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350
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 2600 (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.
Mining the Network Value of Customers
 In Proceedings of the Seventh International Conference on Knowledge Discovery and Data Mining
, 2002
"... One of the major applications of data mining is in helping companies determine which potential customers to market to. If the expected pro t from a customer is greater than the cost of marketing to her, the marketing action for that customer is executed. So far, work in this area has considered only ..."
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Cited by 568 (11 self)
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One of the major applications of data mining is in helping companies determine which potential customers to market to. If the expected pro t from a customer is greater than the cost of marketing to her, the marketing action for that customer is executed. So far, work in this area has considered only the intrinsic value of the customer (i.e, the expected pro t from sales to her). We propose to model also the customer's network value: the expected pro t from sales to other customers she may inuence to buy, the customers those may inuence, and so on recursively. Instead of viewing a market as a set of independent entities, we view it as a social network and model it as a Markov random eld. We show the advantages of this approach using a social network mined from a collaborative ltering database. Marketing that exploits the network value of customersalso known as viral marketingcan be extremely eective, but is still a black art. Our work can be viewed as a step towards providing a more solid foundation for it, taking advantage of the availability of large relevant databases. Categories and Subject Descriptors H.2.8 [Database Management]: Database Applications data mining
A Random Graph Model for Massive Graphs
 STOC 2000
, 2000
"... We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from t ..."
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Cited by 406 (26 self)
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We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.
Mining KnowledgeSharing Sites for Viral Marketing
, 2002
"... Viral marketing takes advantage of networks of influence among customers to inexpensively achieve large changes in behavior. Our research seeks to put it on a firmer footing by mining these networks from data, building probabilistic models of them, and using these models to choose the best viral mar ..."
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Cited by 368 (8 self)
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Viral marketing takes advantage of networks of influence among customers to inexpensively achieve large changes in behavior. Our research seeks to put it on a firmer footing by mining these networks from data, building probabilistic models of them, and using these models to choose the best viral marketing plan. Knowledgesharing sites, where customers review products and advise each other, are a fertile source for this type of data mining. In this paper we extend our previous techniques, achieving a large reduction in computational cost, and apply them to data from a knowledgesharing site. We optimize the amount of marketing funds spent on each customer, rather than just making a binary decision on whether to market to him. We take into account the fact that knowledge of the network is partial, and that gathering that knowledge can itself have a cost. Our results show the robustness and utility of our approach.
The degree sequence of a scalefree random graph process
, 2001
"... Recently, Barabási and Albert [2] suggested modeling complex realworld networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional ..."
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Cited by 243 (2 self)
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Recently, Barabási and Albert [2] suggested modeling complex realworld networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and Albert suggested that after many steps the proportion Pd of vertices with degree d should obey a power law Pdαd−γ. They obtained γ = 29 ± 01 by experiment and gave a simple heuristic argument suggesting that γ = 3. Here we obtain Pd asymptotically for all d ≤ n1/15, where n is the number of vertices, proving as a consequence that γ = 3.
Deeper Inside PageRank
 INTERNET MATHEMATICS
, 2004
"... This paper serves as a companion or extension to the “Inside PageRank” paper by Bianchini et al. [Bianchini et al. 03]. It is a comprehensive survey of all issues associated with PageRank, covering the basic PageRank model, available and recommended solution methods, storage issues, existence, uniq ..."
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Cited by 208 (4 self)
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This paper serves as a companion or extension to the “Inside PageRank” paper by Bianchini et al. [Bianchini et al. 03]. It is a comprehensive survey of all issues associated with PageRank, covering the basic PageRank model, available and recommended solution methods, storage issues, existence, uniqueness, and convergence properties, possible alterations to the basic model, suggested alternatives to the traditional solution methods, sensitivity and conditioning, and finally the updating problem. We introduce a few new results, provide an extensive reference list, and speculate about exciting areas of future research.
Connected Components in Random Graphs with Given Expected Degree Sequences
 ANNALS OF COMBINATORICS
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The phase transition in inhomogeneous random graphs
, 2005
"... 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 ..."
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Cited by 180 (29 self)
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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
Complex contagion and the weakness of long ties
, 2005
"... Complex Contagion and the Weakness of Long Ties The strength of weak ties is that they tend to be long – they connect socially distant locations. Recent research on “small worlds ” shows that remarkably few long ties are needed to give large and highly clustered populations the “degrees of separatio ..."
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Cited by 165 (9 self)
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Complex Contagion and the Weakness of Long Ties The strength of weak ties is that they tend to be long – they connect socially distant locations. Recent research on “small worlds ” shows that remarkably few long ties are needed to give large and highly clustered populations the “degrees of separation ” of a random network, in which information can rapidly diffuse. We test whether this effect of long ties generalizes from simple to complex contagions – those in which the credibility of information or the willingness to adopt an innovation requires independent confirmation from multiple sources. Using Watts and Strogatz’s original small world model, we demonstrate that long ties may not only fail to speed up complex contagions, they can even preclude diffusion entirely. Results suggest that the spread of collective actions, social movements, and risky innovations benefit not from ties that are long but from bridges that are wide enough to transmit strong social reinforcement. Balance theory shows how wide bridges might also form in evolving networks, but this turns out to have surprisingly little effect on the propagation of complex contagions. We find that
Mathematical results on scalefree random graphs
 Handbook of Graphs and Networks
, 2003
"... Recently there has been much interest in studying largescale realworld networks and attempting to model their properties using random graphs. Although the study of realworld networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ..."
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Cited by 147 (3 self)
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Recently there has been much interest in studying largescale realworld networks and attempting to model their properties using random graphs. Although the study of realworld networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ‘smallworld