Rocky Mountain Journal of Mathematics

On the number of isolated vertices in a growing random graph

Anatolii A. Puhalskii

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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.

Article information

Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1941-1989.

First available in Project Euclid: 25 February 2014

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60G99: None of the above, but in this section
Secondary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles

Random graphs stochastic processes isolated vertices large deviations weak convergence


Puhalskii, Anatolii A. On the number of isolated vertices in a growing random graph. Rocky Mountain J. Math. 43 (2013), no. 6, 1941--1989. doi:10.1216/RMJ-2013-43-6-1941.

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