Condensation in/of Networks Jae Dong Noh NSPCS08, 1-4 July, 2008, KIAS Getting wired Moving and Interacting Being rewired References Random walks Noh and Rieger, PRL92, (2004). Noh and Kim, JKPS48, S202 (2006). Zero-range processes Noh, Shim, and Lee, PRL94, (2005). Noh, PRE72, (2005). Noh, JKPS50, 327 (2007). Coevolving networks Kim and Noh, PRL100, (2008). Kim and Noh, in preparation (2008). Networks Basic Concepts Network = {nodes} [ {links} Adjacency matrix A Degree of a node i : Degree distribution Scale-free networks : Random Walks Definition Random motions of a particle along links Random spreading 1/5 Stationary State Property Detailed balance : Stationary state probability distribution Relaxation Dynamics Return probability SF networks w/o loops SF networks with many loops Mean First Passage Time MFPT Zero Range Process Model Interacting particle system on networks Each site may be occupied by multiple particles Dynamics : At each node i, A single particle jumps out of i at the rate u i (n i ), and hops to a neighboring node j selected randomly with the probability W ji. Model Jumping rate u i (n ) 1.depends only on the occupation number at the departing site. 2.may be different for different sites (quenched disorder) Hopping probability W ji independent of the occupation numbers at the departing and arriving sites Note that [ZRP with M=1 particle] = [ single random walker] [ZRP with u(n) = n ] = [ M indep. random walkers] transport capacity particle interactions Stationary State Property Stationary state probability distribution : product state PDF at node i : where e.g., [M.R. Evans, Braz. J. Phys. 30, 42 (2000)] Condensation in ZRP Condensation : single (multiple) node(s) is (are) occupied by a macroscopic number of particles Condition for the condensation in lattices 1.Quenched disorder (e.g., u imp. = 1) : repulsion (=1) : non-interacting ( Phase Diagram ballistic growth of hubsub-linear growth of hub A Variant Model Weighted undirected network + diffusing particles Particles dynamics : random diffusion Weight dynamics Link dynamics : Rewiring with probability 1/w e Weight regularization : A Simplified Theory i 1 2 K potential candidate for the hub Rate equations for K and w Flow Diagram hub condensation no hub no condensation Numerical Data Summary Dynamical systems on networks random walks zero range process Coevolving network models Network heterogeneity $ Condensation