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Diffusion in sparse networks: linear to semilinear crossover.

Yaron de Leeuw1, Doron Cohen

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|December 11, 2012
PubMed
Summary
This summary is machine-generated.

In random networks, a diffusion coefficient (D) transition from diffusion to subdiffusion, seen in 1D, does not occur in higher dimensions. An effective-range-hopping method confirms D remains finite, contrasting prior renormalization-group predictions.

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Area of Science:

  • Statistical Physics
  • Complex Networks
  • Dynamical Systems

Background:

  • Random networks with symmetric transition rates exhibit dynamics governed by rate equations.
  • The system's long-time evolution is characterized by a diffusion coefficient (D).
  • In one dimension, a percolation-like transition from diffusion (D>0) to subdiffusion (D=0) is well-established.

Purpose of the Study:

  • To investigate whether the diffusion-subdiffusion transition observed in 1D random networks occurs in higher dimensions.
  • To determine the behavior of the diffusion coefficient (D) in multi-dimensional random networks.
  • To propose and validate a new method for evaluating D in complex network systems.

Main Methods:

  • Numerical evaluation of the diffusion coefficient (D) using resistor network calculations.
  • Deduction of D from the spectral properties of the system.
  • Development and application of an "effective-range-hopping" procedure for D evaluation.

Main Results:

  • The study deduces that the diffusion coefficient (D) remains finite in higher dimensions, contrary to expectations from renormalization-group analysis.
  • An "effective-range-hopping" procedure is proposed and utilized for evaluating D.
  • Results are contrasted with a linear estimate, and the approach is shown to be effective for quasi-one-dimensional sparse banded matrices.

Conclusions:

  • The abrupt percolation-like transition from diffusion to subdiffusion does not extend to higher dimensions in these random networks.
  • The proposed "effective-range-hopping" method provides a reliable way to calculate the finite diffusion coefficient in complex network dynamics.
  • The findings challenge previous theoretical expectations and offer a new perspective on transport phenomena in higher-dimensional random systems.