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13
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
LikelihoodBased Inference for Stochastic Models of Sexual Network Formation
 Popul. Biol
, 2004
"... SexuallyTransmitted Diseases (STDs) constitute a major public health concern. Mathematical models for the transmission dynamics of STDs indicate that heterogeneity in sexual activity level allow them to persist even when the typical behavior of the population would not support endemicity. This insi ..."
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Cited by 22 (5 self)
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SexuallyTransmitted Diseases (STDs) constitute a major public health concern. Mathematical models for the transmission dynamics of STDs indicate that heterogeneity in sexual activity level allow them to persist even when the typical behavior of the population would not support endemicity. This insight focuses attention on the distribution of sexual activity level in a population. In this paper, we develop several stochastic process models for the f'ormation of sexual partnership networks. Using likelihoodbased model selection procedures, we assess the fit of the different models to three large distributions of sexual partner counts: (1) Rakai, Uganda, (2) Sweden, and (3) the USA. Five of' the six singlesex networks were fit best by the negative binomial model. The American women's network was best fit by a powerlaw model, the Yule. For most networks, several competing models fit approximately equally well. These results sug gest three conclusions: (1) no single unitary process clearly underlies the formation of these sexual networks, (2) behavioral heterogeneity plays an essential role in network structure, (3) substantial model uncertainty exists for sexual network degree distributions. Behavioral research focused on the mechanisms of partnership f'ormation will play an essential role in specifying the best model for empirical degree distributions. We discuss the limitations of inferences f'rom such data, and the utility of degreebased epidemiological models more generally.
Search in the Formation of Large Networks: How Random are Socially Generated Networks?
, 2005
"... We present a model of network formation where entering nodes find other nodes to link to both completely at random and through search of the neighborhoods of these randomly met nodes. We show that this model exhibits the full spectrum of features that have been found to characterize large socially g ..."
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Cited by 11 (3 self)
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We present a model of network formation where entering nodes find other nodes to link to both completely at random and through search of the neighborhoods of these randomly met nodes. We show that this model exhibits the full spectrum of features that have been found to characterize large socially generated networks. Moreover, we derive the distribution of degree (number of links) across nodes, and show that while the upper tail of the distribution is approximately “scalefree,” the lower tail may exhibit substantial curvature, just as in observed networks. We then fit the model to data from six networks. Besides offering a close fit of these diverse networks, the model allows us to impute the relative importance of search versus random attachment in link formation. We find that the fitted ratio of random meetings to searchbased meetings varies dramatically across these applications. Finally, we show that as this random/search ratio varies, the resulting degree distributions can be completely ordered in the sense of second order stochastic dominance. This allows us to infer how the relative randomness in the formation process affects average utility in the network.
The Strategic Formation of Large Networks: When and Why do We See Power Laws and Small Worlds?
 Proceedings of the P2P Conference
, 2004
"... We introduce a searchbased economic model of network formation. Individuals enter over time and find others at random and through a local search process, and then decide which links to form based on myopic selfinterested utility maximization. This model simultaneously accounts for three stylized f ..."
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Cited by 8 (0 self)
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We introduce a searchbased economic model of network formation. Individuals enter over time and find others at random and through a local search process, and then decide which links to form based on myopic selfinterested utility maximization. This model simultaneously accounts for three stylized features of a number of observed large networks: (i) connections tend to be much more highly clustered than one would see in a random network formation process, (ii) the maximal distance between nodes is relatively small (on the order of log[network size]/log[log[network size]] which is small even compared to a random network), and (iii) the distribution of node degrees obeys a power law in the upper tail (there are many more highly linked nodes than one should see in a purely random network, and in particular proportions), but not necessarily for smaller degrees.
A stochastic evolutionary model exhibiting powerlaw behaviour with an exponential cutoff
 in the Condensed Matter Archive, condmat/0209463
, 2005
"... Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed sto ..."
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Cited by 8 (3 self)
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Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein and email networks. Such networks may in fact exhibit an exponential cutoff in the powerlaw scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after, for example, a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will be discarded. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a powerlaw distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model by analysing the protein yeast network, whose distribution is known to follow a power law with an exponential cutoff. 1
Clustering the Chilean Web
, 2003
"... We perform a clustering of the Chilean Web Graph using a local fitness measure, optimized by simulated annealing, and compare the obtained cluster distribution to that of two models of the Web Graph. Information on web clusters can be employed both to validate generation models and to study the prop ..."
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Cited by 7 (0 self)
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We perform a clustering of the Chilean Web Graph using a local fitness measure, optimized by simulated annealing, and compare the obtained cluster distribution to that of two models of the Web Graph. Information on web clusters can be employed both to validate generation models and to study the properties of the graph. Clusters can also be used in semanticsbased grouping of websites or pages e.g. for indexing and browsing.
Link Structure of Hierarchical Information Networks
"... One feature that seems to have been largely ignored in previous models of the Web is the inherent hierarchy that is evident in the structure of URLs. We provide evidence that this hierarchical structure is closely related to the link structure of the web, and this relationship explains several impor ..."
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Cited by 3 (0 self)
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One feature that seems to have been largely ignored in previous models of the Web is the inherent hierarchy that is evident in the structure of URLs. We provide evidence that this hierarchical structure is closely related to the link structure of the web, and this relationship explains several important features of the web, including the locality and bidirectionality of hyperlinks, and the compressibility of the web graph. We describe how to construct data models of the web that capture both the hierarchical nature of the web as well as some crucial features of the link graph. Our analysis is based on observations from a crawl of over a billion URLs, as well as largescale simulations of models. We also show how this interaction between hierarchical structure and link structure extends to other domains. In particular we describe some analysis on corporate instant messaging, in which there is similar correspondence between the corporate management structure and patterns of communication between individuals.
A Stochastic Model for the Evolution of the Web Allowing Link Deletion
, 2004
"... A stochastic model for the evolution of the web allowing ..."
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Cited by 3 (3 self)
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A stochastic model for the evolution of the web allowing
Locality, Hierarchy, and Bidirectionality in the Web
 In Workshop on Algorithms and Models for the Web Graph
, 2003
"... The World Wide Web has been previously observed to be a "small world network " in which nodes are clustered together. We provide evidence, based on a crawl of over a billion pages, that such a clustering e#ect corresponds very closely to the hierarchical nature of URLs. We also show that bidirect ..."
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Cited by 3 (1 self)
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The World Wide Web has been previously observed to be a "small world network " in which nodes are clustered together. We provide evidence, based on a crawl of over a billion pages, that such a clustering e#ect corresponds very closely to the hierarchical nature of URLs. We also show that bidirectionality on the web graph is much more common than previous models predicted. We then propose a new paradigm for models of the Web that incorporates the hierarchical evolution and structure that is evident in the Web.
A model for collaboration networks giving rise to a power law distribution with an exponential cutoff
 Physics and Society Archive
, 2006
"... Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed sto ..."
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Cited by 2 (1 self)
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Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to powerlaw distributions. These models are based on the notion of preferential attachment leading to the “rich get richer ” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein, email, actor and collaboration networks. Such networks may in fact exhibit an exponential cutoff in the powerlaw scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will become inactive. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a powerlaw distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model empirically by analysing the Mathematical Research collaboration network, the distribution of which is known to follow a power law with an exponential cutoff. 1