Condensation in/of Condensation in/of NetworksNetworks

Jae Dong Noh

NSPCS08, 1-4 July, 2008, KIAS

Getting wired

Moving and Interacting

Being rewired

ReferencesReferences

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).

NetworksNetworks

Basic ConceptsBasic Concepts

Network = {nodes} [ {links}

Adjacency matrix A

Degree of a node i :

Degree distribution

Scale-free networks :

1 , if there's a link between and

0 , otherwise ij

i jA

1

2

3 4

0 1 0 0

1 0 1 1

0 1 0 1

0 1 1 0

A

Random WalksRandom Walks

DefinitionDefinition

Random motions of a particle along links

Random spreading

1/51/5

1/5

1/51/5

Stationary State PropertyStationary State Property

Detailed balance :

Stationary state probability distribution

Relaxation DynamicsRelaxation Dynamics

Return probability

SF networksw/o loops

SF networkswith many loops

Mean First Passage TimeMean First Passage Time

MFPT

Zero Range ProcessZero Range Process

ModelModel

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 ui (ni ), and hops to a neighboring node j selected randomly

with the probability Wji .

ModelModel

Jumping rate ui (n )1. depends only on the

occupation number at the departing site.

2. may be different for different sites (quenched disorder)

Hopping probability Wji independent of the occupation numbersat the departing and arriving sites

i

(3)iu

jiW

j

Note that [ZRP with M=1 particle] = [ single random walker] [ZRP with u(n) = n ] = [ M indep. random walkers]

transport capacityparticle interactions

Stationary State PropertyStationary 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 ZRPCondensation 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., uimp. = <1, ui≠imp. = 1)

2. On-site attractive interaction : if the jumping rate function ui(n) = u(n) decays ‘faster’ than ~(1+2/n)

e.g.,

ZRP on SF NetworksZRP 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 NetworksCondensation on SF Networks

Stationary state probability distribution

Mean occupation number

Phase DiagramPhase Diagram

normal phase

condensed phase

transition line1 /( 1)c

Complete condensation

Coevolving NetworksCoevolving Networks

Synaptic PlasticitySynaptic 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 ModelCo-evolving Network Model

Weighted undirected network + diffusing particles

Particles dynamics : random diffusion

Weight dynamics [LTP]

Link dynamics [LTD]: With probability 1/we, each link e is removed and replaced by a new one

12

3 4

23

4 5

Dynamic InstabilityDynamic 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 kNumerical Data for kmaxmax

[N=1000, <k>=4]

dynamic instability

linear growth sub-linear growthdynamic phase transition

Degree DistributionDegree Distribution

Poissonian

+

Poissonian+

Isolated hubs

Poissonian

+

Fat-tailed

low density high density

Analytic TheoryAnalytic 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 ProcessNon-Markovian Queueing Process

node i $ queue (box) edge $ packet (ball) degree k $ queue size K

queue

i1

2

K

Non-Markovian Queueing ProcessNon-Markovian Queueing Process

Weight of a ball

A ball leaves a queue with the probability

queue

Outgoing Particle Flux ~ uOutgoing Particle Flux ~ uZRPZRP(K)(K)

Upper bound for fout(K,)

Dynamic Phase TransitionDynamic Phase Transition

- queue is trapped at K=K1 for instability time t = - queue grows linearly after t >

Phase DiagramPhase Diagram

ballistic growth of hub sub-linear growth of hub

A Variant ModelA Variant Model

Weighted undirected network + diffusing particles

Particles dynamics : random diffusion

Weight dynamics

Link dynamics : Rewiring with probability 1/we

Weight regularization :

12

3 4

23

4 5

A Simplified TheoryA Simplified Theory

i1

2

K

potential candidate for the hub

Rate equations for K and w

Flow DiagramFlow Diagram

hub

condensation

no hub

no condensation

Numerical DataNumerical Data

SummarySummary

Dynamical systems on networks random walks zero range process

Coevolving network models

Network heterogeneity $ Condensation