1. Lassú, villanásos dinamika komplex hálózatokon Géza Ódor MTA-TTK-MFA Budapest 11/04/2014 „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. Criticality and scaling 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.

3. 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

4. Burstyness observed in nature Brain : PL inter-event time distribution of neuron firing sequences & Autocorrelations Y. Ikegaya et al.: Science, 304 (2004) 559, N. Takahashi et al.: Neurosci. Res. 58 (2007) 219 Internet : Email sequences: And many more …. Models exist to explain internal non-Markovian behavior of agents (Karsai et al.: Sci. Rep. 2 (2012) 397) Can we explain it based on solely the collective behavior of agents ? Mobile call: Inter-event times and Autocorrelations Karsai et al. PRE 83 (2013) 025102 : Small but slow world... J. Eckmann et al.: PNAS 101 (2004) 14333

5. 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  / (  )

6. 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 ?

7. 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 Localization in the steady state  ?  Rare region effects, slow dynamics

8. 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

9. Simulation results for ER graphs Percolative ERFragmented ER Mean-field transition vs. Griffiths phase

10. 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

11. 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 ?

12. 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

13. Bursts in the Contact Process in one- dimension Density decay and seed simulations of CP on a ring at the critical point Power-law inter-event time (  ) distribution among subsequent infection attempts Invariance on the time window and initial conditions The tail distribution decays as the known auto-correlation function for t  

14. Bursts of CP in Generalized Small World networks Density decay and seed simulations of CP on a near the critical point Power-law inter-event time (  ) distribution among subsequent infection attempts Invariance on the time window and initial conditions The tail distribution decays with a dependent exponent around the critical point in an extended Griffiths Phase P(l) ~  l - 2 l 

15. Bursts of CP in aging scale-free networks Barabasi-Albert model with preferential detachment of aging links: P(k) ~ k -3 exp(-ak) Density decay simulations of CP on a near the critical point Power-law inter-event time (  ) distribution among subsequent infection attempts The tail distribution decays with a dependent exponent around the critical point in an extended Griffiths Phase

16. Summary [1] G. Ódor, Phys. Rev. E 87, 042132 (2013) [2] G. Ódor, Phys. Rev. E 88, 032109 (2013) [3] G. Ódor, Phys. Rev. E 89, 042102 (2014) 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 dimensional 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 occurs GP in important models: Q-ER (F2F experiments), aging BA graph Bursty behavior in these extended GP-s as a consequence of heterogenity &   „Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C- 11/1/KONV-2012-0013