A critical point for random graphs with a given degree sequence

Molloy, Michael; Reed, Bruce

doi:10.1002/rsa.3240060204

Cited by 2,068 publications

(2,118 citation statements)

References 10 publications

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“…(1) and (2) have been derived earlier using a different approach by Molloy and Reed [15]. We thank Dr. Mark E. J. Newman for bringing this reference to our attention.…”

Section: Acknowledgmentsmentioning

confidence: 94%

Resilience of the Internet to Random Breakdowns

Cohen¹,

Erez²,

Havlinl³

2011

The Structure and Dynamics of Networks

284

415

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A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P (k) = ck −α . We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, pc, that need to be removed before the network disintegrates. We show analytically and numerically that for α ≤ 3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (α ≈ 2.5), we find that it is impressively robust, with pc > 0. 99. 02.50.Cw, 05.40.a, 05.50.+q, 64.60.Ak

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“…(1) and (2) have been derived earlier using a different approach by Molloy and Reed [15]. We thank Dr. Mark E. J. Newman for bringing this reference to our attention.…”

Section: Acknowledgmentsmentioning

confidence: 94%

Resilience of the Internet to Random Breakdowns

Cohen¹,

Erez²,

Havlinl³

2011

The Structure and Dynamics of Networks

284

415

View full text Add to dashboard Cite

show abstract

“…This approach has also been applied in assessing the importance of such features for dynamical processes. The most widely applied reference model is the configuration model [21], where the links of the original network are rewired pairwise randomly. This reference model preserves the original degree sequence but yields networks that are as random as the degree sequence allows.…”

Section: Reference Modelsmentioning

confidence: 99%

Multiscale analysis of spreading in a large communication network

et al. 2012

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Abstract. In temporal networks, both the topology of the underlying network and the timings of interaction events can be crucial in determining how some dynamic process mediated by the network unfolds. We have explored the limiting case of the speed of spreading in the SI model, set up such that an event between an infectious and susceptible individual always transmits the infection. The speed of this process sets an upper bound for the speed of any dynamic process that is mediated through the interaction events of the network. With the help of temporal networks derived from large-scale time-stamped data on mobile phone calls, we extend earlier results that point out the slowing-down effects of burstiness and temporal inhomogeneities. In such networks, links are not permanently active, but dynamic processes are mediated by recurrent events taking place on the links at specific points in time. We perform a multi-scale analysis and pinpoint the importance of the timings of event sequences on individual links, their correlations with neighboring sequences, and the temporal pathways taken by the networkscale spreading process. This is achieved by studying empirically and analytically different characteristic relay times of links, relevant to the respective scales, and a set of temporal reference models that allow for removing selected time-domain correlations one by one.

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“….). We then construct the network describing the global structure of the population accoring to the 'configuration model' (see [12,13]). This model works by assigning each individual in the population a number of 'half-edges' (that individual's degree in the global network) according to independent samples from some distribution D with P(D = k) = p k (k = 0, 1, .…”

Section: Modelmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

133

148

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This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

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A critical point for random graphs with a given degree sequence

Cited by 2,068 publications

References 10 publications

Resilience of the Internet to Random Breakdowns

Resilience of the Internet to Random Breakdowns

Multiscale analysis of spreading in a large communication network

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

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