TY - GEN

T1 - On the Growth of a Superlinear Preferential Attachment Scheme

AU - Sethuraman, Sunder

AU - Venkataramani, Shankar C.

N1 - Funding Information:
Acknowledgements S.V. acknowledges the support from the Simons Foundation through award 524875. S.S. acknowledges the support from the grant ARO W911NF-14-1-0179.
Publisher Copyright:
© Springer Nature Switzerland AG 2019.

PY - 2019

Y1 - 2019

N2 - We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as n ↑ ∞, for the counts of the number of nodes with degree k ≥ 1 at time n ≥ 1. These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.

AB - We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as n ↑ ∞, for the counts of the number of nodes with degree k ≥ 1 at time n ≥ 1. These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.

KW - Degree distribution

KW - Fluctuations

KW - Growth

KW - Preferential attachment

KW - Random graphs

KW - Superlinear

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U2 - 10.1007/978-3-030-15338-0_9

DO - 10.1007/978-3-030-15338-0_9

M3 - Conference contribution

AN - SCOPUS:85068959028

SN - 9783030153373

T3 - Springer Proceedings in Mathematics and Statistics

SP - 243

EP - 265

BT - Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016

A2 - Friz, Peter

A2 - König, Wolfgang

A2 - Mukherjee, Chiranjib

A2 - Olla, Stefano

PB - Springer New York LLC

T2 - Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016

Y2 - 15 August 2016 through 19 August 2016

ER -