1. Dynamics of epicenters in the Olami-Feder-Christensen Model Carmen P. C. Prado Universidade de São Paulo (prado@if.usp.br)

2. Tiago P. Peixoto (USP, PhD st) Osame Kinouchi (Rib. Preto, USP) Suani T. R. Pinho (UFBa ) Josué X. de Carvalho (USP, pos-doc)

3. Introduction: Earthquakes, SOC and the Olami-Feder- Christensen model (OFC) Recent results on earthquake dynamics: Epicenter distribution (real earthquakes) Epicenters in the OFC model (our results) Epicenters in the OFC model (our results)

4. Self-organized criticality “Punctuated equilibrium” Extended systems that, under some slow external drive (instead of evolving in a slow and continuous way) Remain static (equilibrium) for long periods; That are “punctuated” by very fast events that leads the systems to another “equilibrium” state; Statistics of those fast events shows power-laws indicating criticality Bak, Tang, Wisenfeld, PRL 59,1987/ PRA 38, 1988 Sand pile model

5. Earthquake dynamics is probably the best “experimental ” realization of SOC ideas... Exhibits universal power - laws Gutemberg-Richter ’s law Gutemberg-Richter ’s law ( energy) P(E)  E -B Omori ’s law Omori ’s law (aftershocks and foreshocks) n(t) ~ t -A Two distinct time scales& punctuated equilibrium Two distinct time scales & punctuated equilibrium Slow: Slow: movement of tectonic plates (years) Fast: Fast: earthquakes (seconds) The relationship between SOC concepts and the dynamics of earthquakes was pointed out from the beginning ( Bak and Tang, J. Geophys. Res. B (1989); Sornette and Sornette, Europhys. Lett. (1989); Ito and Matsuzaki, J. Geophys. Res. B (1990) )

6. By the 20 ies scientists already knew that most of the earthquakes occurred in definite and narrow regions, where different tectonic plates meet each other...

7. Burridge-Knopoff model (1967) Burridge-Knopoff model (1967) Fixed plate Moving plate V  k i - 1 i i + 1 friction Olami et al, PRL68 (92); Christensen et al, PRA 46 (92)

8. If some site becomes “active”, that is, if F > F th, the system relaxes:Relaxation: Perturbation: If any of the 4 neighbors exceeds F th, the relaxation rule is repeated. This process goes on until F < F th again for all sites of the lattice

9. The size distribution of avalanches obeys a power-law, reproducing the Gutemberg-Richter law and Omori’s Law Simulation for lattices of sizes L = 50,100 e 200. Conservative case:  = 1/4 SOC even in the non conservative regime N( t ) ~ t -  Hergarten, H. J. Neugebauer, PRL 88, 2002 showed that the OFC model exhibits sequences of foreshocks and aftershocks, consistent with Omori’ s law, but only in the non-conservative regime!

10. While there are almost no doubts about the efficiency of this model to describe real earthquakes, the precise behavior of the model in the non conservative regime has raised a lot of controversy, both from a numerical or a theoretical approach. The nature of its critical behavior is still not clear. The model shows many interesting features, and has been one of the most studied SOC models

11.  = 0 model is non-critical  = 0.25 model is critical at which value of  =  c the system changes its behavior ??? First simulations where performed in very small lattices ( L ~ 15 to 50 ) No clear universality class P(s) ~ s - ,  =  (  ) No simple FSS, scaling of the cutoff High sensibility to small changes in the rules (boundaries, randomness) Theoretical arguments, connections with branching process, absence of criticality in the non conservative random neighbor version of the model has suggested conservation as an essential ingredient. Where is the cross-over ?

12. Branching rate approach Most of the analytical progress on the RN -OFC used a formalism developed by Lise & Jensen which uses the branching rate  (  ). cc Almost critical O. Kinouchi, C.P.C. Prado, PRE 59 (1999) J. X. de Carvalho, C. P. C. Prado, Phys. Rev. Lett. 84, 006, (2000). Almost critical Remains controversial

13. Dynamics of the epicenters S. Abe, N. Suzuki, cond-matt / 0210289 Instead of the spatial distribution (that is fractal), the looked at the time evolution of epicenters Instead of the spatial distribution (that is fractal), the looked at the time evolution of epicenters Found a new scaling law for earthquakes (Japan and South California) Found a new scaling law for earthquakes (Japan and South California) Fractal distribution

14. S. Abe, N. Suzuki, cond-matt / 0210289 Time sequence of epicenters from earthquake data of a district of southern California and Japan small cubic cells area was divided into small cubic cells, each of which is regarded as vertex of a graph if an epicenter occurs in it; the seismic data was mapped into na evolving random graph; Free-scale behavior of Barabási-Albert type

16. Complex networks describe a wide range of systems in nature and society R. Albert, A-L. Barabási, Rev. Mod. Phys. 74 (2002) Free-scale network connectivity of the node P(k) ~ k -  Random graph distribution is Poisson

17. We studied the OFC model in this context, to see if it was able to predict also this behavior Tiago P. Peixoto, C. P. C. Prado, 2003 L = 200, transients of 10 7, statistics of 10 5 Clear scaling ( Curves were shifted upwards for the sake of clarity ) 0.240 0.249

18. The exponent  that characterizes the power-law behavior of P(k), for different values of 

19. L = 200, 1 X 1 L = 400, 2 X 2 The size of the cell does not affect the connectivity distribution P(k)...

20. But surprisingly, 0..25

21. L = 300 L = 200 We need a growing network...

22. Distribution of connectivity L = 200,  = 0.249 L = 200,  = 0.25

23. Spatial distribution of connectivity, ( non-conservative) (b) is a blow up of (a); The 20 sites closer to the boundaries have not been plotted and the scale has been changed in order to show the details. It is not a boundary effect

24. Spatial distribution of connectivity, ( conservative) In (a) we use the same scale of the previous case In (b) The scale has been changed to show the details of the structure Much more homogeneous

25. Conclusions Robustness of OFC model to describe real earthquakes, since its able to reproduce the scale free network observed in real data New dynamical mechanism to generate a free-scale network, The preferential attachment present in the network is not a rule but a signature of the dynamics Indicates (in agreement with many previous works) qualitatively different behavior between conservative and non-conservative models Many open questions...