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 !