CROWN Kickoff NKUA

1. CROWN Kickoff NKUA Power Control in Random Networks with N. Bambos, P. Mertikopoulos, L. Lampiris

2. Resource Allocation • Power – Frequency allocation • Random Networks • min “P” subject to “SINR” • “P”: total power – power per user – power per user <Pmax • “SINR”: SINR contraints on all (some) connections = connectivity • Impediments in the analysis of Power Control • Randomness in • Distance between Tx-Rx • Fading coefficients • Interference location/strength • Interference (interaction – domino effect) • Constraints (max power makes problem non-convex)

3. Power control • Simplify (enough) problem so that can obtain analytic solution • Take • Minimize (total) power subject to power constraints • Linear SINR constraints result to conical section or • Good news: If solution exists, can be reached using distributed algorithms (e.g. Foschini-Miljanich) • Simplify neglect random fading, • Problem still non-trivial due to randomness in positions and interference • Two specific examples of randomness

4. Models of Randomness: The Femto-Cell Paradigm • Start with ordered (square) lattice of transmitter-receiver pairs. • (a) With probability p erase transmitter • Intermittency of transmission • Randomness of network • (b) With probability p/(1-p) locate users at distance a1/a2 • Models randomness of location of users

5. Solution Approach • Both represent simple models with all important ingredients: • PC, randomness, interference • After some algebra: • Ei = 0,1 with probability p/(1-p) (erasures) • Assume M circulant : eigenvalues • Using Random Matrix Theory: where • βplays role of shift (β=0, when p=0)

6. Analysis • When p=0 blowup at a given γ • p>0 moves singularity to the right. • Pave does not diverge • Var diverges Hint: a finite number (1?) of nodes diverges Metastable state?

7. Analysis • In reality system is unstable (max/ave) • One – two dimensional systems very accurate • Questions: • Probability of instability as a function of γ? • Fluctuations btw samples?

8. Resource Allocation • Introduce max-power constraint • Distributed version: • λ=1/Pmax • Use 3 methods to find optimum: • Foschini-Miljanic • Best-Response • Nash

9. Introduction Type of problem No Pure Nash Equilibrium Players Best Respond 3 General Categories Payoff: Throughput:

10. One Mixed Nash Equilibrium

11. 3 mixed Nash equilibria

12. Single & Double Pure Nash Equilibria Single Equilibrium Two Equilibria

13. Average Payoffs & Throughput Comparison 1 Nash: FM: Best Response: Throughput: Pure Nash: 3 Nash: FM: FM: Best Response: BR Throughput:

14. Questions • Max-power constraint brings new features • Nash – game on restricted power feasible and better than other cases • BR not bad • Generalisable to more users? • Analytic estimates?

15. Goals • Optimize network connectivity using collaborative methods inspired by statistical mechanics (Task 2.1) • Power – connectivity fundamental trade-off: • Tradeoff between connectivity and number of frequency bands. • Design and validation of distributed message passing algorithms • Develop distributed message passing methods to achieve fundamental limits of detection and localization of a network of primary sources through a network of secondary sensors (Task 2.2) • Detection of sources using compressed sensing on random graphs and Cayley trees • Effect of additive and multiplicative noise on detection • Application of compressed sensing on two-dimensional graphs with realistic channel statistics • Develop decentralized coordinated optimization approaches (Task 2.3). • design self-coordinated, fast-convergent wireless resource management techniques • convergence, stability and the impact of operation on different time scales on the performance.

16. Power Control – Connectivity tradeoff • Minimize power subject to constraints • Interactions due to interference • Simplifications: • Random graphs (1d-2d-inft d) • gij = 0,1 • Power levels • Use replica theory

17. Connectivity – Frequency Bandwitdth • Minimum number of colors needed to color network with interference constraints • E.g. no adjacent nodes in same color • Simplifications: • Random graphs (Bethe lattice / Erdos-Renyi) • gij = 0,1 • Use replica theory and • Graph coloring • Message passing algorithms

18. Cooperative Sensing Transmitter Node Sensor Node Signal Sensor Communication

19. Collaborative Sensing and Localization • Minimize power subject to constraints • Models: • H known and P discrete (on-off) • Random graphs • H random valued • H in a given geometry • possible locations of sources • With/without noise • Use replica theory • Compressed sensing (sparsity) • Message passing

20. Collaborative Sensing and Localization • Minimize power subject to constraints • Models: • H known and P discrete (on-off) • Random graphs • H random valued • H in a given geometry • possible locations of sources • With/without noise • Use replica theory • Compressed sensing (sparsity) • Message passing