Abstract
A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.
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This research was mainly done at Institute Mittag-Leffler, Djursholm, Sweden, during the program Discrete Probability, 2009. We thank other participants, in particular Oliver Riordan, for helpful comments. We thank Will Perkins for the numerical calculations in Remark 3.6.
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Janson, S., Spencer, J. Phase transitions for modified Erdős–Rényi processes. Ark Mat 50, 305–329 (2012). https://doi.org/10.1007/s11512-011-0157-1
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DOI: https://doi.org/10.1007/s11512-011-0157-1