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Area of Science:

  • Graph theory
  • Statistical physics
  • Network science

Background:

  • Percolation transitions are critical phenomena in network science.
  • Understanding graph evolution is key to network analysis.
  • Discontinuous transitions in evolving networks are not well-understood.

Purpose of the Study:

  • To introduce simple models of graph evolution with choice.
  • To analyze these models using mathematical evolution equations.
  • To investigate the emergence of the giant component in evolving graphs.

Main Methods:

  • Developing local graph evolution models where edges are added sequentially.
  • Utilizing mathematical evolution equations to describe graph dynamics.
  • Analyzing the emergence of the giant component and scaling behaviors.

Main Results:

  • The proposed models demonstrate discontinuous percolation transitions.
  • Graph evolution is accurately described by differential equations.
  • A discontinuous emergence of the giant component is observed.
  • Scaling behaviors characteristic of continuous transitions are present.

Conclusions:

  • Simple, local models of graph evolution with choice can exhibit discontinuous percolation.
  • These models provide a tractable framework for studying complex network phenomena.
  • The observed scaling behaviors offer new insights into network transitions.