Evolutionary behaviour of ‘inflating’ random real matrices for economy or biology: stasis statistics of vector iterations upon growth

Abstract

Abstract A scheme is proposed for describing stasis and transitions in evolutionary contexts defined by a growing number of interrelated items. These items could be genes/species in biology, or tools/products in economy. The target is a frame to describe the advent of stasis marked by dominant species or dominant objects (car, smartphone) between sharp transitions (quakes). The tool of random matrices is adapted to add an explicit varietal growth, through an ‘inflation’ of a real random matrix (Ginibre set), by regularly adding a line and a column, such a matrix operating at each unit time step on the evolving vector, U(t + 1) = M(t)U(t). In this view, U(t) ∝ log(C(t)) with C(t) the vector of abundances of genes in a gene pool, or of abundance of tools in a multi-sector production economy (as in Leontieff matrices of sector-wise productions). U(t) is trending toward the eigenvector with the largest-modulus eigenvalue (ev) U (N) for the current N(t). Most times, the next such eigenvector U (N+1) of the inflated matrix is close and mostly colinear to U (N). But, as time goes and N also grows, a wholly unrelated eigenvector may acquire a larger modulus ev and become the new attractor. Thus, there are slowly-moving stases punctuated by quakes. The leading-ev-modulus competition is elucidated, as well as the main features of the stasis duration distribution it entails, akin to a q-exponential law. This is done by means of a two-particles model of Brownian motion taking place with an N-dependent drift and diffusion. To minimally flesh the issue, a set of bibliographic data (yearly number of papers among all journals of a domain) is used, lending support to the vector-colinearity-based methods used for the detection of stases vs quakes. Hints are given for various developments tackling the appropriateness of the model to various growth contexts, e.g. with sparse network or with death and not only novelty/birth.