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

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