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Growth Quakes and Stasis Using Iterations of Inflating Complex Random Matrices.

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This study extends random matrix theory to complex systems, revealing punctuated growth with "quakes" and "stasis." Complex eigenvalues offer more flexibility for modeling real-world systems like economies and ecosystems.

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Area of Science:

  • Complex Systems Dynamics
  • Random Matrix Theory
  • Mathematical Modeling

Background:

  • Prior studies focused on real matrices for growing systems.
  • Complex systems often exhibit periods of stability punctuated by sudden changes.

Purpose of the Study:

  • Extend the "inflating random matrix" method to complex matrices.
  • Describe punctuated growth dynamics in complex systems.
  • Investigate the role of complex eigenvalues in system evolution.

Main Methods:

  • Iterative application of an "inflating random matrix" to a state vector.
  • Analysis of dominant eigenvectors and eigenvalues.
  • Assessment of vector shifts under matrix inflation.

Main Results:

  • The complex matrix model replicates punctuated growth with "quakes" and "stasis."
  • Vector shifts (quakes) occur when inflated matrices have dominant new eigenvectors.
  • A bimodal distribution of dominant eigenvalue changes is observed across update schemes.

Conclusions:

  • Complex eigenvalues provide greater degrees of freedom for modeling real-world systems.
  • The model offers a framework for understanding growth in systems with historical weight and sudden events.
  • Random matrices and non-ergodic tools can be applied to ecological and economic systems.