Thursday, January 14, 2016

How fast does the space of possibilities expand? (replicating Tria, et al 2014)

How fast does the space of possibilities expand?  This question is explored in the following paper (free download):

From the abstract:
Novelties are a familiar part of daily life. They are also fundamental to the evolution of biological systems, human society, and technology. By opening new possibilities, one novelty can pave the way for others in a process that Kauffman has called “expanding the adjacent possible”. The dynamics of correlated novelties, however, have yet to be quantified empirically or modeled mathematically. Here we propose a simple mathematical model that mimics the process of exploring a physical, biological, or conceptual space that enlarges whenever a novelty occurs. The model, a generalization of Polya's urn, predicts statistical laws for the rate at which novelties happen (Heaps' law) and for the probability distribution on the space explored (Zipf's law), as well as signatures of the process by which one novelty sets the stage for another.
I've written a NetLogo program to replicate their model, available here.  The code for the model is quite simple.  A majority of my code is for a "pretty layout", which is a schematic version of a "top-down view" of the urn.  Here's a video of a single run

Full screen with controls. (click to enlarge)
The charts on the top and center right show the frequency distribution by ball type (a.k.a. "color").  These are log-log plots, so a straight line (declining) is signature of a power law distribution, while a gradually curving (concave) is signature of lognormal or similar distribution with somewhat thinner tail.  Sharply declining curve is signature of a thin tailed distribution such as Gaussian.

So what?

This model will be useful in my dissertation because I need mechanisms to endogenously add novelty -- i.e. expand the possibility space based on the actions of agents in the simulated world, and not simply as external "shocks".

This is essential for modeling cyber security because some people claim that quantitative risk management is impossible in principle because of intelligent adversaries who can generate and exploit novel strategies and capabilities.

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