spectral analysis and slow dynamics on quenched complex networks géza Ódor mta-ttk-mfa budapest...
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![Page 1: Spectral analysis and slow dynamics on quenched complex networks Géza Ódor MTA-TTK-MFA Budapest 19/09/2013 „Infocommunication technologies and the society](https://reader036.vdocuments.site/reader036/viewer/2022062421/56649de85503460f94ae21ea/html5/thumbnails/1.jpg)
Spectral analysis and slow dynamics on quenched
complex networks
Géza ÓdorMTA-TTK-MFA Budapest
19/09/2013
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Partners:R. Juhász BudapestM. A. Munoz GranadaC. Castellano RomaR. Pastor-Satorras Barcelona
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Scaling and universality classes appear in complex system due to : x i.e: near critical points, due to currents ...
Basic models are classified by universal scaling behavior in Euclidean, regular system
Why don't we see universality classes in models defined on networks ? Power laws are frequent in nature « Tuning to critical point ?
I'll show a possible way to understand these
Scaling in nonequilibrium system
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Criticality in networks ?
Brain : PL size distribution of neural avalanches G. Werner : Biosystems, 90 (2007) 496,
Internet: worm recovery time is slow:
Can we expect slow dynamics in small-world network models ?
Correlation length (x) diverges Haimovici et al PRL (2013) : Brain complexity born out of criticality.
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Network statphys research Expectation: small world topology mean-field behavior & fast dynamics
Prototype: Contact Process (CP) or Susceptible-Infected-Susceptible (SIS) two-state models:
For SIS : Infections attempted for all nn
Order parameter : density of active ( ) sites Regular, Euclidean lattice: DP critical point : l
c > 0 between inactive
and active phases
Infect: l / (1+l) Heal: 1 / (1+l)
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Rare Region theory for quench disordered CP
Fixed (quenched) disorder/impurity
changes the local birth rate lc > l
c0
Locally active, but arbitrarily large Rare Regions in the inactive phase due to the inhomogeneities Probability of RR of size L
R:
w(L
R ) ~ exp (-c L
R )
contribute to the density: r(t) ~ ∫ dLR L
R w(L
R ) exp [-t /t (L
R)]
For <l lc0 : conventional (exponentially fast) decay
At lc0 the characteristic time scales as: t (L
R) ~ L
R Z saddle point analysis:
ln r(t) ~ t d / ( d + Z) stretched exponential For l
c0 < l < l
c : t (L
R) ~ exp(b L
R): Griffiths Phase
r(t) ~ t - c / b continuously changing exponents At l
c : b may diverge ® r(t) ~ ln(t) -a Infinite randomness fixed point scaling
In case of correlated RR-s with dimension > d - : smeared transition
lc
lc0
Act.
Abs.
GP
„dirty critical point”
„clean critical point”
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Networks considered From regular to random networks:
Erdős-Rényi (p = 1)
Degree (k) distribution in N→ node limit: P(k) = e-<k> <k>k / k!
Topological dimension: N(r) r ∼ d
Above perc. thresh.: d = Below percolation d = 0
Scale free networks:
Degree distribution: P(k) = k - ( 2< g < 3)
Topological dimension: d =
Example: Barabási-Albert lin. prefetential attachment
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Rare active regions below l
c with: t(A)~ eA
→ slow dynamics (Griffiths Phase) ?
M. A. Munoz, R. Juhász, C. Castellano and G. Ódor, PRL 105, 128701 (2010)
1. Inherent disorder in couplings 2. Disorder induced by topology
Optimal fluctuation theory + simulations: YES
In Erdős-Rényi networks below the percolation threshold In generalized small-world networks with finite topological dimension
A
Rare region effects in networks ?
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Spectral Analysis of networks – Quenched Mean-Field method
Weighted (real symmetric) Adjacency matrix:
Express ri on orthonormal eigenvector ( f
i (L) ) basis:
Master (rate) equation of SIS for occupancy prob. at site i:
Total infection density vanishes near lc as :
Mean-field estimate
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QMF results for Erdős-Rényi
Percolative ER Fragmented ER
IPR ~ 1/N → delocalization → multi-fractal exponent → Rényi entropyL1 = 1/l
c → 5.2(2) « L
1 = ákñ =4
IPR → 0.22(2) → localization
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Simulation results for ER graphs
Percolative ER Fragmented ER
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Weighted SIS on ER graph
C. Buono, F. Vazquez, P. A. Macri, and L. A. Braunstein,PRE 88, 022813 (2013)
QMF supports these results
G.Ó.: PRE 2013
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Quenched Mean-Field method for scale-free BA graphs
Barabási-Albert graph attachment prob.:
IPR remains small but exhibits large fluctuations as N ® ¥ (wide distribution)Lack of clustering in the steady state, mean-field transition
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Contact Process on Barabási-Albert (BA) network
Heterogeneous mean-field theory: conventional critical point, with linear density decay:
with logarithmic correction
Extensive simulations confirm this
No Griffiths phase observed
Steady state density vanishes at lc »1
linearly, HMF: b = 1 (+ log. corrections)
G. Ódor, R. Pastor-Storras PRE 86 (2012) 026117
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SIS on weigthed Barabási-Albert graphs
Excluding loops slows down the spreading + Weights: WBAT-II: disassortative weight scheme
l dependent density decay exponents: Griffiths Phases or Smeared phase transition ?
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Do power-laws survive the thermodynamic limit ?
Finite size analysis shows the disappearance of a power-law scaling:
Power-law → saturation explained by smeared phase transition: High dimensional rare sub-spaces
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Rare-region effects in aging BA graphs
BA followed by preferential edge removal p
ij µ k
i k
j
dilution is repeated until 20% of links are removed
No size dependence → Griffiths Phase
IPR → 0.28(5)
QMF
Localization in the steady state
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Summary
[1] M. A. Munoz, R. Juhasz, C. Castellano, and G, Ódor, Phys. Rev. Lett. 105, 128701 (2010)[2] G. Ódor, R. Juhasz, C. Castellano, M. A. Munoz, AIP Conf. Proc. 1332, Melville, New York
(2011) p. 172-178.[3] R. Juhasz, G. Ódor, C. Castellano, M. A. Munoz, Phys. Rev. E 85, 066125 (2012)[4] G. Ódor and Romualdo Pastor-Satorras, Phys. Rev. E 86, 026117 (2012)[5] G. Ódor, Phys. Rev. E 87, 042132 (2013)[6] G. Ódor, Phys. Rev. E 88, 032109 (2013)
Quenched disorder in complex networks can cause slow (PL) dynamics : Rare-regions → Griffiths phases → no tuning or self-organization needed ! GP can occur due to purely topological disorder In infinite dim. Networks (ER, BA) mean-field transition of CP with logarithmic corrections (HMF + simulations, QMF) In weighted BA trees non-universal, slow, power-law dynamics can occur for finite N, but in the N ® ¥ limit saturation was observed Smeared transition can describe this, percolation analysis confirms the existence of arbitrarily large dimensional sub-spaces with (correlated) large weights Quenched mean-field approximation gives correct description of rare-region effects and the possibility of phases with extended slow dynamics GP in important models: Q-ER (F2F experiments), aging BA graph Acknowledgements to : HPC-Europa2, OTKA, Osiris FP7, FuturICT.hu
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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CP + Topological disorder results Generalized Small World networks: P(l) ~ b l - 2
(link length probability) Top. dim: N(r) ∼ r d d(b ) finite:
lc(b ) decreases monotonically from
lc(0 )= 3.29785 (1d CP) to:
limb lc( b ) = 1 towards mean-field CP value
l < lc( b ) inactive, there can be
locally ordered, rare regions due to more than average, active, incoming links
Griffiths phase: l - dep. continuously changing dynamical power laws: for example : r(t) t ∼ - a (l)
Logarithmic corrections !
Ultra-slow (“activated”) scaling: r ln(t)- a at lc
As b 1 Griffiths phase shrinks/disappears
Same results for: cubic, regular random nets higher dimensions ?
lGP
l
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Percolation analysis of the weighted BA tree
We consider a network of a given size N,and delete all the edges with a weight smaller than a threshold w
th.
For small values of wth, many edges remain
in the system, and they form a connected network with a single cluster encompassing almost all the vertices in the network. When increasing the value of w
th, the network
breaks down into smaller subnetworks of connected edges, joined by weights larger than w
th.
The size of the largest ones grows linearly with the network size N
« standard percolation transition. These clusters, which can become arbitrarilylarge in the thermodynamic limit, play the roleof correlated RRs, sustaining independently activity and smearing down the phase transition.