Growth Quakes and Stasis Using Iterations of Inflating Complex Random Matrices

I extend to the case of complex matrices, rather than the case of real matrices as in a prior study, a method of iterating the operation of an inflating random matrix onto a state vector to describe complex growing systems. I show that the process also describes in this complex case a punctuated growth with quakes and stasis. I assess that under one such inflation step, the vector will shift to a really different one (quakes) only if the inflated matrix has sufficiently dominant new eigenvectors. The vector shall prefer stasis (a similar vector) otherwise, similar to the real-valued matrices discussed in a prior study. Specifically, in order to extend the model relevance, I assess that under various update schemes of the system's representative vector, the bimodal distribution of the changes of the dominant eigenvalue remains the core concept. Overall, I contend that the punctuations may appropriately address the issue of growth in systems combining a large weight of history and some sudden quake occurrences, such as economic systems or ecological systems, with the advantage that unpaired complex eigenvalues provide more degrees of freedom to suit real systems. Furthermore, random matrices could be the right meeting point for exerting thermodynamic analogies in a reasonably agnostic manner in such rich contexts, taking into account the profusion of items (individuals, species, goods, etc.) and their networked, tangled interactions 50+ years after their seminal use in R.M. May's famous interaction induced instability paradigm. Finally, I suggest that non-ergodic tools could be further applied for tracking the specifics of large-scale evolution paths and for checking the model's relevance to the domains mentioned above.

Automated summary

This study extends the method of iterating the operation of an inflating random matrix onto a state vector to describe complex growing systems, rather than real matrices as in a prior study. The process is shown to describe punctuated growth with quakes and stasis in this complex case. The study assesses that the vector will shift to a different one only if the inflated matrix has dominant new eigenvectors, otherwise preferring stasis. The bimodal distribution of the changes of the dominant eigenvalue remains the core concept despite variations in the update schemes of the system's representative vector.