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Dynamic Networks, Influence Systems, and Renormalization

This work by Bernard Chazelle delves into the interplay between dynamic networks and influence systems, highlighting models where interacting agents operate under distinct physical laws and communication rules. It explores Hegselmann-Krause systems, authoritarian and libertarian roles, and their implications in a dynamic network. With 20,000 agents adapting weights to converge towards the weighted mass center of neighbors, the study examines chaotic behaviors, phase transitions, and the challenging predictability of long-range dynamics within these complex systems.

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Dynamic Networks, Influence Systems, and Renormalization

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  1. Dynamic Networks, Influence Systems, and Renormalization Bernard Chazelle Princeton University

  2. Interacting particles, each one with its own physical laws !

  3. Hegselmann-Krause systems

  4. authoritarian left right libertarian

  5. authoritarian left right libertarian

  6. authoritarian left right libertarian

  7. authoritarian left right libertarian

  8. Each agent chooses weights and moves to weighted mass center of neighbors

  9. Repeat forever

  10. 20,000 agents

  11. Communication rules  network Dynamical rules  here, averaging

  12. Communication rules  network

  13. Communication rules  network

  14. Communication rules  network

  15. Communication rules  network Eliminate quantifiers (Tarski-Collins)

  16. Interacting particles, each with its own communication laws!

  17. Dynamical rules ( must respect network)

  18. Dynamical rules ( must respect network) eg, Ising model, swarm systems, voter model

  19. Influence systems Very general !

  20. Diffusive Influence systems deterministic convexity

  21. Dynamical system in high dimension stochastic matrix Dynamic network associated with P (x)

  22. Phase space

  23. What if all the matrices are the same?

  24. What if all the matrices are the same? fixed-point attractors or limit cycles

  25. Theory of diffusive influence systems Theory of Markov chains

  26. Results Diffusive influence systems can be chaotic All Lyapunov exponents are

  27. 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

  28. The role of deterministic “randomness”

  29. Bounding the topological entropy via algorithmicrenormalization

  30. Incoherent contractive eigenmodes

  31. Language

  32. Grammar Language

  33. Parse tree

  34. Parse tree produced by flow tracker

  35. Parse tree produced by flow tracker

  36. time

  37. Ready for normalization !

  38. We need a recursive language

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