Dynamic Networks, Influence Systems, and Renormalization Bernard Chazelle Princeton University
Feb 22, 2016
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
Communication rules network
Communication rules network
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?
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 chaoticRandom perturbation leads to a limit cycle almost surely
Phase transitions form a Cantor setPredicting long-range behavior is undecidable
The role of deterministic “randomness”
Bounding the topological entropy
via
algorithmic renormalization
Incoherent contractive eigenmodes
Language
Language Grammar
Parse tree
Parse tree produced by flow tracker
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 allowsrecursive estimation of topological
entropyby working on subsystems
The mixing of timescales
-1
1
1
Trio settles quickly
-1
1
1
Duck learns about her
-1
-1
1
1
-1
-1
1
1
Limit cycle means amnesia
-1
-1
1
1
She regains her memoryLimit cycle is destroyed !
Thank you, John, Leonid, Raghu,
and Joel !