# Dynamics of a birth-death process based on combinatorial innovation

## TL;DR

This paper models human creativity and innovation as a stochastic birth-death process with combinatorial growth, revealing a 'hockey-stick' pattern of slow growth followed by rapid expansion, supported by exact mathematical analysis and simulations.

## Contribution

It introduces a simple stochastic model capturing the dynamics of combinatorial innovation with exact analytical results and generalizations for real-world applications.

## Key findings

- The model exhibits a 'hockey-stick' growth pattern.
- Exact expressions for mean and variance of explosion time are derived.
- Simulations confirm the theoretical predictions.

## Abstract

A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth--death model, with non-linear combinatorial terms in the growth coefficients to capture the propensity of subsets of items to give rise to new items. In its simplest form, this model involves just two parameters $(P, \alpha)$. This process exhibits a characteristic 'hockey-stick' behaviour: a long period of relatively little growth followed by a relatively sudden 'explosive' increase. We provide exact expressions for the mean and variance of this time to explosion and compare the results with simulations. We then generalise our results to allow for more general parameter assignments, and consider possible applications to data involving human productivity and creativity.