Random graphs with arbitrary degree distributions and their applications

Newman, M. E. J.; Strogatz, Steven H.; Watts, Duncan J.

doi:10.1103/physreve.64.026118

Cited by 2,928 publications

(2,194 citation statements)

References 32 publications

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“…For part one, we use generating function methods to derive the probability and expected demographic distribution of outbreaks, with and without public health intervention. This is an extension of both epidemiological theory previously developed for undirected contact networks [3] and a general theory of random graphs containing only directed edges [15]. We show that in semidirected networks the probability of an epidemic and the expected fraction of the population infected during such an epidemic may be different.…”

mentioning

confidence: 69%

“…Probabilitiy generating functions for semi-directed networks In the theory of random directed graphs developed by Newman et al [15], one considers the joint probability distribution p jk that a randomly chosen vertex has in-degree j and out-degree k. Then one defines a generating function F (x, y) whose coefficients are the probabilities in this distribution:…”

Section: Derivations Of Epidemic Quantitiesmentioning

confidence: 99%

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Predicting epidemics on directed contact networks

Meyers

Newman

Pourbohloul

2006

Journal of Theoretical Biology

263

245

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Abstract:Contact network epidemiology is an approach to modeling the spread of infectious diseases that explicitly considers patterns of person-to-person contacts within a community. Contacts can be asymmetric, with a person more likely to infect one of their contacts than to become infected by that contact. This is true for some sexually transmitted diseases that are more easily caught by women than men during heterosexual encounters; and for severe infectious diseases that cause an average person to seek medical attention and thereby potentially infect health care workers who would not, in turn, have an opportunity to infect that average person. Here we use methods from percolation theory to develop a mathematical framework for predicting disease transmission through semi-directed contact networks in which some contacts are undirected -the probability of transmission is symmetric between individuals -and others are directed -transmission is possible only in one direction. We find that probability of an epidemic and the expected fraction of a population infected during an epidemic can be different in semi-directed networks, in contrast to the routine assumption that these two quantities are equal. Using these methods, we furthermore demonstrate the vulnerability of health care workers and the importance hospital-based containment during outbreaks of severe respiratory diseases.3

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mentioning

confidence: 69%

Section: Derivations Of Epidemic Quantitiesmentioning

confidence: 99%

Predicting epidemics on directed contact networks

Meyers

Newman

Pourbohloul

2006

Journal of Theoretical Biology

263

245

View full text Add to dashboard Cite

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“…Hence for γ < 3, the gap is large but so are the fluctuations, whereas for γ > 3 the gap decreases and the fluctuations are effectively constant. The best payoff between (5) (5) To test our analytical predictions, we generated networks with fixed P(k) using the configuration model 46 , and then ranked each node according to their pagerank. We perturbed the network by rewiring every edge (keeping the degree of each node unchanged) and determined the pagerank after each rewiring, helping us identify nodes whose ranking did not change as a result of the perturbation.…”

Section: Resultsmentioning

confidence: 99%

Ranking stability and super-stable nodes in complex networks

2011

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Pagerank, a network-based diffusion algorithm, has emerged as the leading method to rank web content, ecological species and even scientists. Despite its wide use, it remains unknown how the structure of the network on which it operates affects its performance. Here we show that for random networks the ranking provided by pagerank is sensitive to perturbations in the network topology, making it unreliable for incomplete or noisy systems. In contrast, in scale-free networks we predict analytically the emergence of super-stable nodes whose ranking is exceptionally stable to perturbations. We calculate the dependence of the number of super-stable nodes on network characteristics and demonstrate their presence in real networks, in agreement with the analytical predictions. These results not only deepen our understanding of the interplay between network topology and dynamical processes but also have implications in all areas where ranking has a role, from science to marketing.

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“…The analytical framework is based on a generating-function formalism widely used for studies of percolation and structure within a single network [73][74][75] . The framework for interdependent networks enables us to follow the dynamics of the cascading failures as well as to derive the analytic solutions for the final steady state.…”

Section: Figure 1 | Schematic Demonstration Of First-and Second-ordermentioning

confidence: 99%

“…We begin by describing the generating-function formalism 74 for a single network that will also be useful in studying interdependent networks. We assume that all N i nodes in network i are randomly assigned a degree k from a probability distribution P i (k), and are randomly connected with the only constraint that the node with degree k has exactly k links 91 .…”

Section: Generating Functions For a Single Networkmentioning

confidence: 99%

Networks formed from interdependent networks

et al. 2011

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Complex networks appear in almost every aspect of science and technology. Although most results in the field have been obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of 'networks of networks' formed by interdependent random networks.T he interdisciplinary field of network science has attracted a great deal of attention in recent years . This development is based on the enormous number of data that are now routinely being collected, modelled and analysed, concerning social 31-39 , economic 14,36,40,41 , technological 40,42-48 and biological 9,13,49,50 systems. The investigation and growing understanding of this extraordinary volume of data will enable us to make the infrastructures we use in everyday life more efficient and more robust.The original model of networks, random graph theory, was developed in the 1960s by ErdAEs and Rényi, and is based on the assumption that every pair of nodes is randomly connected with the same probability, leading to a Poisson degree distribution. In parallel, in physics, lattice networks, where each node has exactly the same number of links, have been studied to model physical systems. Although graph theory is a well-established tool in the mathematics and computer science literature, it cannot describe well modern, real-life networks. Indeed, the pioneering 1999 observation by Barabasi 2 , that many real networks do not follow the ErdAEs-Rényi model but that organizational principles naturally arise in most systems, led to an overwhelming accumulation of supporting data, new models and computational and analytical results, and to the emergence of a new science, that of complex networks.Complex networks are usually non-homogeneous structures that in many cases obey a power-law form in their degree (that is, number of links per node) distribution. These systems are called scale-free networks. Real networks that can be approximated as scale-free networks include the Internet 3 , the World Wide Web 4 , social networks 31-39 representing the relations between individuals, infrastructure networks such as those of airlines 51 , networks in biology 9,13,49,50 , in particular networks of proteinprotein interactions 10 , gene regulation and biochemical pathways, and networks in physics, such as polymer networks or the potentialenergy-landscape network. The discovery of scale-free networks led to a re-evaluation of the basic properties of networks, such as their robustness, which exhibit a drastically different character than those of ErdAEs-Rényi networks. For example, whereas homogeneous ErdAEs-Rényi networks are extremely vulnerable to random fa...

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Random graphs with arbitrary degree distributions and their applications

Cited by 2,928 publications

References 32 publications

Predicting epidemics on directed contact networks

Predicting epidemics on directed contact networks

Ranking stability and super-stable nodes in complex networks

Networks formed from interdependent networks

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