Open Access
2013 On the number of isolated vertices in a growing random graph
Anatolii A. Puhalskii
Rocky Mountain J. Math. 43(6): 1941-1989 (2013). DOI: 10.1216/RMJ-2013-43-6-1941


This paper studies the properties of the number of isolated vertices in a random graph where vertices arrive one-by-one at times $1,2,\ldots$\,. They are connected by edges to the previous vertices independently with the same probability. Assuming that the probability of an edge tends to zero, we establish the asymptotics of large, normal, and moderate deviations for the stochastic process of the number of the isolated vertices considered at times inversely proportional to that probability. In addition, we identify the most likely trajectory for that stochastic process to follow conditioned on the event that at a large time the graph is found with a large number of isolated vertices.


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Anatolii A. Puhalskii. "On the number of isolated vertices in a growing random graph." Rocky Mountain J. Math. 43 (6) 1941 - 1989, 2013.


Published: 2013
First available in Project Euclid: 25 February 2014

zbMATH: 1288.60010
MathSciNet: MR3178451
Digital Object Identifier: 10.1216/RMJ-2013-43-6-1941

Primary: 60C05 , 60G99
Secondary: 60F10 , 60F17

Keywords: isolated vertices , large deviations , Random graphs , Stochastic processes , weak convergence

Rights: Copyright © 2013 Rocky Mountain Mathematics Consortium

Vol.43 • No. 6 • 2013
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