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