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Following the Dynamics of Structural Variants in Experimentally Evolved Populations
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Simple evolving random graphs.

P L Krapivsky1

  • 1Department of Physics, <a href="https://ror.org/05qwgg493">Boston University</a>, Boston, Massachusetts 02215, USA and <a href="https://ror.org/01arysc35">Santa Fe Institute</a>, Santa Fe, New Mexico 87501, USA.

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Summary
This summary is machine-generated.

Simple random graphs evolve differently from classical ones. Their growth freezes when trees vanish, unlike classical graphs that grow indefinitely.

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Area of Science:

  • Graph theory
  • Statistical physics

Background:

  • Classical random graphs exhibit continuous growth and a giant component formation.
  • Understanding graph evolution dynamics is crucial in network science.

Purpose of the Study:

  • To investigate the unique evolution and phase transition of simple random graphs.
  • To analyze the properties of the frozen state in these graphs.

Main Methods:

  • Modeling random graph densification by adding edges probabilistically.
  • Analyzing graph structures (trees and unicycles) and their transitions.
  • Investigating behavior in the thermodynamic limit and the frozen state.

Main Results:

  • Simple random graphs form only trees and unicycles.
  • A phase transition occurs in the thermodynamic limit, similar to percolation.
  • Unlike classical graphs, their evolution freezes when trees disappear.

Conclusions:

  • Simple random graphs exhibit a distinct freezing phenomenon.
  • The frozen state properties differ significantly from classical random graph models.