Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models

@article{Pachn2015RandomGA,
  title={Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models},
  author={Ang{\'e}lica Pach{\'o}n and Federico Polito and Laura Sacerdote},
  journal={Journal of Statistical Physics},
  year={2015},
  volume={162},
  pages={1608-1638}
}
We give a common description of Simon, Barabási–Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási–Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter $$\alpha $$α) goes to infinity, a… 
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References

SHOWING 1-10 OF 25 REFERENCES
The degree sequence of a scale‐free random graph process
TLDR
Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.
Structure of growing networks with preferential linking.
TLDR
The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites and it is shown that the relation beta(gamma-1) = 1 between the exponents is universal.
Generalized Preferential Attachment: Tunable Power-Law Degree Distribution and Clustering Coefficient
TLDR
This work proposes a common framework for analysis of a wide class of preferential attachment models, which includes LCD, Buckley–Osthus, Holme–Kim and many others, and shows that both the parameter of the power-law degree distribution and the clustering coefficient can be controlled via variation of the model parameters.
World Wide Web scaling exponent from Simon's 1955 model.
  • S. Bornholdt, Holger Ebel
  • Physics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2001
TLDR
A simple and elegant model for scaling phenomena in general copy- and growth-processes as proposed by Simon in 1955 is recalled and when combined with an experimental measurement of network growth in the World Wide Web, this classical model is able to model the in-link dynamics and predicts the scaling exponent gamma=2.1 in accordance with observation.
The role of detachment of links in scale-free networks
TLDR
A tractable extension of Yule model to account for detachment phenomenon determined by the cancelling of previously existing connections is discussed and the agreement of the proposed model on very recent data is shown.
A general theory of bibliometric and other cumulative advantage processes
  • D. Price
  • Mathematics
    J. Am. Soc. Inf. Sci.
  • 1976
TLDR
It is shown that such a stochastic law is governed by the Beta Function, containing only one free parameter, and this is approximated by a skew or hyperbolic distribution of the type that is widespread in bibliometrics and diverse social science phenomena.
A general model of web graphs
TLDR
A very general model of a random graph process whose proportional degree sequence obeys a power law is described, which has recently been observed in graphs associated with the world wide web.
Emergence of scaling in random networks
TLDR
A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS
It is the purpose of this paper to analyse a class of distribution functions that appears in a wide range of empirical data-particularly data describing sociological, biological and economic
The Structure and Function of Complex Networks
TLDR
Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
...
...