• Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
615
On Slideshare
0
From Embeds
0
Number of Embeds
1

Actions

Shares
Downloads
7
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 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
  • 8. Two properties of diagraph nodes
    • Out-degree: Number of connections going out from a node
    • In-degree: Number of connections going in to a node
    • Edge: Connection between any two nodes
    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
  • 12. Continuous Time System
    • Each node if assumed to have simple integrator dynamics, for -th node,
    • Input
    • Resultant Dynamics of the graph with all node
    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
  • 24. Eigenvalue properties
    • For stability all the eigenvalues should be in the left half of the plane
    • The second largest eigenvalue is of a standard laplacian matrix is known as Fiedler eigenvalue
    • Fiedler eigenvalue determines the speed of the whole network, thus it is important to maximize its value
    • Note that Fiedler eigenvalue in general can not be determined from the dominant eigenvalue of the inverse of laplacian matrix
    11/07/11 ARRI, UTA
  • 25. Gershgorin disk of a network 11/07/11 ARRI, UTA
  • 26. More comments
    • Fiedler eigenvalue is also known as algebraic connectivity or spectral gap of a graph
    • Algebraic connectivity is different from connectivity or vertex-connectivity
    • Network synchronization speed does NOT depend on vertex-connectivity
    • Number of zero eigenvalues in a laplacian matrix reveals, number of connected components in a graph
    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 .
  • 29. Comment on reducibility
    • A connected graph (strongly/balanced) is generally have irreducible adjacency and laplacian matrix
    • A tree network generally posses a reducible adjacency and laplacian matrix
    11/07/11 ARRI, UTA
  • 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