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. libertarian authoritarian left right

5. libertarian authoritarian left right

6. libertarian authoritarian left right

7. libertarian authoritarian left right

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

13. Repeat forever

14. 20,000 agents

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

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

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

25. Dynamical rules ( must respect network)

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

27. Influence systems Very general !

28. Diffusive Influence systems convexity deterministic

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

30. Phase space

32. What if all the matrices are the same?

33. fixed-point attractors or limit cycles

34. Theory of Markov chains Theory of diffusive influence systems

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

36. 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

37. The role of deterministic “randomness”

38. Bounding the topological entropy via algorithmic renormalization

39. Incoherent contractive eigenmodes

40. Language

41. Grammar

42. Parse tree

43. Parse tree produced by flow tracker

48. time

49. Ready for normalization !

50. We need a recursive language

51. Direct sum Direct product

52. Renormalized dynamical subsystems

53. What’s the point of all this ? Algorithmic renormalization allows recursive estimation of topological entropy by working on subsystems

54. The mixing of timescales

55. 1 1 Trio settles quickly

56. 1 1 Duck learns about her

57. 1 1

58. 1 1 Limit cycle means amnesia

59. 1 1 She regains her memoryLimit cycle is destroyed !

60. Thank you, John, Leonid, Raghu, and Joel !