1. Towards a theory of large scale networks Massimo Franceschetti

2. Where are we heading year Log ( people per computer) Number Crunching Data Storage Networking Communication Interaction with the physical world

3. Building Comfort, Smart Alarms Great Duck Island Elder Care Fire Response Factories Wind Response Of Golden Gate Bridge Vineyards Redwoods Instrumenting the world Soil monitoring

4. Connected ? Information throughput ? Transmission power ? routing ? delay ? reliability ? Extreme scaling Large scale networks theory

5. Spatial stochastic networks Connectivity Sensor placement algorithms Throughput capacity Closing the loop Physics of propagation Distributed Algorithms Talk Outline

6. Poisson distribution of points of density λ Points are connected if their distance is less than 2r Studies the formation of connected components Continuum percolation S D Gilbert J.SIAM (1961) Book: Meester & Roy (1995)

7. There is a phase transition at a critical node density value Phase transition

8. Simulation of phase transition Gradually increase the node density on the plane

9. λcλc λ2λ2 1 0 λ P λ1λ1 P = Prob(exists unbounded connected component) Phase transition Theorem Gilbert J.SIAM (1961)

10. Networks with interference Communication range is different from connectivity Nodes in close range might not be connected Dependent percolation model What happens to the phase transition?

11. A percolation model for wireless networks Node j can receive data from node i if the signal to noise plus interference ratio is above a threshold  All nodes transmit at power P

12. A percolation model for wireless networks   Gupta Kumar, IEEE Trans. IT, 2000]   Gilbert, J. SIAM, 1961]  [Dousse Baccelli Thiran, IEEE Trans. Net, 2004] [Dousse Franceschetti Meester Thiran, preprint, 2004] 10  All nodes transmit at power P Node j can receive data from node i if the signal to noise plus interference ratio is above a threshold 

13. Phase transition theorem (Dousse Baccelli Thiran)

14. Can we prove a stronger result ? c super-critical sub-critical No interference model (Gilbert)  Interference model  super-critical

15. Theorem c super-critical sub-critical  (Dousse Franceschetti Meester Thiran) No interference model (Gilbert) Interference model  super-critical

16. Bottom line An ideal network with perfect interference cancellation (  independent percolation model) exhibits a phase transition for  c A network where nodes cause interference (  dependent percolation model) exhibits a phase transition for  c,   

17. More work on connectivity Gilbert’s model Interference model Spread out, unreliable connections

18. Prob(correct reception) Let’s look at some real data 168 nodes on a 12x14 grid grid spacing 2 feet only one node transmits “I’m Alive” (no interference) surrounding nodes try to receive message http://localization.millennium.berkeley.edu

19. Absence of sharp threshold 1 Connection probability ||x i -x j || 1 Connection probability How does the critical density change with the shape of the connection probability? c

20. Connection probability ||x i -x j || Basic transformation: spreading probability

21. Example Connection probability 1 ||x||

22. longer links are trading off for the unreliability of the connection Theorem Franceschetti Booth Cook Bruck Meester (2003) It is easier to reach connectivity with unreliable spread out connections

23. Conjecture More complex spreading of g also helps percolation 1 Disc is hardest shape to percolate

24. More work on connectivity Gilbert’s model Extension with interference Spread out, unreliable connections Sensor placement algorithms

25. Clustered wireless networks Random point process Algorithm Connectivity each point is covered by at least a red disc and each red disc covers at least a point Franceschetti PhD Thesis, CIT 2003

26. Theorem if for any covering algorithm, with probability one then for  c percolation occurs 1  if then some covering algorithms may avoid percolation for any value of λ, with probability one 1  R radius blue disc r radius red disc r

27. Covering algorithms “Covering Algorithms, continuum percolation, and the geometry of wireless networks” Annals of Applied Probability, 2003 Booth Bruck Franceschetti Meester PhD Thesis, CIT, 2003 Franceschetti When which classes of algorithms form an unbounded connected component, a.s., When is high?

28. From connectivity to network capacity Gilbert’s model Extension with interference Spread out, unreliable connections Sensor placement algorithms Throughput capacity

29. How much information can flow through the network?

30. Throughput capacity without interference How many disjoint paths there are that traverse the network? Nodes closer than a given range are connected

31. High density = more disjoint paths Interference !

32. Main claim Operate the network near percolation threshold not in the high density regime

33. Previous results, routing in high density regime Gupta Kumar (2000) Kulkarni Viswanath (2002) El Gamal, Mammen, Prabhakar, Shah (2004)

34. High density regime routing Divide area into small boxes Scale down power to allow only transmission to adjacent boxes Route along almost straight lines

35. Previous results, high density regime

36. Our strategy, percolation regime Franceschetti Dousse Tse Thiran (2004)

37. Adopt thermodynamic scaling Some boxes are empty Our strategy

38. Adopt thermodynamic scaling Take c large to have many crossing paths of adjacent full boxes (by percolation) Use these paths as the “wireless backbone” to relay traffic crossing path Our strategy

39. Theorem crossing path This strategy achieves maximum throughput, minumum delay

40. Throughput per node vs Range Interference limited network No interference ideal network 0  c Percolation =log n =n High density regime Interference

41. Bottom line Scaling power at a slow rate, order, can use straight line routing Scaling power at a fast rate, disorder, no backbone forms Phase transition, backbone forms, rich in crossing paths, not straight lines, carry most traffic over short hops

42. Gilbert’s model Extension with interference Spread out, unreliable connections Sensor placement algorithms Throughput capacity Closing the loop Let’s look at the Application level

43. Pursuit evasion games at Berkeley

44. Random losses in the feedback loop Sinopoli Schenato Franceschetti Poolla Sastry Jordan IEEE Trans-AC (2004) System Sensor web Controller State estimator Wireless Multi-hop What happens to the Kalman filter when some sensor readings are lost? Can we bound the error covariance

45. Theorem Sinopoli Schenato Franceschetti Poolla Sastry Jordan IEEE Trans-AC (2004) c 1 0

46. Theorem Sinopoli Schenato Franceschetti Poolla Sastry Jordan IEEE Trans-AC (2004)

47. The road ahead Towards a system theory of large scale networks Spatial stochastic networks as a core discipline Intellectual unification across disciplines

48. New branches Time varying stochastic networks Impact of mobility Games on graphs

49. Phase transitions Phase transitions are a fundamental effect in engineering systems with randomness Optimal operation regions are often at the boundary of these transitions

50. A random walk model of wave propagation IEEE Trans.-AP Franceschetti Bruck Schulman Stochastic rays propagation IEEE Trans.-AP Franceschetti Some more work… Interaction with physical level Small world networks Small-world networks a continuum model Franceschetti Meester A geometric theorem for network design IEEE Trans.-Comp. Franceschetti Cook Bruck Lower bounds on data collection times in sensory networks IEEE-JSAC Florens Franceschetti McEliece A group membership algorithm with a practical specification IEEE Trans.-PDS Franceschetti Bruck Algorithms and protocols