Spectral analysis and slow dynamics on
complex networks
Géza Ódor
MTA-TTK-MFA Budapest
R. Juhász
M. A. Munoz
C. Castellano
R. Pastor-Satorras
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-20120013
Scaling in nonequilibrium system
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
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:
Infect: l / (1+l)
Heal: 1 / (1+l)
For SIS : Infections attempted for all nn
Order parameter : density of active ( ) sites
Regular, Euclidean lattice: DP critical point : lc > 0 between inactive
and active phases
Rare Region theory for quench disordered CP
Fixed (quenched) disorder/impurity
„dirty critical point”
changes the local birth rate  l c > lc0
Locally active, but arbitrarily large
Rare Regions
„clean critical point”
in the inactive phase
due to the inhomogeneities
Probability of RR of size LR:
w(LR ) ~ exp (-c LR )
contribute to the density:
r(t) ~
∫ dLR LR w(LR ) exp [-t /t (LR)]
For l < lc0 : conventional (exponentially fast) decay
At lc0 the characteristic time scales as: t (LR) ~ LR Z
ln r(t) ~ t d
/ ( d + Z)
For lc0 < l < lc :
stretched exponential
t (LR) ~ exp(b LR):
r(t) ~ t - c / b
r(t) ~ ln(t) -a
saddle point analysis:
Griffiths Phase
continuously changing exponents
At lc : b may diverge 
In case of correlated RR-s with dimension > d - : smeared transition
Infinite randomness fixed point scaling
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
Scale free networks:
Degree distribution:
P(k) = k - ( 2<  < 3)
Topological dimension: d =
Example: Barabási-Albert
lin. prefetential attachment
Rare region effects in networks ?
Rare active regions below lc 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
Spectral Analysis of networks – Quenched Mean-Field method
Master (rate) equation of SIS for occupancy prob. at site i:
Weighted (real symmetric) Adjacency matrix:
Express ri on orthonormal eigenvector ( fi (L) ) basis:
Mean-field estimate
Total infection density vanishes near lc as :
QMF results for Erdős-Rényi
Percolative ER
IPR ~ 1/N → delocalization
→ multi-fractal exponent → Rényi entropy
L1 = 1/lc → 5.2(2)  L1 = k =4
Fragmented ER
IPR → 0.22(2) → localization
Simulation results for ER graphs
Percolative ER
Fragmented ER
Weighted SIS on ER graph
G.Ó.: PRE 2013
C. Buono, F. Vazquez,
P. A. Macri, and L. A. Braunstein,
PRE 88, 022813 (2013)
QMF supports these results
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
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
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
pij  ki kj
dilution is repeated until 20% of links are removed
IPR → 0.28(5)
No size dependence → Griffiths Phase
Localization in the steady state
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
[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)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C11/1/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 ?
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 wth.
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 wth, the network
breaks down into smaller subnetworks of
connected edges, joined by weights larger
than wth.
The size of the largest ones grows linearly
with the network size N
« standard percolation transition.
These clusters, which can become arbitrarily
large in the thermodynamic limit, play the role
of correlated RRs, sustaining independently
activity and smearing down the phase transition.

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