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Lass, villansos dinamika komplex hlzatokon Gza dor MTA-TTK-MFA Budapest 11/04/2014 Infocommunication technologies and the society of future (FuturICT.hu) TMOP-4.2.2.C-11/1/KONV-2012- 0013 Partners: R. Juhsz Budapest M. A. Munoz Granada C. Castellano Roma R. Pastor-Satorras Barcelona Slide 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. Slide 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 Slide 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 Slide 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 / ( ) Slide 6 Rare active regions below c with: (A)~ e A slow dynamics (Griffiths Phase) ? M. A. Munoz, R. Juhsz, C. Castellano and G. dor, PRL 105, 128701 (2010) 1. Inherent disorder in couplings 2. Disorder induced by topology Optimal fluctuation theory + simulations: YES In Erds-Rnyi networks below the percolation threshold In generalized small-world networks with finite topological dimension A Rare region effects in networks ? Slide 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 Slide 8 QMF results for Erds-Rnyi Percolative ER Fragmented ER IPR ~ 1/N delocalization multi-fractal exponent Rnyi entropy = 1/ c 5.2(2) 1 = k =4 IPR 0.22(2) localization Slide 9 Simulation results for ER graphs Percolative ERFragmented ER Mean-field transition vs. Griffiths phase Slide 10 Quenched Mean-Field method for scale-free BA graphs Barabsi-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 Slide 11 SIS on weigthed Barabsi-Albert graphs Excluding loops slows down the spreading + Weights: WBAT-II: disassortative weight scheme dependent density decay exponents: Griffiths Phases or Smeared phase transition ? Slide 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 Slide 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 Slide 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 Slide 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 Slide 16 Summary [1] G. dor, Phys. Rev. E 87, 042132 (2013) [2] G. dor, Phys. Rev. E 88, 032109 (2013) [3] G. dor, 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) TMOP-4.2.2.C- 11/1/KONV-2012-0013