Dynamics over Random Networks Mehran Mesbahi Aeronautics and Astronautics University of Washington August 2005 Napa Valley power gridYeast Proteins networked vehicles Sinoatrial Node TPF-like missions Gene networks dynamic processes over networks spread of information, diseases, rumors diffusion process, agreement Ising model, spin systems, Landhu theory cellular auotmata iterated games weakly coupled oscillators, synchronization Laplacians/agreement dynamics example: possibly dynamic A mathematician is a device for turning coffee into theorems. Paul Erdos random graphs agreement over random networks n = 10 P = 0.1 G k (n,p) is disconnected for all k G k (n,p) updates every second 10 sec dynamics over random graphs: why bother? I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive. Plato 1. robustness analysis 2. ease of use 3. quasi-randomness 4. probabilistic method Laplacians dynamics over random graphs k=0k=1 G 0 (n,p)G 1 (n,p)G 2 (n,p)G 3 (n,p) k=2k=3 agreement over random networks agreement on heading direction edge probability p=0.02 random graph updates every 2 seconds : agent with heading vector : information link Erdos meets Lyapunov Television is something the Russians invented to destroy American education. towards agreement from supermatringales to agreement rate of convergence Rate of convergence depends on random matrices and their spectrum Juvan and Mohar (1993) Juhasz (1991) Furedi & Komlos (1981) Wigner (1955) 2 (n,p) for large networks sharper bounds for when n large : increasing function of n and p 2 (n,p) for large networks Recall: 2 (n,p) # of times observation: for fixed p convergence is improved for large n on deterministic uses of randomness in the spirit of randomized algorithms probabilistic method probabilistic checkable proofs statistical mechanics and so on 1. robustness analysis 2. ease of use 3. quasi-randomness 4. probabilistic method why study dynamics over random graphs state-dependent graphs density/regularity n n. Szemeredi Regularity A. Szemeredi 1 n a dynamic graph controllability concept extensions relaxing the independence assumption in time effect of noise forcing terms random directed networks special classes of nonlinear systems (attitude dynamics, power systems) referencesAristotle The probable is what usually happens. 1. robustness analysis 2. ease of use 3. quasi-randomness 4. probabilistic method dynamics over random graphs: recap Acknowledgement: NSF joint with Y. Hatano