Where Does the Statistics of Complex Systems Come From?

Explore the origins of complex systems statistics in this comprehensive lecture by Stefan Thurner at the International Centre for Theoretical Sciences. Delve into sample space reducing (SSR) processes and their connection to Zipf's law, examining applications in sentence formation, network diffusion, and driven non-equilibrium systems. Learn about the robustness of Zipf's law, conservation laws in SSR processes, and the role of driving rates in determining system statistics. Gain insights into various examples of SSR-nature phenomena and their implications for understanding complex systems across multiple disciplines.

Where does the statistics of complex systems come from? Stefan Turner
Relaxation processes are sample space reducing
Example: History-dependent SSR processes
Sentence-formation is SSR
SSR lead to exact Zipf's law!
Proof
What if restart SSR before it is fully relaxed?
Driven SSR
History-dependent processes with fast driving
SSR-based Zipf law is extremely robust
SSR lead to exact Zipf's law!
prior probabilities are practically irrelevant!
Zipf-law is extremely robust-accelerated SSR
SSR and diffusion on networks
SSR is a random walk on directed ordered NW
SSR = targeted random walk on networks
All diffusion processes on DAG are SSR
What happens if introduce weights on links?
What happens if we introduce cycles?
Zipf's law is an immense attractor
Conservation laws in SSR processes
Complex systems are driven non-equilibrium
Where do all the distributions come from? - Assume that driving rate depends on state 1i b
Special cases Ax = -x4, log px
Driving determines statistics of driven systems
Examples that are of SSR-nature
Conclusions
More power laws ...