2. Dynamics of epicenters in the Olami-Feder-Christensen Model Carmen P. C. Prado Universidade de São Paulo Workshop on Transport and Self Organization in Complex Systems TSOCS’2004 - Porto Alegre, Brazil, 2004

3. Tiago P. Peixoto (USP, PhD st) Osame Kinouchi (Rib. Preto, USP) Suani T. R. Pinho (UFBa ) Josué X. de Carvalho (USP, pos-doc) prado@if.usp.br

4. Brief Introduction: Earthquakes, SOC and the Olami-Feder- Christensen model (OFC)Earthquakes, SOC and the Olami-Feder- Christensen model (OFC) Recent results on earthquake dynamics: Recent results on earthquake dynamics: connection between SOC and free scale networks Epicenter distribution (real earthquakes) Epicenter distribution (real earthquakes) Epicenters in the OFC model (our results) Epicenters in the OFC model (our results) Recent data Recent data

6. Self-organized criticality “Punctuated equilibrium” extended systems that, under 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

7. 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 -t Fractal distribution of epicenters N(r) ~ r d 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) )

8. 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...

9. 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)

10. 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

11. 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

12. How well the model describes real earthquakes? Predictions, lot of recent new results (networks) How well the model describes real earthquakes? Predictions, lot of recent new results (networks) 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 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 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

13. 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 ?  = 0 model is non-critical  = 0.25 model is critical at which value of  =  c the system changes its behavior ???

14. Branching rate approach Most of the analytical progress on the RN -OFC used a formalism 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

15. Dynamics of the epicenters & Complex networks S. Abe, N. Suzuki, Europhys. Lett. 65, 581 (2004) 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

16. 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

17. S. Abe, N. Suzuki’ s result connectivity K Degree distribution

18. Free-scale network connectivity of the node P(k) ~ k -  Complex networks describe a wide range of systems in nature and society R. Albert, A-L. Barabási, Rev. Mod. Phys. 74 (2002) Small world network: Despite their large size, in many networks there is a short path between any two nodes. High clustering coefficient Random graph P(k) distribution is Poisson Low clustering

19. We studied the OFC model in this context, to see if it was able to predict also this behavior Clear scaling (Curves were shifted upwards for the sake of clarity ) Tiago P. Peixoto, C. P. C. Prado, 2004 PRE 69, 025101(R) (2004) L = 200, transients of 10 7, statistics of 10 5 0.240 0.249

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

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

22. L = 300 L = 200 We still need a growing network, as in Barabasi-Albert model... But there is no preferential attachment rule. The preferential attachment is a signature of the dynamics Finite size effect

23. But surprisingly, There is a qualitative difference between conservative and non-conservative regimes ! 0..25

24. 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

25. 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

26. Network structure Non - conservative: clustering c ~ 0.15 Cell size = 5; L=200,  = 0.14 Conservative: no clustering, c ~ 0.0 Cell size = 5; L=200,  = 0.25

27. 200 sites/links, no bordes (b=2 cells = 10 sites) c ~ 0.13 500 sites/links, no bordes ( b=2 cells = 10 sites) c ~ 0.23 Non - conservative Conservative 200 sites/links, no bordes (b=2 cells = 10 sites) c ~ 0.0

28. Distribution of distances Distribution of distances between two consecutive epicenters 200 X 200 lattice,  = 0.25 and  = 0.14 Distribution of distances between two consecutive epicenters 50 X 50 X 50 lattice,  = 1/6 = 0.1666..,  = 0.160 and  = 0.12. P. Peixoto, C. P. C. Prado, T. P. Peixoto, C. P. C. Prado, to appear in Physica A (2004), corrected proof available on line

29. Conclusions Robustness of OFC model to describe real earthquakes, since its able to reproduce the scale free network observed in real data 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, in which the preferential attachment, present in all free scale networks is not a pre-defined rule but a signature of the dynamics New dynamical mechanism to generate a free-scale network, in which the preferential attachment, present in all free scale networks is not a pre-defined rule but a signature of the dynamics Indicates (in agreement with many previous works) qualitatively different behavior between conservative and non-conservative models Indicates (in agreement with many previous works) qualitatively different behavior between conservative and non-conservative models Many open questions... Many open questions... The theory of networks: way of studying de particular entanglement of spatio-temporal distribution of earthquakes, new promising approach to the problem of prediction... The theory of networks: way of studying de particular entanglement of spatio-temporal distribution of earthquakes, new promising approach to the problem of prediction...