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# Graph Consensus: A Review

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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
• 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
• 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