1. Graph Consensus: Autonomus and Controlled Prepared by Abhijit Das

2. Many of the beautiful pictures are from a lecture by Ron Chen , City U. Hong Kong Pinning Control of Graphs Natural and biological structures

3. Airline Route Systems

4. Distribution of galaxies in the universe

5. Motions of biological groups Fish school Birds flock Locusts swarm Fireflies synchronize

6. J.J. Finnigan, Complex science for a complex world The internet ecosystem Professional Collaboration network Barcelona rail network

7. Graph Directed Graph or Diagraph Un-directed Graph 11/07/11 ARRI, UTA

9. Important types of Diagraphs Balanced Strongly Connected Tree 11/07/11 ARRI, UTA

10. What is Consensus among nodes Consensus in the English language is defined firstly as unanimous or general agreement Before Consensus After Consensus 11/07/11 ARRI, UTA

11. Graph Dynamics (Diagraph) Adjacency Matrix or Diagonal Matrix Laplacian matrix Note that is row stochastic 11/07/11 ARRI, UTA

13. Comment As is row stochastic The first eigenvalue of will be 0 The right eigenvector corresponding to 0 eigenvalue will be At steady state all state values will be equal 11/07/11 ARRI, UTA

14. State solution Eigen decomposition and Left and right eigenvector Right eigenvector Left eigenvector 11/07/11 ARRI, UTA

15. State solution (Contd..) 11/07/11 ARRI, UTA

16. State solution (Contd..) At Steady state 11/07/11 ARRI, UTA

17. State solution (Contd..) with 11/07/11 ARRI, UTA

18. Finding consensus value for SC graph Considering only the first line of the equation For balanced graph 11/07/11 ARRI, UTA

19. Simulation results (SC graph) 11/07/11 ARRI, UTA

20. What if there is one leader in the graph Assuming rest of the graph is connected The Laplacian matrix of a graph with a leader with may be anything Left eigenvector 11/07/11 ARRI, UTA

21. Consensus value for one leader graph Note that if there is more than one leaders then no single solution is possible 11/07/11 ARRI, UTA

22. Simulation result (one leader case) For tree network the result will be equivalent 11/07/11 ARRI, UTA

23. Graph contains a spanning tree How the value of can be determined ? 11/07/11 ARRI, UTA

25. Gershgorin disk of a network 11/07/11 ARRI, UTA

27. Reducibility Consider a matrix with . If is reducible, there exist an integer and a Permutation matrix such that 11/07/11 ARRI, UTA

28. Irreducibility 11/07/11 ARRI, UTA Consider a matrix . Then, is irreducible if and only if For any scalar .

30. Discrete time system Murray-Saber, 2004 Continuous time system Max out-degree Discretized Perron matrix 11/07/11 ARRI, UTA

31. Definition 11/07/11 ARRI, UTA

32. Perron-Frobenius Theorem 11/07/11 ARRI, UTA

33. Comment 11/07/11 ARRI, UTA

34. State Solution- DT system 11/07/11 ARRI, UTA

35. Comparison 11/07/11 ARRI, UTA Courtesy: Fax-Murray-Saber, 2006

36. Performance – Murray-Saber 2007 11/07/11 ARRI, UTA

37. Theorems 11/07/11 ARRI, UTA

38. Alternative Laplacian-Structure: Fax-Murray 2004 11/07/11 ARRI, UTA

39. Based on Vicsek model: Jadbabaie-Lin-Morse 11/07/11 ARRI, UTA

40. Example: Bipartite graph 11/07/11 ARRI, UTA

41. Trust Consensus: Ballal-Lewis-2008 11/07/11 ARRI, UTA

42. Bilinear trust Dynamics 11/07/11 ARRI, UTA

43. Simulations 11/07/11 ARRI, UTA

44. Comment 11/07/11 ARRI, UTA

45. Zhihua Qu’s formulation 11/07/11 ARRI, UTA

46. Comment 11/07/11 ARRI, UTA

47. Passive system: Definition 11/07/11 ARRI, UTA

48. Mark Spong’s Lyapunov formulation 11/07/11 ARRI, UTA

49. Can we change for which 11/07/11 ARRI, UTA

50. Zhihua Qu’s Lyapunov formulation 11/07/11 ARRI, UTA

51. Comments: Zhihua Qu 11/07/11 ARRI, UTA

52. Lihua Xie’s Lyapunov formulation 11/07/11 ARRI, UTA

53. Lihua Xie’s formulation contd… 11/07/11 ARRI, UTA

54. Scale free network 11/07/11 ARRI, UTA Courtesy Wikipedia

55. Ron Chen’s pinning control 11/07/11 ARRI, UTA

56. Ron Chen’s Lyapunov formulation 11/07/11 ARRI, UTA

57. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA

58. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA

59. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA

60. Controlled consensus 11/07/11 ARRI, UTA

61. Some case studies Consensus time approx 7.5 sec 11/07/11 ARRI, UTA 1 4 2 3

62. Some case studies contd… 1 4 2 3 L Consensus time approx 8 sec 11/07/11 ARRI, UTA

63. Some case studies contd… 1 4 2 3 L Consensus time approx: 3 sec 11/07/11 ARRI, UTA

64. A special case 11/07/11 ARRI, UTA L L

65. A special case contd… L 11/07/11 ARRI, UTA

66. Mathematical formulation: Lewis, 09 11/07/11 ARRI, UTA

67. Controlled consensus: Lewis-’09 11/07/11 ARRI, UTA

68. Leader-Graph network Leader network Graph network Connection may be from both way 11/07/11 ARRI, UTA

69. One case study: based on Z. Qu’s Laplacian Lower triangularly complete 11/07/11 ARRI, UTA N1 N3 N2

70. One case study 11/07/11 ARRI, UTA

71. Case study: contd… 11/07/11 ARRI, UTA

72. Jadbabaie-Lin-Morse’s leader network 11/07/11 ARRI, UTA

73. Noisy information exchange: Ren-Beard-Kingston-2005 11/07/11 ARRI, UTA

74. Estimator dynamics 11/07/11 ARRI, UTA

75. Das-Lewis contribution 11/07/11 ARRI, UTA Select from Lyapunov

76. 11/07/11 ARRI, UTA Thank you

77. Addendum: Zhihong Man 11/07/11 ARRI, UTA

78. Addendum: Lihua Xie 11/07/11 ARRI, UTA

79. Addendum: Courtesy Fang-Antsaklis 11/07/11 ARRI, UTA