1. MASSIMO FRANCESCHETTI University of California at San Diego Information-theoretic and physical limits on the capacity of wireless networks TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAA P. Minero (UCSD), M. D. Migliore (U. Cassino)

2. Standing on the shoulder of giants

3. The problem Computers equipped with power constrained radios Randomly located Random source-destination pairs Transmit over a common wireless channel Possible cooperation among the nodes Maximum per-node information rate (bit/sec) ?

4. Scaling approach All pairs must achieve the same rate Consider the limit IEEE Trans-IT (2000)

5. Information-theoretic limits Provide the ultimate limits of communication Independent of any scheme used for communication

6. Assume physical propagation model Allow arbitrary cooperation among nodes Xie Kumar IEEE Trans-IT (2004) Xue Xie Kumar IEEE Trans-IT (2005) Leveque, Telatar IEEE Trans-IT (2005) Ahmad Jovicic Viswanath IEEE Trans-IT (2006) Gowaikar Hochwald Hassibi IEEE Trans-IT (2006) Xie Kumar IEEE Trans-IT (2006) Aeron Saligrama IEEE Trans-IT (2007) Franceschetti IEEE Trans-IT (2007) Ozgur Leveque Preissmann IEEE Trans-IT (2007) Ozgur Leveque Tse IEEE Trans-IT (2007) Classic Approach

7. Information theoretic truths High attenuation regime Low attenuation regime without fading Low attenuation regime with fading No attenuation regime, fading only

8. Good research should shrink the knowledge tree

9. There is only one scaling law This is a degrees of freedom limitation dictated by Maxwells physics and by Shannons theory of information. It is independent of channel models and cannot be overcome by any cooperative communication scheme.

10. Approach

11. ... Approach...

12. Information flow decomposition A D V

13. First flow component...

14. Second flow component...

15. Second flow component D O M

16. Singular values have a phase transition at the critical value Hilbert-Schmidt decomposition of operator G

17. Singular values of operator G

18. Degrees of freedom theorem O

19. The finishing touches O

20. Understanding the space resource Space is a capacity bearing object Geometry plays a fundamental role in determining the number of degrees of freedom and hence the information capacity

21. Geometrical configurations In 2D the network capacity scales with the perimeter boundary of the network In 3D the network capacity scales with the surface boundary of the network

22. A different configuration Distribute nodes in a 3D volume of size Nodes are placed uniformly on a 2D surface inside the volume

23. Different configurations

24. The endless enigma (Salvador Dali) A hope beyond a shadow of a dream (John Keats) To be continued…