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Graph Consensus: Autonomus and Controlled Prepared by Abhijit Das
Many of the beautiful pictures are from a lecture by  Ron Chen , City U. Hong Kong Pinning Control of Graphs Natural and b...
Airline Route Systems
Distribution of galaxies in the universe
Motions of biological groups Fish school Birds flock Locusts swarm Fireflies synchronize
J.J. Finnigan, Complex science for a complex world The internet ecosystem Professional Collaboration network Barcelona rai...
Graph Directed Graph or Diagraph Un-directed Graph 11/07/11 ARRI, UTA
Two properties of diagraph nodes <ul><li>Out-degree: Number of connections going out from a node </li></ul><ul><li>In-degr...
Important types of Diagraphs Balanced Strongly Connected Tree 11/07/11 ARRI, UTA
What is Consensus among nodes Consensus  in the English language is defined firstly as unanimous or general agreement Befo...
Graph Dynamics (Diagraph) Adjacency Matrix or Diagonal Matrix Laplacian matrix Note that  is row stochastic 11/07/11 ARRI,...
Continuous Time System <ul><li>Each node if assumed to have simple integrator dynamics, for  -th node, </li></ul><ul><li>I...
Comment As  is row stochastic The first eigenvalue of  will be 0 The right eigenvector corresponding to 0 eigenvalue will ...
State solution Eigen decomposition and Left and right eigenvector Right eigenvector  Left eigenvector  11/07/11 ARRI, UTA
State solution (Contd..) 11/07/11 ARRI, UTA
State solution (Contd..) At Steady state  11/07/11 ARRI, UTA
State solution (Contd..) with 11/07/11 ARRI, UTA
Finding consensus value for SC graph Considering only the first line of the equation For balanced graph 11/07/11 ARRI, UTA
Simulation results (SC graph) 11/07/11 ARRI, UTA
What if there is one leader in the graph Assuming rest of the graph is connected The Laplacian matrix of a graph with a le...
Consensus value for one leader graph Note that if there is more than one leaders then no single solution is possible 11/07...
Simulation result (one leader case) For tree network the result will be  equivalent 11/07/11 ARRI, UTA
Graph contains a spanning tree How the value of can be determined ? 11/07/11 ARRI, UTA
Eigenvalue properties <ul><li>For stability all the eigenvalues should be in the left half of the  plane </li></ul><ul><li...
Gershgorin disk of a network 11/07/11 ARRI, UTA
More comments <ul><li>Fiedler eigenvalue is also known as  algebraic connectivity or spectral gap  of a graph </li></ul><u...
Reducibility Consider a matrix  with  . If  is reducible, there exist an integer  and a Permutation matrix  such that  11/...
Irreducibility 11/07/11 ARRI, UTA Consider a matrix  . Then,  is  irreducible if and only if  For any scalar  .
Comment on reducibility <ul><li>A connected graph (strongly/balanced) is generally have irreducible adjacency and laplacia...
Discrete time system  Murray-Saber, 2004 Continuous time system  Max out-degree Discretized Perron matrix 11/07/11 ARRI, UTA
Definition 11/07/11 ARRI, UTA
Perron-Frobenius Theorem 11/07/11 ARRI, UTA
Comment 11/07/11 ARRI, UTA
State Solution- DT system 11/07/11 ARRI, UTA
Comparison 11/07/11 ARRI, UTA Courtesy: Fax-Murray-Saber, 2006
Performance –  Murray-Saber 2007 11/07/11 ARRI, UTA
Theorems 11/07/11 ARRI, UTA
Alternative Laplacian-Structure:  Fax-Murray 2004 11/07/11 ARRI, UTA
Based on Vicsek model:  Jadbabaie-Lin-Morse 11/07/11 ARRI, UTA
Example: Bipartite graph 11/07/11 ARRI, UTA
Trust Consensus:  Ballal-Lewis-2008 11/07/11 ARRI, UTA
Bilinear trust Dynamics 11/07/11 ARRI, UTA
Simulations 11/07/11 ARRI, UTA
Comment 11/07/11 ARRI, UTA
Zhihua Qu’s formulation 11/07/11 ARRI, UTA
Comment 11/07/11 ARRI, UTA
Passive system: Definition 11/07/11 ARRI, UTA
Mark Spong’s Lyapunov formulation 11/07/11 ARRI, UTA
Can we change  for which  11/07/11 ARRI, UTA
Zhihua Qu’s Lyapunov formulation 11/07/11 ARRI, UTA
Comments:  Zhihua Qu  11/07/11 ARRI, UTA
Lihua Xie’s Lyapunov formulation 11/07/11 ARRI, UTA
Lihua Xie’s formulation contd… 11/07/11 ARRI, UTA
Scale free network 11/07/11 ARRI, UTA Courtesy Wikipedia
Ron Chen’s pinning control 11/07/11 ARRI, UTA
Ron Chen’s Lyapunov formulation 11/07/11 ARRI, UTA
Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
Controlled consensus 11/07/11 ARRI, UTA
Some case studies Consensus time approx 7.5 sec 11/07/11 ARRI, UTA 1 4 2 3
Some case studies contd… 1 4 2 3 L Consensus time approx 8 sec 11/07/11 ARRI, UTA
Some case studies contd… 1 4 2 3 L Consensus time approx: 3 sec 11/07/11 ARRI, UTA
A special case 11/07/11 ARRI, UTA L L
A special case contd… L 11/07/11 ARRI, UTA
Mathematical formulation:  Lewis, 09 11/07/11 ARRI, UTA
Controlled consensus:  Lewis-’09 11/07/11 ARRI, UTA
Leader-Graph network Leader network Graph network Connection may  be from both way 11/07/11 ARRI, UTA
One case study:  based on Z. Qu’s Laplacian Lower triangularly  complete 11/07/11 ARRI, UTA N1 N3 N2
One case study 11/07/11 ARRI, UTA
Case study: contd… 11/07/11 ARRI, UTA
Jadbabaie-Lin-Morse’s leader network 11/07/11 ARRI, UTA
Noisy information exchange:  Ren-Beard-Kingston-2005 11/07/11 ARRI, UTA
Estimator dynamics 11/07/11 ARRI, UTA
Das-Lewis contribution 11/07/11 ARRI, UTA Select from Lyapunov
11/07/11 ARRI, UTA Thank you
Addendum:  Zhihong Man 11/07/11 ARRI, UTA
Addendum:  Lihua Xie 11/07/11 ARRI, UTA
Addendum:  Courtesy Fang-Antsaklis 11/07/11 ARRI, UTA
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Transcript of "Graph Consensus: A Review"

  1. 1. Graph Consensus: Autonomus and Controlled Prepared by Abhijit Das
  2. 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. 3. Airline Route Systems
  4. 4. Distribution of galaxies in the universe
  5. 5. Motions of biological groups Fish school Birds flock Locusts swarm Fireflies synchronize
  6. 6. J.J. Finnigan, Complex science for a complex world The internet ecosystem Professional Collaboration network Barcelona rail network
  7. 7. Graph Directed Graph or Diagraph Un-directed Graph 11/07/11 ARRI, UTA
  8. 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. 9. Important types of Diagraphs Balanced Strongly Connected Tree 11/07/11 ARRI, UTA
  10. 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. 11. Graph Dynamics (Diagraph) Adjacency Matrix or Diagonal Matrix Laplacian matrix Note that is row stochastic 11/07/11 ARRI, UTA
  12. 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. 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. 14. State solution Eigen decomposition and Left and right eigenvector Right eigenvector Left eigenvector 11/07/11 ARRI, UTA
  15. 15. State solution (Contd..) 11/07/11 ARRI, UTA
  16. 16. State solution (Contd..) At Steady state 11/07/11 ARRI, UTA
  17. 17. State solution (Contd..) with 11/07/11 ARRI, UTA
  18. 18. Finding consensus value for SC graph Considering only the first line of the equation For balanced graph 11/07/11 ARRI, UTA
  19. 19. Simulation results (SC graph) 11/07/11 ARRI, UTA
  20. 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. 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. 22. Simulation result (one leader case) For tree network the result will be equivalent 11/07/11 ARRI, UTA
  23. 23. Graph contains a spanning tree How the value of can be determined ? 11/07/11 ARRI, UTA
  24. 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. 25. Gershgorin disk of a network 11/07/11 ARRI, UTA
  26. 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. 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. 28. Irreducibility 11/07/11 ARRI, UTA Consider a matrix . Then, is irreducible if and only if For any scalar .
  29. 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. 30. Discrete time system Murray-Saber, 2004 Continuous time system Max out-degree Discretized Perron matrix 11/07/11 ARRI, UTA
  31. 31. Definition 11/07/11 ARRI, UTA
  32. 32. Perron-Frobenius Theorem 11/07/11 ARRI, UTA
  33. 33. Comment 11/07/11 ARRI, UTA
  34. 34. State Solution- DT system 11/07/11 ARRI, UTA
  35. 35. Comparison 11/07/11 ARRI, UTA Courtesy: Fax-Murray-Saber, 2006
  36. 36. Performance – Murray-Saber 2007 11/07/11 ARRI, UTA
  37. 37. Theorems 11/07/11 ARRI, UTA
  38. 38. Alternative Laplacian-Structure: Fax-Murray 2004 11/07/11 ARRI, UTA
  39. 39. Based on Vicsek model: Jadbabaie-Lin-Morse 11/07/11 ARRI, UTA
  40. 40. Example: Bipartite graph 11/07/11 ARRI, UTA
  41. 41. Trust Consensus: Ballal-Lewis-2008 11/07/11 ARRI, UTA
  42. 42. Bilinear trust Dynamics 11/07/11 ARRI, UTA
  43. 43. Simulations 11/07/11 ARRI, UTA
  44. 44. Comment 11/07/11 ARRI, UTA
  45. 45. Zhihua Qu’s formulation 11/07/11 ARRI, UTA
  46. 46. Comment 11/07/11 ARRI, UTA
  47. 47. Passive system: Definition 11/07/11 ARRI, UTA
  48. 48. Mark Spong’s Lyapunov formulation 11/07/11 ARRI, UTA
  49. 49. Can we change for which 11/07/11 ARRI, UTA
  50. 50. Zhihua Qu’s Lyapunov formulation 11/07/11 ARRI, UTA
  51. 51. Comments: Zhihua Qu 11/07/11 ARRI, UTA
  52. 52. Lihua Xie’s Lyapunov formulation 11/07/11 ARRI, UTA
  53. 53. Lihua Xie’s formulation contd… 11/07/11 ARRI, UTA
  54. 54. Scale free network 11/07/11 ARRI, UTA Courtesy Wikipedia
  55. 55. Ron Chen’s pinning control 11/07/11 ARRI, UTA
  56. 56. Ron Chen’s Lyapunov formulation 11/07/11 ARRI, UTA
  57. 57. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
  58. 58. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
  59. 59. Ron Chen’s formulation contd… 11/07/11 ARRI, UTA
  60. 60. Controlled consensus 11/07/11 ARRI, UTA
  61. 61. Some case studies Consensus time approx 7.5 sec 11/07/11 ARRI, UTA 1 4 2 3
  62. 62. Some case studies contd… 1 4 2 3 L Consensus time approx 8 sec 11/07/11 ARRI, UTA
  63. 63. Some case studies contd… 1 4 2 3 L Consensus time approx: 3 sec 11/07/11 ARRI, UTA
  64. 64. A special case 11/07/11 ARRI, UTA L L
  65. 65. A special case contd… L 11/07/11 ARRI, UTA
  66. 66. Mathematical formulation: Lewis, 09 11/07/11 ARRI, UTA
  67. 67. Controlled consensus: Lewis-’09 11/07/11 ARRI, UTA
  68. 68. Leader-Graph network Leader network Graph network Connection may be from both way 11/07/11 ARRI, UTA
  69. 69. One case study: based on Z. Qu’s Laplacian Lower triangularly complete 11/07/11 ARRI, UTA N1 N3 N2
  70. 70. One case study 11/07/11 ARRI, UTA
  71. 71. Case study: contd… 11/07/11 ARRI, UTA
  72. 72. Jadbabaie-Lin-Morse’s leader network 11/07/11 ARRI, UTA
  73. 73. Noisy information exchange: Ren-Beard-Kingston-2005 11/07/11 ARRI, UTA
  74. 74. Estimator dynamics 11/07/11 ARRI, UTA
  75. 75. Das-Lewis contribution 11/07/11 ARRI, UTA Select from Lyapunov
  76. 76. 11/07/11 ARRI, UTA Thank you
  77. 77. Addendum: Zhihong Man 11/07/11 ARRI, UTA
  78. 78. Addendum: Lihua Xie 11/07/11 ARRI, UTA
  79. 79. Addendum: Courtesy Fang-Antsaklis 11/07/11 ARRI, UTA
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