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, 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.,

ZRP on SF Networks  Scale-free networks  Jumping rate (δ>1) : repulsion (δ=1) : non-interacting (δ<1) : attraction  Hopping probability : random walks

Condensation on SF Networks  Stationary state probability distribution  Mean occupation number

Phase Diagram normal phase condensed phase transition line Complete condensation

Coevolving Networks

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

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

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

Numerical Data for k max [N=1000, =4] dynamic instability linear growthsub-linear growth  dynamic phase transition 

Degree Distribution Poissonian + Poissonian + Isolated hubs Poissonian + Fat-tailed   low density high density

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

Non-Markovian Queueing Process  node i $ queue (box)  edge $ packet (ball)  degree k $ queue size K queue i 1 2 K 

Non-Markovian Queueing Process  Weight of a ball   A ball  leaves a queue with the probability queue

Outgoing Particle Flux ~ u ZRP (K)  Upper bound for f out (K, )

Dynamic Phase Transition - queue is trapped at K=K 1 for instability time t =  - queue grows linearly after t > 

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