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Dynamics over Random Networks

Dynamics over Random Networks. Mehran Mesbahi Aeronautics and Astronautics University of Washington August 2005 Napa Valley. Sinoatrial Node. TPF-like missions. Gene networks. power grid. Yeast Proteins. networked vehicles. dynamic processes over networks.

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Dynamics over Random Networks

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  1. Dynamics over Random Networks Mehran Mesbahi Aeronautics and Astronautics University of Washington August 2005 Napa Valley

  2. Sinoatrial Node TPF-like missions Gene networks power grid Yeast Proteins networked vehicles

  3. dynamic processes over networks • spread of information, diseases, rumors • diffusion process, agreement • Ising model, spin systems, Landhu theory • cellular auotmata • iterated games • weakly coupled oscillators, synchronization

  4. Laplacians/agreement dynamics possibly dynamic example:

  5. random graphs Paul Erdos “A mathematician is a device for turning coffee into theorems.”

  6. agreement over random networks n = 10 P = 0.1 Gk(n,p) updates every second Gk(n,p) is disconnected for all k 10 sec

  7. dynamics over random graphs:why bother? robustness analysis ease of use quasi-randomness probabilistic method “I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive.” Plato

  8. Laplacians dynamics over random graphs k=0 k=1 k=2 k=3 G0(n,p) G1(n,p) G2(n,p) G3(n,p)

  9. agreement over random networks • agreement on heading direction • edge probability p=0.02 • random graph updates every 2 seconds : agent with heading vector : information link

  10. Erdos meets Lyapunov “Television is something the Russians invented to destroy American education.”

  11. towards agreement

  12. from supermatringales to agreement

  13. rate of convergence Rate of convergence depends on

  14. random matrices and their spectrum Juvan and Mohar (1993) Juhasz (1991) Furedi & Komlos (1981) Wigner (1955)

  15. l2(n,p) for large networks

  16. l2(n,p) for large networks • sharper bounds for when n large • : increasing function of n and p Recall:

  17. l2(n,p) # of times 100 0.8 50 0 observation: for fixed p convergence is improved for large n

  18. on deterministic uses of randomness why study dynamics over random graphs • robustness analysis • ease of use • quasi-randomness • probabilistic method • in the spirit of • randomized algorithms • probabilistic method • probabilistic checkable proofs • statistical mechanics • and so on …

  19. state-dependent graphs

  20. density/regularity 1 1 2 . . 2 . . . . n n .

  21. Szemeredi Regularity n 1 A. Szemeredi

  22. a dynamic graph controllability concept

  23. extensions • relaxing the independence assumption in time • effect of noise • forcing terms • random directed networks • special classes of nonlinear systems • (attitude dynamics, power systems)

  24. references dynamics over random graphs: recap robustness analysis ease of use quasi-randomness probabilistic method www.aa.washington.edu/faculty/mesbahi Acknowledgement: NSF joint with Y. Hatano “The probable is what usually happens.” Aristotle

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