Dynamic Networks, Influence Systems, and Renormalization Bernard Chazelle Princeton University
Interacting particles, each one with its own physical laws !
Hegselmann-Krause systems
libertarian authoritarian left right
libertarian authoritarian left right
libertarian authoritarian left right
libertarian authoritarian left right
Each agent chooses weights and moves to weighted mass center of neighbors
Repeat forever
20,000 agents
Dynamical rules here, averaging Communication rules network
Eliminate quantifiers (Tarski-Collins) Communication rules network
Interacting particles, each with its own communication laws !
Dynamical rules ( must respect network)
eg, Ising model, swarm systems, voter model Dynamical rules ( must respect network)
Influence systems Very general !
Diffusive Influence systems convexity deterministic
stochastic matrix Dynamical system in high dimension Dynamic network associated with P (x)
Phase space
What if all the matrices are the same?
fixed-point attractors or limit cycles
Theory of Markov chains Theory of diffusive influence systems
Results Diffusive influence systems can be chaotic All Lyapunov exponents are
Results Diffusive influence systems can be chaotic Random perturbation leads to a limit cycle almost surely Phase transitions form a Cantor set Predicting long-range behavior is undecidable
The role of deterministic “randomness”
Bounding the topological entropy via algorithmic renormalization
Incoherent contractive eigenmodes
Language
Grammar
Parse tree
Parse tree produced by flow tracker
time
Ready for normalization !
We need a recursive language
Direct sum Direct product
Renormalized dynamical subsystems
What’s the point of all this ? Algorithmic renormalization allows recursive estimation of topological entropy by working on subsystems
The mixing of timescales
1 1 Trio settles quickly
1 1 Duck learns about her
1 1
1 1 Limit cycle means amnesia
1 1 She regains her memoryLimit cycle is destroyed !
Thank you, John, Leonid, Raghu, and Joel !