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255
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 2591 (7 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.
Power laws, Pareto distributions and Zipf’s law
"... Many of the things that scientists measure have a typical size or “scale”—a typical value around which individual measurements are centred. A simple example would be the heights of human beings. Most adult human beings are about 180cm tall. There is some variation around this figure, notably dependi ..."
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Cited by 392 (0 self)
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Many of the things that scientists measure have a typical size or “scale”—a typical value around which individual measurements are centred. A simple example would be the heights of human beings. Most adult human beings are about 180cm tall. There is some variation around this figure, notably depending on sex, but we never see people who are 10cm tall, or 500cm. To make this observation more quantitative, one can plot a histogram of people’s heights, as I have done in Fig. 1a. The figure shows the heights in centimetres of adult men in the United States measured between 1959 and 1962, and indeed the distribution is relatively narrow and peaked around 180cm. Another telling observation is the ratio of the heights of the tallest and shortest people.
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 251 (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.
Models of the small world
 J. Stat. Phys
, 2000
"... It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the ``six degrees of separation,' ' has been the subject of considerable rece ..."
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Cited by 132 (1 self)
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It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the ``six degrees of separation,' ' has been the subject of considerable recent interest within the physics community. This paper provides a short review of the topic. KEY WORDS: social networks. Small world; networks; disordered systems; graph theory;
Graph mining: laws, generators, and algorithms
 ACM COMPUT SURV (CSUR
, 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 in ..."
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Cited by 131 (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.
Kronecker Graphs: An Approach to Modeling Networks
 JOURNAL OF MACHINE LEARNING RESEARCH 11 (2010) 9851042
, 2010
"... How can we generate realistic networks? In addition, how can we do so with a mathematically tractable model that allows for rigorous analysis of network properties? Real networks exhibit a long list of surprising properties: Heavy tails for the in and outdegree distribution, heavy tails for the ei ..."
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Cited by 126 (4 self)
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How can we generate realistic networks? In addition, how can we do so with a mathematically tractable model that allows for rigorous analysis of network properties? Real networks exhibit a long list of surprising properties: Heavy tails for the in and outdegree distribution, heavy tails for the eigenvalues and eigenvectors, small diameters, and densification and shrinking diameters over time. Current network models and generators either fail to match several of the above properties, are complicated to analyze mathematically, or both. Here we propose a generative model for networks that is both mathematically tractable and can generate networks that have all the above mentioned structural properties. Our main idea here is to use a nonstandard matrix operation, the Kronecker product, to generate graphs which we refer to as “Kronecker graphs”. First, we show that Kronecker graphs naturally obey common network properties. In fact, we rigorously prove that they do so. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. We then present KRONFIT, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take superexponential
PowerLaws and the ASlevel Internet Topology
 IEEE/ACM Transactions on Networking
, 2003
"... In this paper, we study and characterize the topology of the Internet at the Autonomous System level. First, we show that the topology can be described efficiently with powerlaws. The elegance and simplicity of the powerlaws provide a novel perspective into the seemingly uncontrolled Internet struc ..."
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Cited by 106 (11 self)
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In this paper, we study and characterize the topology of the Internet at the Autonomous System level. First, we show that the topology can be described efficiently with powerlaws. The elegance and simplicity of the powerlaws provide a novel perspective into the seemingly uncontrolled Internet structure. Second, we show that powerlaws appear consistently over the last 5 years. We also observe that the powerlaws hold even in the most recent and more complete topology [10] with correlation coefficient above 99% for the degree powerlaw. In addition, we study the evolution of the powerlaw exponents over the 5 year interval and observe a variation for the degree based powerlaw of less than 10%. Third, we provide relationships between the exponents and other topological metrics.
SmallWorld FileSharing Communities
, 2003
"... Web caches, content distribution networks, peertopeer file sharing networks, distributed file systems, and data grids all have in common that they involve a community of users who generate requests for shared data. In each case, overall system performance can be improved significantly if we can fi ..."
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Cited by 90 (10 self)
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Web caches, content distribution networks, peertopeer file sharing networks, distributed file systems, and data grids all have in common that they involve a community of users who generate requests for shared data. In each case, overall system performance can be improved significantly if we can first identify and then exploit interesting structure within a community's access patterns. To this end, we propose a novel perspective on file sharing based on the study of the relationships that form among users based on the files in which they are interested. We propose a new structure that captures common user interests in datathe datasharing graph and justify its utility with studies on three datadistribution systems: a highenergy physics collaboration, the Web, and the Kazaa peertopeer network. We find smallworld patterns in the datasharing graphs of all three communities. We analyze these graphs and propose some probable causes for these emergent smallworld patterns. The significance of smallworld patterns is twofold: it provides a rigorous support to intuition and, perhaps most importantly, it suggests ways to design mechanisms that exploit these naturally emerging patterns.
THE SHAPE OF PRODUCTION FUNCTIONS AND THE DIRECTION OF TECHNICAL CHANGE
, 2004
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Scalable modeling of real graphs using Kronecker multiplication
 IN 24TH ICML
, 2007
"... Given a large, real graph, how can we generate a synthetic graph that matches its properties, i.e., it has similar degree distribution, similar (small) diameter, similar spectrum, etc? We propose to use “Kronecker graphs”, which naturally obey all of the above properties, and we present KronFit, a f ..."
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Cited by 76 (9 self)
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Given a large, real graph, how can we generate a synthetic graph that matches its properties, i.e., it has similar degree distribution, similar (small) diameter, similar spectrum, etc? We propose to use “Kronecker graphs”, which naturally obey all of the above properties, and we present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to real networks. A naive approach to fitting would take superexponential time. In contrast, KronFit takes linear time, by exploiting the structure of Kronecker product and by using sampling. Experiments on large real and synthetic graphs show that KronFit indeed mimics very well the patterns found in the target graphs. Once fitted, the model parameters and the resulting synthetic graphs can be used for anonymization, extrapolations, and graph summarization.