Aris Moustakas, University of Athens CROWN Kickoff NKUA Power Control in Random Networks with N. Bambos, P. Mertikopoulos, L. Lampiris

Aris Moustakas, University of Athens 2 Power – Frequency allocation Random Networks min “P” subject to “SINR” –“P”: total power – power per user – power per user <P max –“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) Resource Allocation

Aris Moustakas, University of Athens 3 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 Power control

Aris Moustakas, University of Athens 4 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 a 1 /a 2 –Models randomness of location of users Models of Randomness: The Femto-Cell Paradigm

Aris Moustakas, University of Athens 5 Both represent simple models with all important ingredients: –PC, randomness, interference After some algebra: –E i = 0,1 with probability p/(1-p) (erasures) – Assume M circulant : eigenvalues Using Random Matrix Theory: where β plays role of shift (β=0, when p=0) Solution Approach

Aris Moustakas, University of Athens 6 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? Analysis

Aris Moustakas, University of Athens 7 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? Analysis

Aris Moustakas, University of Athens 8 Introduce max-power constraint Distributed version: λ=1/P max Use 3 methods to find optimum: –Foschini-Miljanic –Best-Response –Nash Resource Allocation

Aris Moustakas, University of Athens 9 Type of problem No Pure Nash Equilibrium Players Best Respond 3 General Categories Payoff: Introduction Throughput:

Aris Moustakas, University of Athens 10 One Mixed Nash Equilibrium

Aris Moustakas, University of Athens 11 3 mixed Nash equilibria

Aris Moustakas, University of Athens Single & Double Pure Nash Equilibria 12 Single Equilibrium Two Equilibria

Aris Moustakas, University of Athens 13 Average Payoffs & Throughput Comparison 1 Nash: Throughput: FM: Best Response: 3 Nash: FM: BR Pure Nash: Throughput: FM: Best Response:

Aris Moustakas, University of Athens 14 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? Questions

Aris Moustakas, University of Athens 15 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. Goals

Aris Moustakas, University of Athens 16 Minimize power subject to constraints Interactions due to interference Simplifications: –Random graphs (1d-2d-inft d) –g ij = 0,1 –Power levels Use replica theory Power Control – Connectivity tradeoff

Aris Moustakas, University of Athens 17 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) –g ij = 0,1 Use replica theory and –Graph coloring Message passing algorithms Connectivity – Frequency Bandwitdth

Aris Moustakas, University of Athens 18 Cooperative Sensing Sensor Node Transmitter Node Signal Sensor Communication

Aris Moustakas, University of Athens 19 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 Collaborative Sensing and Localization

Aris Moustakas, University of Athens 20 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 Collaborative Sensing and Localization