Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase transitions for modified Erd\"os-R\'enyi processes   A fundamental and very well studied region of the Erd\"os-R\'enyi process is
the phase transition at 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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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
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The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
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The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
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The categories of this paper are: patt-sol