1. Spectral analysis and slow dynamics on quenched complex networks Géza Ódor MTA-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 Budapest M. A. Munoz Granada C. Castellano Roma R. Pastor-Satorras Barcelona

2. Scaling and universality classes appear in complex system due to :    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

3. 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 (  ) diverges Haimovici et al PRL (2013) : Brain complexity born out of criticality.

4. 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 : c > 0 between inactive and active phases Infect  / (  )Heal  / (  )

5. Rare Region theory for quench disordered CP Fixed (quenched) disorder/impurity changes the local birth rate  c  c  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:  (t) ~ ∫ dL R L R w(L R ) exp [-t /  (L R )] For  c   : conventional (exponentially fast) decay At c  the characteristic time scales as:  L R ) ~ L R Z  saddle point analysis: ln  (t) ~ t d / ( d + Z) stretched exponential For c 0 < < c :  L R ) ~ exp(b L R ): Griffiths Phase   (t) ~ t - c / b continuously changing exponents At c : b may diverge   (t) ~ ln(t   Infinite randomness fixed point scaling In case of correlated RR-s with dimension > d - : smeared transition c c  Act. Abs. GP „dirty critical point” „clean critical point”

6. Networks considered From regular to random networks: Erdős-Rényi (p = 1) Degree (k) distribution in N→  node limit: P(k) = e - k / k! Topological dimension: N(r) ∼ r d Above perc. thresh.: d =  Below percolation d = 0 Scale free networks: Degree distribution: P(k) = k -    Topological dimension: d =  Example: Barabási-Albert lin. prefetential attachment

7. Rare active regions below  c with:  (A)~ e A → 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 ?

8. Spectral Analysis of networks – Quenched Mean-Field method Weighted (real symmetric) Adjacency matrix: Express  i on orthonormal eigenvector ( f i (  ) ) basis: Master (rate) equation of SIS for occupancy prob. at site i: Total infection density vanishes near c as : Mean-field estimate

9. QMF results for Erdős-Rényi Percolative ER Fragmented ER IPR ~ 1/N → delocalization → multi-fractal exponent → Rényi entropy   = 1/ c → 5.2(2)   1 =  k  =4 IPR → 0.22(2) → localization

10. Simulation results for ER graphs Percolative ERFragmented ER

11. 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

12. 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

13. 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 c  1 linearly, HMF:  = 1 (+ log. corrections) G. Ódor, R. Pastor-Storras PRE 86 (2012) 026117

14. SIS on weigthed Barabási-Albert graphs Excluding loops slows down the spreading + Weights: WBAT-II: disassortative weight scheme dependent density decay exponents: Griffiths Phases or Smeared phase transition ?

15. 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

16. Rare-region effects in aging BA graphs BA followed by preferential edge removal p ij  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

17. 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

18. CP + Topological disorder results Generalized Small World networks: P(l) ~  l - 2 (link length probability) Top. dim: N(r) ∼ r d d(  ) finite: c (  ) decreases monotonically from c (0 )= 3.29785 (1d CP) to: lim b   c (  ) = 1 towards mean-field CP value < c (  ) inactive, there can be locally ordered, rare regions due to more than average, active, incoming links Griffiths phase: - dep. continuously changing dynamical power laws: for example : r(t) ∼ t - a (l) Logarithmic corrections ! Ultra-slow (“activated”) scaling: r  ln(t) -  at c As   1 Griffiths phase shrinks/disappears Same results for: cubic, regular random nets higher dimensions ? l GP l

19. 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  th. For small values of  th, 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  th, the network breaks down into smaller subnetworks of connected edges, joined by weights larger than  th. 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.