On singularities in combination-driven models of technological innovation

TL;DR

This paper develops a generalized model of technological innovation based on combinatorial growth and aging, demonstrating conditions under which innovation rates can diverge or slow down, revealing phases with or without singularities.

Contribution

It introduces a comprehensive model incorporating invention combination and aging, analyzing conditions for explosive growth or stagnation in technological evolution.

Findings

Singularities can occur with long-range memory in innovation models.

A characteristic aging time scale prevents singularities, leading to a slowdown in innovation.

A 'black hole' of old inventions expands, reducing innovation rate to linear growth.

Abstract

It has been suggested that innovations occur mainly by combination: the more inventions accumulate, the higher the probability that new inventions are obtained from previous designs. Additionally, it has been conjectured that the combinatorial nature of innovations naturally leads to a singularity: at some finite time, the number of innovations should diverge. Although these ideas are certainly appealing, no general models have been yet developed to test the conditions under which combinatorial technology should become explosive. Here we present a generalised model of technological evolution that takes into account two major properties: the number of previous technologies needed to create a novel one and how rapidly technology ages. Two different models of combinatorial growth are considered, involving different forms of ageing. When long-range memory is used and thus old inventions are available for novel innovations, singularities can emerge under some conditions with two phases separated by a critical boundary. If the ageing has a characteristic time scale, it is shown that no singularities will be observed. Instead, a "black hole" of old innovations appears and expands in time, making the rate of invention creation slow down into a linear regime.