The power of interdisciplinarity
On a personal highlight illustrating the power of interdisciplinarity and unity of the mathematical description of natural and social processes
We have found the same fundamental concepts to apply efficiently to model on the one hand the triggering processes between earthquakes leading to their complex space-time statistical organization [Helmstetter et al., 2003; Ouillon and Sornette, 2005; Sornette and Ouillon, 2005] and on the other hand the social response to shocks in such examples aInternet downloads in response to information shocks [Johansen and Sornette, 2000], book sales blockbusters [Sornette et al., 2004; Deschatres and Sornette, 2005], the response to social shocks [Roehner et al., 2004], financial volatility shocks [Sornette et al., 2003], and financial bubbles and their crashes [Johansen et al., 1999; 2000; Sornette, 2003, Andersen and Sornette, 2005; Sornette and Zhou, 2006].
Let me tell a quite interesting story on how the research process developed.
First, the possibility that precursory seismic activity, known as foreshocks, could be intimately related to aftershocks has been entertained by several authors in the past but has not been clearly demonstrated by a combined derivation of the so-called direct Omori law for aftershocks and of the inverse Omori for foreshocks within a consistent model. In a nutshell, the Omori law for aftershocks describes the decay rate of seismicity after a large earthquake (called a mainshock), roughly going as the inverse of time since the mainshock. The inverse Omori law for foreshocks describes the statistically increasing rate of earthquakes going roughly as the inverse of the time till the next mainshock. It has been demonstrated empirically only by stacking many earthquake sequences (see [Helmstetter and Sornette, 2003] and references therein). We had been working for several years on the theoretical understanding of a fashionable statistical seismicity, known at the ETAS model, a self-excited Hawkes conditional point process, in mathematical parlance. We had the intuition that the inverse Omori law for foreshocks could be derived from the direct Omori law by viewing mainshocks as the "aftershocks of foreshocks, conditional on the magnitude of mainshocks being larger than that of their progenitors". But we could not find the mathematical trick to complete the theoretical derivation. In parallel, we have been working on the statistical properties of financial returns and were starting a collaboration with J.-F. Muzy, one of the discoverer of a new stochastic random walk with exact multifractal properties, named the multifractal random walk (MRW) [Bacry et al, 2001; Muzy and Bacry, 2002], which seems a powerful model of financial time series. We then realized that similar questions could be asked on the precursory as well as posterior behavior of financial volatility around shocks. The analysis of the data showed clear Omori-like and inverse Omori-like behavior. It turned out that we were able to formulate the solution mathematically within the formalism of the MRW and we showed the deep link between the precursory increase and posterior behavior around financial shocks [Sornette et al., 2003]. In particular, we showed a clear quantitative relationship between the relaxation after an exogenously caused shock and that arising spontaneously (termed "endogenous"). Then, inspired by the conceptual path used to solve the problem in the financial context, we were able to derive the solution in the context of the ETAS model, demonstrating mathematically the deep link between the inverse Omori law for foreshocks and the direct Omori law for aftershocks [Helmstetter et al., 2003; Helmstetter and Sornette, 2003]. The path was simpler and clearer for financial time series and their study clarified the methodology to be used for the more complicated specific point processed modeling earthquakes.
This remarkable back-and-forth thought process between two a priori very different fields will remain a personal highlight of my scientific life.
References:
Andersen, J.V. and D. Sornette, A Mechanism for Pockets of Predictability in Complex Adaptive Systems, Europhys. Lett., 70 (5), 697-703 (2005)
Bacry, E, J. Delour and, J.-F. Muzy, Multifractal random walk, Phys. Rev. E 64, 026103 (2001).
Deschatres, F. and D. Sornette, The Dynamics of Book Sales: Endogenous versus Exoge-nous Shocks in Complex Networks, Phys. Rev. E 72, 016112 (2005).
Helmstetter, A. and D. Sornette, Foreshocks explained by cascades of triggered seismic-ity, J. Geophys. Res. (Solid Earth) 108 (B10), 2457 10.1029/2003JB002409 01 (2003)
Helmstetter, A., D. Sornette and J.-R. Grasso, Mainshocks are Aftershocks of Conditional Foreshocks: How do foreshoc statistical properties emerge from aftershock laws, J. Geophys. Res., 108 (B10), 2046, doi:10.1029/2002JB001991 (2003).
Johansen, A., O. Ledoit and D. Sornette, Crashes as critical points, International Journal of Theoretical and Applied Finance 3 (2), 219-255 (2000).
Johansen, A. and D. Sornette, Download relaxation dynamics on the WWW following newspaper publication of URL, Physica A 276/1-2, 338-345 (2000).
Johansen, A., D. Sornette and O. Ledoit, Predicting Financial Crashes using discrete scale invariance, Journal of Risk 1 (4), 5-32 (1999).
Muzy, J.-F. and E. Bacry, Multifractal stationary random measures and multifractal ran-dom walks with log infinitely divisible scaling laws, Phys. Rev. E 66, 056121 (2002).
Ouillon, G. and D. Sornette. Magnitude-Dependent Omori Law: Theory and Empirical Study, J. Geophys. Res., 110, B04306, doi:10.1029/2004JB003311 (2005).
Roehner, B.M., D. Sornette and J.V. Andersen, Response Functions to Critical Shocks in Social Sciences: An Empirical and Numerical Study, Int. J. Mod. Phys. C 15 (6), 809-834 (2004).
Sornette, D., Why Stock Markets Crash (Critical Events in Complex Financial Systems)
(Princeton University Press, Princeton, NJ, 2003).
Sornette, D., F. Deschatres, T. Gilbert and Y. Ageon, Endogenous Versus Exogenous Shocks in Complex Networks: an Empirical Test Using Book Sale Ranking, Phys. Rev. Letts. 93 (22), 228701 (2004).
Sornette, D., Y. Malevergne and J.-F. Muzy, What causes crashes? Risk 16 (2), 67-71 (2003). http://arXiv.org/abs/cond-mat/0204626
Sornette, D. and G. Ouillon, Multifractal Scaling of Thermally-Activated Rupture Proc-esses, Phys. Rev. Lett. 94, 038501 (2005)
Sornette, D. and W.-X. Zhou, Predictability of Large Future Changes in Complex Sys-tems, International Journal of Forecasting 22, 153-168 (2006).