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Condensation in/of Networks Jae Dong Noh NSPCS08, 1-4 July, 2008, KIAS
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Getting wired Moving and Interacting Being rewired
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References Random walks Noh and Rieger, PRL92, 118701 (2004). Noh and Kim, JKPS48, S202 (2006). Zero-range processes Noh, Shim, and Lee, PRL94, 198701 (2005). Noh, PRE72, 056123 (2005). Noh, JKPS50, 327 (2007). Coevolving networks Kim and Noh, PRL100, 118702 (2008). Kim and Noh, in preparation (2008).
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Networks
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Basic Concepts Network = {nodes} [ {links} Adjacency matrix A Degree of a node i : Degree distribution Scale-free networks :
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Random Walks
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Definition Random motions of a particle along links Random spreading 1/5
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Stationary State Property Detailed balance : Stationary state probability distribution
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Relaxation Dynamics Return probability SF networks w/o loops SF networks with many loops
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Mean First Passage Time MFPT
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Zero Range Process
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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.
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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
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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)]
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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, u i≠imp. = 1) 2.On-site attractive interaction : if the jumping rate function u i (n) = u(n) decays ‘faster’ than ~(1+2/n) e.g.,
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ZRP on SF Networks Scale-free networks Jumping rate (δ>1) : repulsion (δ=1) : non-interacting (δ<1) : attraction Hopping probability : random walks
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Condensation on SF Networks Stationary state probability distribution Mean occupation number
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Phase Diagram normal phase condensed phase transition line Complete condensation
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Coevolving Networks
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Synaptic Plasticity In neural networks Bio-chemical signal transmission from neural to neural through synapses Synaptic coupling strength may be enhanced (LTP) or suppressed (LTD) depending on synaptic activities Network evolution
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Co-evolving Network Model Weighted undirected network + diffusing particles Particles dynamics : random diffusion Weight dynamics [LTP] Link dynamics [LTD]: With probability 1/w e, each link e is removed and replaced by a new one 12 3 4 23 4 5
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Dynamic Instability Due to statistical fluctuations, a node ‘hub’ may have a higher degree than others Particles tend to visit the ‘hub’ more frequently Links attached to the ‘hub’ become more robust, hence the hub collects more links than other nodes Positive feedback dynamic instability toward the formation of hubs
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Numerical Data for k max [N=1000, =4] dynamic instability linear growthsub-linear growth dynamic phase transition
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Degree Distribution Poissonian + Poissonian + Isolated hubs Poissonian + Fat-tailed low density high density
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Analytic Theory Separation of time scales particle dynamics : short time scale network dynamics : long time scale Integrating out the degrees of freedom of particles Effective network dynamics : Non-Markovian queueing (balls-in-boxes) process
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Non-Markovian Queueing Process node i $ queue (box) edge $ packet (ball) degree k $ queue size K queue i 1 2 K
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Non-Markovian Queueing Process Weight of a ball A ball leaves a queue with the probability queue
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Outgoing Particle Flux ~ u ZRP (K) Upper bound for f out (K, )
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Dynamic Phase Transition - queue is trapped at K=K 1 for instability time t = - queue grows linearly after t >
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Phase Diagram ballistic growth of hubsub-linear growth of hub
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A Variant Model Weighted undirected network + diffusing particles Particles dynamics : random diffusion Weight dynamics Link dynamics : Rewiring with probability 1/w e Weight regularization : 12 3 4 23 4 5
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A Simplified Theory i 1 2 K potential candidate for the hub Rate equations for K and w
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Flow Diagram hub condensation no hub no condensation
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Numerical Data
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Summary Dynamical systems on networks random walks zero range process Coevolving network models Network heterogeneity $ Condensation
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