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

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

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.

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)

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”

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

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 ?

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

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

Simulation results for ER graphs

Percolative ER Fragmented ER

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

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

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

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 ?

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

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

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

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

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.