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This study introduces a novel "cascade" model for anomalous diffusion, mimicking Lévy flights with local step cascades. This mechanism can generate superdiffusion and normal diffusion, offering new insights into search and navigation strategies.

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Area of Science:

  • Physics
  • Complex Systems
  • Mathematical Biology

Background:

  • Anomalous diffusion, particularly superdiffusion, is crucial for efficient search and navigation in biological and human mobility.
  • Lévy flights, characterized by power-law distributed jump lengths, are a common model for achieving superdiffusion.

Purpose of the Study:

  • To present a new model for anomalous diffusion using local "cascade" mechanisms instead of long jumps.
  • To investigate how power-law distributed cascade lengths and waiting times influence diffusion dynamics.

Main Methods:

  • A one-dimensional lattice model where walkers perform "cascades" of n unitary steps.
  • The number of steps 'n' is drawn from a power-law distributed cascade distribution p(n).
  • Incorporation of power-law distributed waiting times.

Main Results:

  • Demonstrated that the local cascade mechanism can produce both superdiffusion and normal diffusion.
  • Showed that the scaling of the probability distribution is governed by the exponents of the power-law distributions for cascades and waiting times.

Conclusions:

  • The proposed cascade model offers a viable alternative to Lévy flights for generating anomalous diffusion.
  • This approach may generalize anomalous diffusion to complex networks and temporal networks where distances are ill-defined.