlation pattern to reallife social networks. I.
Coauthorship networks and patterns of scientific collaboration
 In Proceedings of the National Academy of Sciences
, 2004
"... Using data from three bibliographic databases in biology, physics, and mathematics respectively, networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper together. We use these networks to answer a broad variety of questions about co ..."
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Cited by 121 (0 self)
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Using data from three bibliographic databases in biology, physics, and mathematics respectively, networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper together. We use these networks to answer a broad variety of questions about collaboration patterns, such as the numbers of papers authors write, how many people they write them with, what the typical distance between scientists is through the network, and how patterns of collaboration vary between subjects and over time. We also summarize a number of recent results by other authors on coauthorship patterns. 1
The phase transition in inhomogeneous random graphs, preprint available from http://www.arxiv.org/abs/math.PR/0504589
"... Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. ..."
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Cited by 101 (30 self)
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Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with
Random Evolution in Massive Graphs
, 2001
"... Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for generating p ..."
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Cited by 89 (7 self)
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Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for generating power law graphs by adding one node/edge at a time. We will show that for any given edge density and desired distributions for indegrees and outdegrees (not necessarily the same, but adhered to certain general conditions), the resulting graph will almost surely satisfy the power law and the in/outdegree conditions. We will show that our most general directed and undirected models include nearly all known models as special cases. In addition, we consider another crucial aspects of massive graphs that is called "scalefree" in the sense that the f requency of sampling (w.r.t. the growth rate) is independent of the parameter of the resulting power law graphs. We will show that our evolution models generate scalefree power law graphs. 1
Characterization of complex networks: A survey of measurements
 Advances in Physics
"... Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of mea ..."
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Cited by 89 (7 self)
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Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements organized into classes. Special attention is given to relating complex network analysis with the areas of pattern recognition and feature selection, as well as on surveying some concepts and measurements from traditional graph theory which are potentially useful for complex network research. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the
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 70 (6 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.