21st Century Science Initiative Grant: Studying Complex Systems
A central property of a complex system is the possible occurrence of coherent large-scale collective behaviors with a very rich structure, resulting from the repeated non-linear interactions among its constituents: the whole turns out to be much more than the sum of its parts. It is widely believed that most complex systems are not amenable to mathematical, analytic descriptions and can only be explored by means of "numerical experiments". In the context of the mathematics of algorithmic complexity , many complex systems are said to be computationally irreducible, i.e. the only way to decide about their evolution is to actually let them evolve in time. Accordingly, the "dynamical" future time evolution of complex systems would be inherently unpredictable. This unpredictability does not prevent however the application of the scientific method for the prediction of novel phenomena as exemplified by many famous cases (prediction of the planet Neptune by Leverrier from calculations of perturbations in the orbit of Uranus, the prediction by Einstein of the deviation of light by the sun's gravitation field, the prediction of the helical structure of the DNA molecule by Watson and Crick based on earlier predictions by Pauling and Bragg, etc.). In contrast, it refers to the frustration to satisfy the quest for the knowledge of what tomorrow will be made of, often filled by the vision of prophets" who have historically terrified or inspired the masses.
The view that complex systems are unpredictable has recently been defended persuasively in concrete prediction applications, such as the socially important issue of earthquake prediction (see the contributions in ). In addition to the persistent failures at reaching a reliable earthquake predictive scheme, this view is rooted theoretically in the analogy between earthquakes and self-organized criticality . In this "fractal" framework, there is no characteristic scale and the power law distribution of sizes reflects the fact that the large earthquakes are nothing but small earthquakes that did not stop. They are thus unpredictable because their nucleation is not different from that of the multitude of small earthquakes which obviously cannot be all predicted.
Does this really hold for all features of complex systems? Take our personal life. We are not really interested in knowing in advance at what time we will go to a given store or drive to a highway. We are much more interested in forecasting the major bifurcations ahead of us, involving the few important things, like health, love and work that count for our happiness. Similarly, predicting the detailed evolution of complex systems has no real value and the fact that we are taught that it is out of reach from a fundamental point of view does not exclude the more interesting possibility to predict phases of evolutions of complex systems that really count.
It turns out that most complex systems in natural and social sciences do exhibit rare and sudden transitions, that occur over time intervals that are short compared to the characteristic time scales of their posterior evolution. Such extreme events express more than anything else the underlying "forces" usually hidden by almost perfect balance and thus provide the potential for a better scientific understanding of complex systems.
These crises have fundamental societal impacts and range fro large natural catastrophes such as earthquakes, volcanic eruptions, hurricanes and tornadoes, landslides, avalanches, lightning strikes, meteorite/asteroid impacts, catastrophic events of environmental degradation, to the failure of engineering structures, crashes in the stock market, social unrest leading to large-scale strikes and upheaval, economic drawdowns on national and global scales, regional power blackouts, traffic gridlock, diseases and epidemics, etc. It is essential to realize that the long-term behavior of these complex systems is often controlled in large part by these rare catastrophic events: the universe was probably born during an extreme explosion (the "big-bang"); the nucleosynthesis of all important atomic elements constituting our matter results from the colossal explosion of supernovae; the largest earthquake in California repeating about once every two centuries accounts for a significant fraction of the total tectonic deformation; landscapes are more shaped by the "millenium" flood that moves large boulders rather than the action of all other eroding agents; the largest volcanic eruptions lead to major topographic changes as well as severe climatic disruptions; according to some contemporary views, evolution is probably characterized by phases of quasi-statis interrupted by episodic bursts of activity and destruction; financial crashes, which can destroy in an instant trillions of dollars, loom over and shape the psychological state of investors; political crises and revolutions shape the long-term geopolitical landscape; even our personal life is shaped on the long run by a few key decisions or happening.
The outstanding scientific question is thus how such large-scale patterns of catastrophic nature might evolve from a series of interactions on the smallest and increasingly larger scales. In complex systems, it has been found that the organization of spatial and temporal correlations do not stem, in general, from a nucleation phase diffusing across the system. It results rather from a progressive and more global cooperative process occurring over the whole system by repetitive interactions. An instance would be the many occurrences of simultaneous scientific and technical discoveries signaling the global nature of the maturing process.
Standard models and simulations of scenario of extreme events are subject to numerous sources of error, each of which may have a negative impact on the validity of the predictions . Some of the uncertainties are under control in the modeling process; they usually involve trade-offs between a more faithful description and manageable calculations. Other sources of errors are beyond control as they are inherent in the modeling methodology of the specific disciplines. The two known strategies for modeling are both limited in this respect: analytical theoretical predictions are out of reach for most complex problems. Brute force numerical resolution of the equations (when they are known) or of scenarii is reliable in the "center of the distribution", i.e. in the regime far from the extremes where good statistics can be accumulated. Crises are extreme events that occur rarely, albeit with extraordinary impact, and are thus completely under-sampled and thus poorly constrained. Even the introduction of teraflop (or even pentaflops in the future) supercomputers does not change qualitatively this fundamental limitation.
Notwithstanding these limitations, we believe that the progress of science and of its multidisciplinary enterprises make the time ripe for a full-fledge effort towards the prediction of complex systems. We propose to develop a set of novel approaches for modeling and predicting certain catastrophic events, or "ruptures," that is, sudden transitions from a quiescent state to a crisis or catastrophic event. Such ruptures involve interactions between structures at many different scales. Our approach consists in combining ideas and tools from statistical physics and artificial/computational intelligence, to identify and classify possible universal structures that occur at different scales, and to develop application-specific methodologies to use these structures for prediction of the "crises" known to arise in each application of interest. We shall investigate the following concrete applications: (i) the hierarchical patterns and precursors that precede damage and rupture in complex heterogeneous materials, such as composites with multiple components and other elaborated engineering structures as well as earthquakes; (ii) the premonitory processes before financial crashes or "bubble" corrections in the stock market; and (iii) precursors to abrupt changes in climate and weather regimes.
We will develop a new set of computational methods which are capable of searching and comparing patterns, simultaneously and iteratively, at multiple scales in hierarchical systems. We will use these patterns to improve the understanding of the dynamical state before and after ruptures and to enhance the statistical modeling of hierarchical systems with the goal of developing reliable forecasting skills for these large-scale ruptures. We live in a planet and society with intermittent dynamics rather than at equilibrium, and so there is a growing and urgent need to sensitize students to the importance and impacts of ruptures in their multiple forms. This research will thus have important implications for our graduate and undergraduate instructions at UCLA and in our outreach to a broader audience, that transcends a disciplinary community of scholars.
 Chaitin, G.J., Algorithmic information theory, Cambridge, New York : Cambridge University Press, 1987.
 Is the reliable prediction of individual earthquakes a realistic scientific goal? Nature debates, http://helix.nature.com/debates/earthquake/
 Bak, P., How nature works : the science of self-organized criticality, New York,
NY, USA : Copernicus, 1996.
 Karplus, W.J., The Heavens are Falling: The Scientific Prediction of Catastrophes
in Our Time, Plenum, 1992.