TY - GEN
T1 - On the Growth of a Superlinear Preferential Attachment Scheme
AU - Sethuraman, Sunder
AU - Venkataramani, Shankar C.
N1 - 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
UR - http://www.scopus.com/inward/record.url?scp=85068959028&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85068959028&partnerID=8YFLogxK
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 -