1 / 29

Spreading dynamics on small-world networks with a power law degree distribution

Spreading dynamics on small-world networks with a power law degree distribution. Alexei Vazquez The Simons Center for Systems Biology Institute for Advanced Study. Epidemic outbreak. External source. Population. Population structure. Contact graph. N individuals

belindae
Download Presentation

Spreading dynamics on small-world networks with a power law degree distribution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Spreading dynamics on small-world networks with a power law degree distribution Alexei Vazquez The Simons Center for Systems Biology Institute for Advanced Study

  2. Epidemic outbreak External source Population

  3. Population structure Contact graph N individuals pk connectivity distribution D average distance

  4. Sexual contacts Sweden 1 year lifetime -1 -1 Sexually transmitted diseases • pk~k- , 2<<5 • Liljeros et al. Nature (2001) • Jones & Handcook, Nature (2003) • Schneeberger et al, Sex Transm Dis (2004)

  5. Sexual contacts STD Colorado Springs HIV network Potterat et al, Sex. Transm Infect 2002 k-2 N=250 D 8

  6. Physical contact or proximity 1 day Portland k-1.8 USA D4.37 city Eubank et al, Nature 2004 nation/world Barrat et al, PNAS 2004 N=3,880

  7. Physical contact or proximity

  8. Branching process model Generation 0 root 1 2 3 4 Spanning tree kpk/<k> k-1 pk k

  9. Branching process model Timming dd+1 generation Generation time T Distribution G()=Pr(T) tt+T1t+T2t+T3 time

  10. Branching process model • The process start with a node (d=0) that generates k sons with probability distribution pk. • Each son at generation 0<d<D generates k-1 new sons with probability kpk/<k>. • Nodes at generation D does not generate any son. • The generation times are independent random variables with distribution function G(). Note: Galton-Watson, Newman Bellman-Harris, Crum-Mode-Jagers

  11. Recursive calculation T1 t=0 T2 d d+1 t=0 d

  12. Results Reproductive number Time scale Vazquez, Phys. Rev. Lett. 2006 Constant transmission rate : G()=1-e- Incidence I(t): expected rate of new infections at time t

  13. pk~k -, kmax~N >3, t<<t0 (t0 when N ) <3, t>>t0 (t00 when N ) Vazquez, Phys. Rev. Lett. 2006

  14. Numerical simulations • Network: random graph with a given degree distribution. • pk~k - • Constant transmission rate  • N=1000, 10000, 100000 • 100 graph realizations, 10000 outbreaks

  15. Numerical simulations log-log linear-log e(K-1)t I(t)/N I(t)/N  1,000  10,000  100,000 tD-1e-t t t

  16. Case study: AIDS epidemics • New York - HOM • New York - HET • San Francisco - HOM • South Africa • Kenya • Georgia • Latvia • Lithuania t3 t3 t3 Cumulative number t2 t2 exponential t (years) Szendroi & Czanyi, Proc. R Soc. Lond. B 2004

  17. Generalizations Degree correlations Multitype

  18. Degree correlations k’ k

  19. Degree correlations Kk Kk k k

  20. Degree correlations N( t)D-1e-t k’ k   e(R*-1)t  Vazquez, Phys. Rev. E 74, 056101 (2006)

  21. Multi-type i=1,…,M types Ni number of type i agents p(i)k type i degree distribution eijmixing matrix D average distance Reproductive number matrix : largest eigenvalue

  22. Multi-type Strongly connected type-networks Vazquez, Phys. Rev. E (In press); http://arxiv.org/q-bio.PE/0605001  Type 1  Type 2  Type 3  Type 4 Type-network eij eii

  23. Generalizations Non-exponential generating time distributions

  24. Intermediate states Vazquez, DIMACS Series in Discrete Mathematics… 70, 163 (2006)

  25. Long time behavior: Email worms Generating time probability density Receive infected email Sent infected emails time generating time (residual waiting time) In collaboration with R. Balazs, L. Andras and A.-L. Barabasi

  26. Email activity patterns Left: University server 3,188 users 129,135 emails sent <  >~1 day E~25 days Right: Comercial email server ~1,7 millions users ~39 millions emails sent <  >~4 days E~9 months T T

  27. Incidence: model

  28. Prevalence: http://www.virusbtn.com I(t) I(t) I(t) Prevalence data Decay time ~ 1 year Poisson model <  >~1 day - University <  >~4 days - Comercial Email data E~25 days - University E~9 months - Comercial

  29. Conclusions • Truncated branching processes are a suitable framework to model spreading processess on real networks. • There are two spreading regimes. • Exponential growth. • Polynomial growth followed by an exponential decay. • The time scale separating them is determined by D/R. • The small-world property and the connectivity fluctuations favor the polynomial regime. • Intermediate states favor the exponential regime.

More Related