The Interacting Branching Process as a Simple Model of Innovation

TL;DR

This paper models innovation using a generalized branching process, revealing continuous survival probability, super-exponential growth, and accelerating returns, with detailed analysis of behavior as the innovation rate approaches zero.

Contribution

It introduces a novel interacting branching process model for innovation, showing no phase transition and detailed asymptotic analysis as the innovation rate diminishes.

Findings

No phase transition; survival probability is finite for all p > 0.

Processes exhibit super-exponential growth and accelerating returns.

Detailed asymptotic behavior as the innovation rate approaches zero.

Abstract

We describe innovation in terms of a generalized branching process. Each new invention pairs with any existing one to produce a number of offspring, which is Poisson distributed with mean p. Existing inventions die with probability p/\tau at each generation. In contrast to mean field results, no phase transition occurs; the chance for survival is finite for all p > 0. For \tau = \infty, surviving processes exhibit a bottleneck before exploding super-exponentially - a growth consistent with a law of accelerating returns. This behavior persists for finite \tau. We analyze, in detail, the asymptotic behavior as p \to 0.