Condensation in/of Networks
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Transcript of Condensation in/of Networks
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Condensation in/of Condensation in/of NetworksNetworks
Jae Dong Noh
NSPCS08, 1-4 July, 2008, KIAS
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Getting wired
Moving and Interacting
Being rewired
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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).
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NetworksNetworks
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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
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Random WalksRandom Walks
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DefinitionDefinition
Random motions of a particle along links
Random spreading
1/51/5
1/5
1/51/5
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Stationary State PropertyStationary State Property
Detailed balance :
Stationary state probability distribution
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Relaxation DynamicsRelaxation Dynamics
Return probability
SF networksw/o loops
SF networkswith many loops
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Mean First Passage TimeMean First Passage Time
MFPT
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Zero Range ProcessZero Range Process
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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 .
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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
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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)]
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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.,
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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
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Condensation on SF NetworksCondensation on SF Networks
Stationary state probability distribution
Mean occupation number
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Phase DiagramPhase Diagram
normal phase
condensed phase
transition line1 /( 1)c
Complete condensation
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Coevolving NetworksCoevolving Networks
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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
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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
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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
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Numerical Data for kNumerical Data for kmaxmax
[N=1000, <k>=4]
dynamic instability
linear growth sub-linear growthdynamic phase transition
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Degree DistributionDegree Distribution
Poissonian
+
Poissonian+
Isolated hubs
Poissonian
+
Fat-tailed
low density high density
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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
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Non-Markovian Queueing ProcessNon-Markovian Queueing Process
node i $ queue (box) edge $ packet (ball) degree k $ queue size K
queue
i1
2
K
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Non-Markovian Queueing ProcessNon-Markovian Queueing Process
Weight of a ball
A ball leaves a queue with the probability
queue
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Outgoing Particle Flux ~ uOutgoing Particle Flux ~ uZRPZRP(K)(K)
Upper bound for fout(K,)
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Dynamic Phase TransitionDynamic Phase Transition
- queue is trapped at K=K1 for instability time t = - queue grows linearly after t >
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Phase DiagramPhase Diagram
ballistic growth of hub sub-linear growth of hub
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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
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A Simplified TheoryA Simplified Theory
i1
2
K
potential candidate for the hub
Rate equations for K and w
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Flow DiagramFlow Diagram
hub
condensation
no hub
no condensation
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Numerical DataNumerical Data
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SummarySummary
Dynamical systems on networks random walks zero range process
Coevolving network models
Network heterogeneity $ Condensation