Core Concepts
The Theory of the Adjacent Possible (TAP) equation, a model for combinatorial innovation, reveals how complex systems can experience sudden, explosive growth after periods of relative stability, with potential applications in fields ranging from economics to cosmology.
Abstract
Research Paper Summary
Bibliographic Information: Cortˆes, M., Kauffman, S. A., Liddle, A. R., & Smolin, L. (2024). The TAP equation: evaluating combinatorial innovation in biocosmology. arXiv preprint arXiv:2204.14115v3.
Research Objective: This paper investigates the dynamics of the TAP equation, a mathematical model based on the Theory of the Adjacent Possible, to understand how combinatorial innovation leads to explosive growth in various systems.
Methodology: The authors analyze different versions of the TAP equation, both analytically and numerically. They explore the impact of parameters like the initial number of elements, extinction rate, and combinatorial efficiency on the system's evolution.
Key Findings:
- The TAP equation typically exhibits a "hockey-stick" pattern, characterized by an initial plateau phase followed by a rapid, super-exponential growth phase termed "blow-up."
- Analytical formulas for predicting the time to blow-up are presented and validated through numerical simulations.
- A new "two-scale" variant of the TAP equation is introduced, incorporating both individual object evolution and multi-object combinations, resulting in an initial exponential growth phase before the blow-up.
Main Conclusions: The TAP equation provides a powerful framework for understanding how combinatorial innovation drives explosive growth in complex systems. The sudden transition to blow-up highlights the difficulty in predicting such events and emphasizes the potential risks associated with uncontrolled growth.
Significance: This research has significant implications for fields like economics, evolutionary biology, and cosmology, offering insights into phenomena like technological singularity, economic booms, and the evolution of the universe.
Limitations and Future Research: The paper primarily focuses on deterministic models. Future research could explore the impact of stochasticity and investigate the potential for a "logistic TAP equation" to model growth limitations.
Stats
The sequence for M0 = 2 in the simplest TAP equation with α = 1 and µ = 0 is {2, 3, 7, 127, ~10^38, ~exp(10^38)}.
The blow-up time for the constant αi model with M0 = 2, α = 10^-2, and µ = 0 is numerically determined to be 85 timesteps, while the analytical estimate is 200 timesteps.
In the two-scale TAP equation with α1 = 0.01, α = 10^-6, and no extinction, the blow-up occurs at Mt ≃ 17 and around timestep 210.
Quotes
"The TAP equation permits dramatically-explosive behaviour, much faster than exponential, because not only do the combinatoric terms rapidly become vast, but so too does the number of terms in the summation."
"The TAP equation is under active investigation as a possible model for a variety of processes... the explosive recent growth of GDP per capita, the rapidly-developing diversity of manufactured goods, and the family tree of US patent applications."
"Our findings show that the transition to the TAP blow-up is sudden, explosive, and is not foreshadowed by any features of the curves until its onset. It is hence essentially impossible to predict in advance the timing of a TAP blow-up, until it is already underway."