1. MASSIMO FRANCESCHETTI University of California at Berkeley Phase transitions an engineering perspective

2. when small changes in certain parameters of a system result in dramatic shifts in some globally observed behavior of the system. Phase transition effect

3. Example percolation theory, Broadbent and Hammersley (1957)

4. Example Broadbent and Hammersley (1957) H. Kesten (1980) pcpc 0 p P 1

5. Gilbert (1961) Mathematics Physics Percolation theory Random graphs Random Coverage Processes Continuum Percolation Models of the internet Impurity Conduction Ferromagnetism… Universality, Ken Wilson Nobel prize Grimmett (1989) Bollobas (1985) Hall (1985) Meester and Roy (1996) Broadbent and Hammersley (1957) Erdös and Rényi (1959) Phase transitions in random graphs

6. Large scale networks of embedded devices Ad-Hoc networks Sensor networks Where are the phase transitions?

7. Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component First model Continuum percolation, Gilbert (1961)

8. First model Continuum percolation, Gilbert (1961) The first paper in ad hoc wireless networks ! A B

9.  0.3  0.4 c  0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000] Example

10. Unit area Penrose (1997) Penrose Pisztora (1996) Xue Kumar (2003) Gilbert (1961) [Meester, Roy book (1996)] Haggstrom Meester (1996) Dousse Baccelli Thiran (2003) Booth, Bruck, Franceschetti, Meester (2003) Density of points in Continuum Percolation Nearest neighbors percolation Percolation with interference Clustered networks More wireless models

11. Towards a less idealistic model Franceschetti, Booth, Cook, Meester, Bruck (2003)

12. Prob(correct reception) Experiment

13. 1 Connection probability d Continuum percolation 2r Our model Our model d 1 Connection probability Connectivity model

14. Connection probability 1 x A first order question How does the percolation threshold c change?

15. Squishing and Squashing Connection probability x

16. Theorem For all “longer links are trading off for the unreliability of the connection” “it is easier to reach connectivity in an unreliable network”

17. Shifting and Squeezing Connection probability x

18. Example Connection probability x 1

19. Mixture of short and long edges Edges are made all longer Do long edges help percolation?

20. Conjecture For all

21. Theorem For all, there exists a finite, such that gss r* (x) percolates, for all It is possible to decrease the percolation threshold by taking a sufficiently large shift ! For all    2 )(0 x xg

22. CNP Squishing and squashing Shifting and squeezing What have we proven? (sporadic) long links help the percolation process

23. CNP Among all convex shapes the hardest to percolate is centrally symmetric Jonasson (2001) Is the disc the hardest shape to percolate overall? What about non-circular shapes?

24. CNP To the engineer: above 4.51 we are fine! To the theoretician: can we prove “disc is hardest” conjecture? can we exploit long links for routing? Bottom line

25. Small World Networks Regular Random Small World Watts Strogatz (1998)

26. Kleinberg (2000)Franceschetti & Meester (2003) Routing in a small world Nodes on the grid Fixed number of contacts Probability scales with distance Nodes on the plane Random number contacts in a given region Density scales with distance each node has only local information of the network connectivity

27. Routing in a small world Connections of z are PPP of density  - delivery occurs when msg is delivered within  to target  S T

28. Routing in a small world  S T d

29.  S T d Scale the number of neighbors as 1/x 2 to obtain efficient routing

30. Not only graphs… One application

31. A pursuit evasion game Sinopoli, Schenato, Franceschetti, Poolla, Sastry (2003)

32. A pursuit evasion game

44. Goal: given observations find the best estimate (minimum variance) for the state But may not arrive at each time step when traveling over a sensor network Intermittent observations Problem formulation

45. System Kalman Filter M z -1 utut etet xtxt M K + + + - x t+1 y t+1

46. Discrete time LTI system and are Gaussian random variables with zero mean and covariance matrices Q and R positive definite. Loss of observation

47. Discrete time LTI system Let it have a “huge variance” when the observation does not arrive Loss of observation

48. The arrival of the observation at time t is a binary random variable Redefine the noise as: Kalman Filter with losses Derive Kalman equations using a “dummy” observation when then take the limit for  t =0

49. Results on mean error covariance P t

50. Special cases C is invertible, or A has a single unstable eigenvalue

51. Conclusion Phase transitions are a fundamental effect in engineering systems with randomness There is plenty of formal work to be done

52. For papers: massimof@EECS.berkeley.edu “Navigation in small world networks, a continuum scale-free model” Franceschetti and Meester Preprint ‘03 “Percolation in wireless multi-hop networks”, Franceschetti, Booth, Cook, Meester, Bruck ISIT ’03 and Submitted to IEEE Trans. Info Theory “Covering algorithm continuum percolation and the geometry of wireless networks” Booth, Bruck, Franceschetti, Meester Annals of Applied Probability, 13(2), May 2003. “Kalman Filtering with intermittent observations” Sinopoli, Schenato, Franceschetti, Poolla, Sastry CDC ’03 and Submitted to IEEE Trans. Automatic Control