1. Conceptual issues in scaling sensor networks Massimo Franceschetti, UC Berkeley

2. UCB Sensor Networks day, Jan 28, 2004 2 State of the art on scaling: PEG ~100 sensor nodes, 1 evader, 2 pursuers Cory Sharp & Shawn Schaffert Shankar Sastry group

3. UCB Sensor Networks day, Jan 28, 2004 3 Can we dramatically scale this? Practical problems Conceptual problems design for complexity

4. UCB Sensor Networks day, Jan 28, 2004 4 Percolation theory Random graphs Distributed computing Distributed control Channel physics Distributed sampling Network information theory Network coding Connectivity Routing Storage Failures Packet loss Malicious behavior Remote operation Theory Practice

5. UCB Sensor Networks day, Jan 28, 2004 5 Some conceptual issues LARGE SCALE CONNECTIVITY ROUTING CAPACITY CONTROL

6. UCB Sensor Networks day, Jan 28, 2004 6 1 Connection probability |x| Continuum percolation 2r Random connection model Random connection model |x| 1 Connection probability Single hop model

7. UCB Sensor Networks day, Jan 28, 2004 7 Multi-hop connectivity model There is a phase transition at a critical node density value

8. UCB Sensor Networks day, Jan 28, 2004 8 1 g 1 (x) |x| 2r |x| 1 g 2 (x) How does the critical density change with the shape of the connection function?

9. UCB Sensor Networks day, Jan 28, 2004 9 General Tendency When the selection mechanism with which nodes are connected to each other is sufficiently “spread out’’, then few links (in the limit one on average) will suffice to obtain global connectivity. Balister, Bollobas, Walters (2003) Franceschetti, Booth, Cook, Bruck, Meester (2003) D. Dubhashi, O. Haggstrom, A. Panconesi (2003) R. Meester, M. Penrose, A. Sarkar (1997) M. Penrose (1993)

10. UCB Sensor Networks day, Jan 28, 2004 10 General Tendency In contrast, when connections do not spread out, few links are not enough for connectivity. Xue and P. R. Kumar (2003) O. Haggstrom and R. Meester (1996)

11. UCB Sensor Networks day, Jan 28, 2004 11 Connection probability 1 |x| Spread out connections (1)

12. UCB Sensor Networks day, Jan 28, 2004 12 Theorem For all connection functions “longer links are trading off for the unreliability of the connection” “it is easier to reach connectivity in this model of unreliable network” Franceschetti, Booth, Cook, Bruck, Meester (2003)

13. UCB Sensor Networks day, Jan 28, 2004 13 Spread out connections (2) Connection probability |x||x| 1

14. UCB Sensor Networks day, Jan 28, 2004 14 Mixture of short and long links Two different spreading strategies Links are made all longer

15. UCB Sensor Networks day, Jan 28, 2004 15 Theorem Consider annuli shapes A(r) of inner radius r, unit area, and critical density For all, there exists a finite, such that A(r*) percolates, for all It is possible to decrease the connectivity threshold by taking a sufficiently large shift ! Balister, Bollobas, Walters (2003) Franceschetti, Booth, Cook, Bruck, Meester (2003)

16. UCB Sensor Networks day, Jan 28, 2004 16 CNL squashing Shifting What have we learned CNL=average number of connections per node needed for connectivity

17. UCB Sensor Networks day, Jan 28, 2004 17 Navigation in the small world Need links at ALL scale lengths ! What about routing?

18. UCB Sensor Networks day, Jan 28, 2004 18 Z Intuition: scale invariance r1r1 r 2 Model of neighbors density:

19. UCB Sensor Networks day, Jan 28, 2004 19 Z Intuition: scale invariance r1r1 r 2 Model of neighbors density:

20. UCB Sensor Networks day, Jan 28, 2004 20 Z Intuition: scale invariance r1r1 r 2 Model of neighbors density:

21. UCB Sensor Networks day, Jan 28, 2004 21 Z Intuition: scale invariance r1r1 r 2 Slow far from destination Slow close to destination

22. UCB Sensor Networks day, Jan 28, 2004 22 Theorem  S T d Franceschetti & Meester (2003)

23. UCB Sensor Networks day, Jan 28, 2004 23 Bottom line  S T d Build routing trees that are scale invariant to route with few hops at all distance scales Want to balance the number of short and long links Need to exploit the “hairy edge” (D. Culler)

24. UCB Sensor Networks day, Jan 28, 2004 24 Summary Towards a system theory of large scale networks Conceptual issues at different levels Design for complexity strategy Close the gap between theory and practice