Nonequilibrium dynamics on complex topologies: models for epidemics and opinions Claudio Castellano (claudio.castellano@roma1.infn.it) Istituto dei Sistemi Complessi (ISC-CNR), Roma, Italy and Dipartimento di Fisica, Sapienza Universita’ di Roma, Italy venerdì 9 luglio 2010
Outline • Introduction • Models for epidemics: SIS and SIR • Disordered contact process: rare region effects • Opinion dynamics: voter model on networks • Conclusions venerdì 9 luglio 2010
Introduction • Statistical physics approach to interdisciplinary research • Complex topologies are the natural substrate • Highly nontrivial interplay between structure and dynamics venerdì 9 luglio 2010
Susceptible-Infected-Susceptible (SIS) model • Two possible states: susceptible and infected • Two possible events for infected nodes: ‣ Recovery (rate 1 ) ‣ Infection to neighbors (rate λ ) venerdì 9 luglio 2010
Heterogeneous Mean-Field theory for SIS Pastor-Satorras and Vespignani (2001) • Degree distribution P(k) ~ k - γ • ρ k = density of infected nodes of degree k � P ( k ′ | k ) ρ k ′ ρ k = − ρ k + λ k [1 − ρ k ] ˙ 1 k ′ λ c = � k 2 � � k � Scale-free networks γ < 3 zero epidemic threshold venerdì 9 luglio 2010
Susceptible-Infected-Removed (SIR) model • Three possible states: susceptible, infected and removed. • Two possible events for infected nodes: ‣ Death/recovery (rate 1 ) ‣ Infection to neighbors (rate λ ) ‣ Transition between healthy and infected venerdì 9 luglio 2010
HMF for SIR • HMF theory � k � λ c = � k 2 � − � k � • Zero epidemic threshold for scale-free networks • Finite epidemic threshold for scale-rich networks venerdì 9 luglio 2010
Beyond HMF for SIS • Wang et al., 2003 Λ N = largest eigenvalue 1 λ c = of adjacency matrix Λ N • Chung et al., 2005 √ k c > � k 2 � � � k � ln 2 ( N ) √ k c c 1 Λ N = � k 2 � � k 2 � � k � > √ k c ln( N ) c 2 � k � k c = largest degree in the network venerdì 9 luglio 2010
Beyond HMF for SIS • Summing up 1 / √ k c � γ > 5 / 2 λ c ≃ � k � 2 < γ < 5 / 2 � k 2 � • In any uncorrelated quenched random network with power-law distributed connectivities, the epidemic threshold goes to zero as the system size goes to infinity. • This has nothing to do with the scale-free nature of the degree distribution. venerdì 9 luglio 2010
SIS γ = 4.5 venerdì 9 luglio 2010
Finite Size Scaling SIS γ = 4.5 venerdì 9 luglio 2010
SIR γ = 4.5 venerdì 9 luglio 2010
Mathematical origin of HMF failure for SIS • HMF is equivalent to using annealed networks with adjacency matrix a ij = k i k j � k � N • This matrix has a unique nonzero eigenvalue Λ N = � k 2 � � k � venerdì 9 luglio 2010
Physical origin of HMF failure for SIS • Star graph with nodes ρ max ∝ ( λ 2 k max − 1) ρ 1 ∝ ( λ 2 k max − 1) • For λ > 1/ √ k max the hub and its neighbors are a self-sustained core of infected nodes, which spread the activity to the rest of the system. venerdì 9 luglio 2010
Star vs full graph same k max venerdì 9 luglio 2010
Fluctuations of k max venerdì 9 luglio 2010
Summary on epidemics • Zero epidemic threshold for SIS on scale- rich networks. • Finite epidemic threshold for SIR on scale- rich networks. • Conjecture: zero threshold for all models with steady state. • Caveat: annealed networks are important for real epidemics. venerdì 9 luglio 2010
Contact Process (CP) • Two possible states: susceptible and infected • Two possible events for infected nodes: ‣ Recovery (rate 1 ) ‣ Infection to neighbors (rate λ /k ) • Phase-transition with finite threshold venerdì 9 luglio 2010
Quenched (disordered) contact process • A fraction q of nodes has reduced infection rate: λ r 0 ≤ r ≤ 1 • A fraction 1-q of nodes has normal infection rate: λ � k � 1 λ c ( q, r ) = 1 − q (1 − r ) . � k � − 1 • For q > q perc nodes with normal infection rate form only small clusters q perc = 1 − 1 / � k � venerdì 9 luglio 2010
q < q perc venerdì 9 luglio 2010
q > q perc venerdì 9 luglio 2010
Rare regions effect • For the “dirty” system, the threshold is larger than the threshold for the pure system. λ dirty > λ pure c c • For λ cpure < λ < λ cdirty , there are rare local clusters of “pure” nodes, which are above the threshold, i.e. in the active phase. • Activity in pure clusters lives until a coherent fluctuation destroys it. This occurs in a time τ ( s ) ≃ t 0 exp[ A ( λ ) s ] venerdì 9 luglio 2010
Rare regions effect • Size distribution of “pure” clusters 1 p = � k � (1 − q ) 2 π ps − 3 / 2 e − s ( p − 1 − ln( p )) P ( s ) ∼ √ • Overall activity decay � ds s P ( s ) exp [ − t/ ( t 0 e A ( λ ) s )] ∼ t − γ ( p, λ ) ρ ( t ) ∼ γ ( p, λ ) = − ( p − 1 − ln( p )) /A ( λ ) Generic power-law decay with continuously varying exponents venerdì 9 luglio 2010
Phase diagram venerdì 9 luglio 2010
Conclusions • Models for epidemics have zero threshold also on scale-rich networks if they have a steady state. HMF may fail. • Quenched disorder may yield generic power law decays. • Voter dynamics is strongly affected by scale-free nature. Heterogeneous pair approximation works. venerdì 9 luglio 2010
• C. Castellano and R. Pastor-Satorras, “Thresholds for epidemic spreading in networks” (soon in arXiv) • M. A. Munoz, R. Juhasz, C. Castellano and G. Odor, “Griffiths phases in networks” (soon in arXiv) • E. Pugliese and C. Castellano, “Heterogeneous pair approximation for voter models on networks”, EPL, 88, 58004 (2009) venerdì 9 luglio 2010
Recommend
More recommend
