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176
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 413 (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.
Heuristically optimized tradeoffs: a new paradigm for power laws in the internet
, 2002
"... Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: "last mile " connection costs, and transmissi ..."
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Cited by 178 (1 self)
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Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: &quot;last mile &quot; connection costs, and transmission delays measured in hops. We also point out a similar phenomenon, anticipated in [2], in the distribution of file sizes. Our results seem to suggest that power laws tend to arise as a result of complex, multiobjective optimization.
Towards a theory of scalefree graphs: Definition, properties, and implications
 Internet Mathematics
, 2005
"... Abstract. There is a large, popular, and growing literature on “scalefree ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale ..."
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Cited by 137 (12 self)
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Abstract. There is a large, popular, and growing literature on “scalefree ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scalefree graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scalefree. This structural view can be related to previously studied graph properties such as the various notions of selfsimilarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theoremproof style of exposition, but who may be unfamiliar with the existing literature on scalefree networks. 1.
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 132 (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 123 (3 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
The Economics of Social Networks.
 In Advances in Economics and Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society.
, 2006
"... Abstract We analyze the problem of optimal monopoly pricing in social networks in order to characterize the influence of the network topology on the pricing rule. It is shown that this influence depends on the type of providers (local versus global monopoly) and of externalities (consumption versus ..."
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Cited by 118 (2 self)
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Abstract We analyze the problem of optimal monopoly pricing in social networks in order to characterize the influence of the network topology on the pricing rule. It is shown that this influence depends on the type of providers (local versus global monopoly) and of externalities (consumption versus price). We identify two situations where the monopolist does not discriminate across nodes in the network (global monopoly with consumption externalities and local monopoly with price externalities) and characterize the relevant centrality index used to discriminate among nodes in the other situations. We also analyze the robustness of the analysis with respect to changes in demand, and the introduction of bargaining between the monopolist and the consumer. JEL Classification Numbers: D85, D43, C69 Keywords: Social Networks, Monopoly Pricing, Network Externalities, Reference Price, Centrality Measures * We dedicate this paper to the memory of Toni CalvóArmengol, a gifted network theorist and a wonderful friend. We thank Coralio Ballester,
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 109 (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.
Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication
 In PKDD
, 2005
"... Abstract. How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the in and outdegree distribution; heavy tails f ..."
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Cited by 107 (25 self)
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Abstract. How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the in and outdegree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law ” (DPL). All published graph generators either fail to match several of the above properties, are very complicated to analyze mathematically, or both. Here we propose a graph generator that is mathematically tractable and matches this collection of properties. The main idea is to use a nonstandard matrix operation, the Kronecker product, to generate graphs that we refer to as “Kronecker graphs”. We show that Kronecker graphs naturally obey all the above properties; in fact, we can rigorously prove that they do so. We also provide empirical evidence showing that they can mimic very well several real graphs. 1
The Structural Cause of File Size Distributions
, 2001
"... We propose a user model that explains the shape of the distribution of file sizes in local file systems and in the World Wide Web. We examine evidence from 562 file systems, 38 web clients and 6 web servers, and find that this model is an accurate description of these systems. We compare this model ..."
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Cited by 90 (1 self)
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We propose a user model that explains the shape of the distribution of file sizes in local file systems and in the World Wide Web. We examine evidence from 562 file systems, 38 web clients and 6 web servers, and find that this model is an accurate description of these systems. We compare this model to an alternative that has been proposed, the Pareto model. Our results cast doubt on the widespread view that the distribution of file sizes is longtailed; we discuss the implications of this conclusion for proposed explanations of selfsimilarity in the Internet. Keywords: File sizes, lognormal distribution, longtailed distribution, selfsimilarity. 1.
Complex systems analysis of series of blackouts: cascading failure, critical points, and selforganization
 Chaos
, 2004
"... We give a comprehensive account of a complex systems approach to large blackouts caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics, dynamics and risk of series of blackouts with approximate global models. North American blackout data sugg ..."
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Cited by 88 (14 self)
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We give a comprehensive account of a complex systems approach to large blackouts caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics, dynamics and risk of series of blackouts with approximate global models. North American blackout data suggests that the frequency of large blackouts is governed by a power law. This result is consistent with the power system being a complex system designed and operated near criticality. The power law makes the risk of large blackouts consequential and implies the need for nonstandard risk analysis. Power system overall load relative to operating limits is a key factor affecting the risk of cascading failure. Blackout models and an abstract model of cascading failure show that there are critical transitions as load is increased. Power law behavior can be observed at these transitions. The critical loads at which blackout risk sharply increases are identifiable thresholds for cascading failure and we discuss approaches to computing the proximity to cascading failure using these thresholds. Approximating cascading failure as a branching process suggests ways to compute and monitor criticality by quantifying how much failures propagate. Inspired by concepts from selforganized criticality, we suggest that power system operating margins evolve slowly to near criticality and confirm this idea using a blackout model. Mitigation of blackout risk should take care to account for counterintuitive effects in complex selforganized critical systems. For example, suppressing small blackouts could lead the system to be operated closer to the edge and ultimately increase the risk of large blackouts. 1