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Minimal spanning trees at the percolation threshold: a numerical calculation.

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Area of Science:

  • Statistical physics
  • Network science
  • Fractal geometry

Background:

  • Percolation theory describes the behavior of connected clusters in random networks.
  • Minimal Spanning Trees (MSTs) are fundamental in network analysis, connecting all points with minimum total edge weight.
  • Fractal dimensions characterize the complexity and space-filling properties of irregular shapes, like those found in critical phenomena.

Purpose of the Study:

  • To estimate the fractal dimension of MSTs on percolation clusters up to d=5.
  • To develop a robust analysis technique for correlated data in fractal structures.
  • To validate theoretical predictions for MST path dimensions on critical percolation clusters.

Main Methods:

  • Utilized a combination of Prim's and Kruskal's algorithms to construct MSTs, optimizing memory usage for larger system simulations.
  • Developed and applied a robust analysis technique specifically designed for correlated data within tree structures.
  • Estimated the path length fractal dimension (d_s) of MSTs on critical percolation clusters.

Main Results:

  • Successfully estimated the fractal dimension of MSTs for system dimensions up to d=5.
  • The developed analysis technique proved robust for handling correlated data inherent in fractal trees.
  • The calculated path length fractal dimension (d_s) showed compatibility with existing theoretical models.

Conclusions:

  • The study provides reliable estimates for MST fractal dimensions on percolation clusters.
  • The novel analysis method is broadly applicable to various randomly generated fractal structures.
  • Findings support the accuracy of perturbation expansion theories for describing MST properties in critical systems.