Growth Quakes and Stasis Using Iterations of Inflating Complex Random Matrices

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.