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D Sornette

Publications and source records attributed to D Sornette.

At least 19 recordsLinked to original sources
PubMed2006

"Universal" distribution of interearthquake times explained.

We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks, and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of interevent times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforce the view that triggering earthquakes by other earthquakes is a key physical mechanism to understand seismicity.

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PubMed2006

Generic multifractality in exponentials of long memory processes.

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent phi+1/2, where phi>0. This generalizes previous studies performed only with phi=0(with a truncation at an integral scale) by showing that multifractality holds over a remarkably large range of dimensionless scales for phi>0. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation phi from 1/2 and of another parameter sigma2 embodying information on the short-range amplitude of the memory kernel, the ultraviolet cutoff ("viscous") scale, and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the "inertial" scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra zeta(q) on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum zeta(q) by different combinations of phi and sigma2.

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PubMed2005

Predicting failure using conditioning on damage history: demonstration on percolation and hierarchical fiber bundles.

We formulate the problem of probabilistic predictions of global failure in the simplest possible model based on site percolation and on one of the simplest models of time-dependent rupture, a hierarchical fiber bundle model. We show that conditioning the predictions on the knowledge of the current degree of damage (occupancy density p or number and size of cracks) and on some information on the largest cluster improves significantly the prediction accuracy, in particular by allowing one to identify those realizations which have anomalously low or large clusters (cracks). We quantify the prediction gains using two measures, the relative specific information gain (which is the variation of entropy obtained by adding new information) and the root mean square of the prediction errors over a large ensemble of realizations. The bulk of our simulations have been obtained with the two-dimensional site percolation model on a lattice of size L x L=20 x 20 and hold true for other lattice sizes. For the hierarchical fiber bundle model, conditioning the measures of damage on the information of the location and size of the largest crack extends significantly the critical region and the prediction skills. These examples illustrate how ongoing damage can be used as a revelation of both the realization-dependent preexisting heterogeneity and the damage scenario undertaken by each specific sample.

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PubMed2005

Vere-Jones' self-similar branching model.

Motivated by its potential application to earthquake statistics as well as for its intrinsic interest in the theory of branching processes, we study the exactly self-similar branching process introduced recently by Vere-Jones. This model extends the ETAS class of conditional self-excited branching point-processes of triggered seismicity by removing the problematic need for a minimum (as well as maximum) earthquake size. To make the theory convergent without the need for the usual ultraviolet and infrared cutoffs, the distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m < m' with exponent beta - d and m' > m with exponent beta + d, where beta and d are two positive parameters. We investigate the condition and nature of the subcritical, critical, and supercritical regime in this and in an extended version interpolating smoothly between several models. We predict that the distribution of magnitudes of events triggered by a mother of magnitude m over all generations has also two branches m' < m with exponent and with exponent beta - h, with h=d squareroot of (1-s), where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. For a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources with a single branch and is blind to the exponents beta, d of the distribution of triggered events. Since the distribution of earthquake magnitudes is usually obtained with catalogs including many sequences, we conclude that the two branches of the distribution of aftershocks are not directly observable and the model is compatible with real seismic catalogs. In summary, the exactly self-similar Vere-Jones model provides an attractive new approach to model triggered seismicity, which alleviates delicate questions on the role of magnitude cutoffs in other non-self-similar models. The new prediction concerning two branches in the distribution of magnitudes of aftershocks could be tested with recently introduced stochastic reconstruction methods, tailored to disentangle the different triggered sequences.

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PubMed2005

Dynamics of book sales: endogenous versus exogenous shocks in complex networks.

We present an extensive study of the foreshock and aftershock signatures accompanying peaks of book sales. The time series of book sales are derived from the ranking system of Amazon.com. We present two independent ways of classifying peaks, one based on the acceleration pattern of sales and the other based on the exponent of the relaxation. They are found to be consistent and reveal the coexistence of two types of sales peaks: exogenous peaks occur abruptly and are followed by a power law relaxation, while endogenous sale peaks occur after a progressively accelerating power law growth followed by an approximately symmetrical power law relaxation which is slower than for exogenous peaks. We develop a simple epidemic model of buyers connected within a network of acquaintances which propagates rumors and opinions on books. The comparison between the predictions of the model and the empirical data confirms the validity of the model and suggests in addition that social networks have evolved to converge very close to criticality (here in the sense of critical branching processes of opinion spreading). We test in detail the evidence for a power law distribution of book sales and confirm a previous indirect study suggesting that the fraction of books (density distribution) P (S) of sales S is a power law P(S) approximately 1/ S(1+mu) with mu approximately equal to 2 .

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PubMed2005

Distribution of the largest aftershocks in branching models of triggered seismicity: theory of the universal Båth law.

Using the epidemic-type aftershock sequence (ETAS) branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Båth's law. Our theory shows that Båth's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +/- 0.1 for Båth's constant value around 1.2, our exact analytical treatment of Båth's law provides new constraints on the productivity exponent alpha and the branching ratio n: 0.9 approximately < alpha < or =1. We propose a method for measuring alpha based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the "second Båth law for foreshocks:" the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude rho.

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PubMed2005

Discrete hierarchical organization of social group sizes.

The 'social brain hypothesis' for the evolution of large brains in primates has led to evidence for the coevolution of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size that depends, in some way, on the volume of neural material available for processing and synthesizing information on social relationships. More recently, work on both human and non-human primates has suggested that social groups are often hierarchically structured. We combine data on human grouping patterns in a comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continuous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3-5, 9-15, 30-45, etc. Such discrete scale invariance could be related to that identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing of social nearness by human brains.

Anthropology, Cultural↗
PubMed2005

Creep ruptures in heterogeneous materials.

We present creep experiments on fiber composite materials with different controlled heterogeneity. All samples exhibit a power-law relaxation of the strain rate in the primary creep regime (Andrade's law) followed by a power-law acceleration up to rupture. We discover that the rupture time is proportional to the duration of the primary creep regime, showing the interplay between the two regimes and offering a method of rupture prediction. These experimental results are rationalized by a mean-field model of representative elements with nonlinear viscoelastic rheology and with a large heterogeneity of strengths.

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PubMed2005

Multifractal scaling of thermally activated rupture processes.

We propose a "multifractal stress activation" model combining thermally activated rupture and long memory stress relaxation, which predicts that seismic decay rates after mainshocks follow the Omori law approximately 1/t(p) with exponents p linearly increasing with the magnitude M(L) of the mainshock. We carefully test this prediction on earthquake sequences in the Southern California earthquake catalog: we find power law relaxations of seismic sequences triggered by mainshocks with exponents p increasing with the mainshock magnitude by approximately 0.1-0.15 for each magnitude unit increase, from p(M(L) = 3) approximately 0.6 to p(M(L) = 7) approximately 1.1, in good agreement with the prediction of the multifractal model.

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PubMed2005

Andrade, Omori, and time-to-failure laws from thermal noise in material rupture.

Using a simple mean-field rupture model with quenched disorder in the presence of thermal fluctuations introduced by S. Ciliberto et al., we provide an analytical theory of three ubiquitous empirical observations obtained in creep (constant applied stress) experiments: the initial Andrade-like and Omori-like 1/t decay of the rate of deformation and of fiber ruptures and the 1/( tc-t) critical time-to-failure behavior of acoustic emissions just prior to the macroscopic rupture. The lifetime of the material is controlled by a thermally activated Arrhenius nucleation process, describing the crossover between these two regimes, as shown by S. Ciliberto et al. Thus tiny thermal fluctuations may actually play an essential role in macroscopic deformation and rupture processes at room temperature. We also discover a reentrant dependence of the lifetime as a function of the amount of quenched disorder.

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PubMed2004

Endogenous versus exogenous shocks in complex networks: an empirical test using book sale rankings.

We study the precursory and recovery signatures accompanying shocks in complex networks, that we test on a unique database of the Amazon.com ranking of book sales. We find clear distinguishing signatures classifying two types of sales peaks. Exogenous peaks occur abruptly and are followed by a power law relaxation, while endogenous peaks occur after a progressively accelerating power law growth followed by an approximately symmetrical power law relaxation which is slower than for exogenous peaks. These results are rationalized quantitatively by a simple model of epidemic propagation of interactions with long memory within a network of acquaintances. The observed relaxation of sales implies that the sales dynamics is dominated by cascades rather than by the direct effects of news or advertisements, indicating that the social network is close to critical.

Bookselling↗
PubMed2004

Anomalous power law distribution of total lifetimes of branching processes: application to earthquake aftershock sequences.

We consider a general stochastic branching process, which is relevant to earthquakes, and study the distributions of global lifetimes of the branching processes. In the earthquake context, this amounts to the distribution of the total durations of aftershock sequences including aftershocks of arbitrary generation number. Our results extend previous results on the distribution of the total number of offspring (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We consider a branching model of triggered seismicity, the epidemic-type aftershock sequence model, which assumes that each earthquake can trigger other earthquakes ("aftershocks"). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake ("productivity" or "fertility"), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime where the distribution of fertilities mu is characterized by a power law approximately 1/ mu(1+gamma) and the bare Omori law for the memory of previous triggering mothers decays slowly as approximately 1/ t(1+theta;) , with 0<theta;<1 relevant for earthquakes. Using the tool of generating probability functions and a quasistatic approximation which is shown to be exact asymptotically for large durations, we show that the density distribution of total aftershock lifetimes scales as approximately 1/ t(1+theta;/gamma) when the average branching ratio is critical (n=1) . The coefficient 1<gamma=b/alpha<2 quantifies the interplay between the exponent b approximately 1 of the Gutenberg-Richter magnitude distribution approximately 10(-bm) and the increase approximately 10(alpham) of the number of aftershocks with mainshock magnitude m (productivity), with 0.5<alpha<1 . The renormalization of the bare Omori decay law approximately 1/ t(1+theta;) into approximately 1/ t(1+theta;/gamma) stems from the nonlinear amplification due to the heavy-tailed distribution of fertilities and the critical nature of the branching cascade process. In the subcritical case n<1 , the crossover from approximately 1/ t(1+theta;/gamma) at early times to approximately 1/ t(1+theta;) at longer times is described. More generally, our results apply to any stochastic branching process with a power-law distribution of offspring per parent and a long memory.

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PubMed2004

Statistical methods of parameter estimation for deterministically chaotic time series.

We discuss the possibility of applying some standard statistical methods (the least-square method, the maximum likelihood method, and the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A "segmentation fitting" maximum likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value x(1) considered as an additional unknown parameter. The segmentation fitting method, called "piece-wise" ML, is similar in spirit but simpler and has smaller bias than the "multiple shooting" previously proposed. Comparisons with different previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size N and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

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PubMed2003

Self-similar factor approximants.

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving an improved type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are called self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Padé approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions, which include a variety of nonalgebraic functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.

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PubMed2002

Diffusion of epicenters of earthquake aftershocks, Omori's law, and generalized continuous-time random walk models.

The epidemic-type aftershock sequence (ETAS) model is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power-law decay approximately 1/t(1+theta) of seismicity after an earthquake) and Gutenberg-Richter law (power-law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power-law distribution approximately 1/r(1+mu) of distances between triggered and triggering earthquakes. The ETAS model has been studied for the last two decades to model real seismicity catalogs and to obtain short-term probabilistic forecasts. Here, we present a mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. After translating into the relevant variable for the ETAS model, we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. The different regimes are fully characterized by the two exponents theta and mu. Our predictions are checked by careful numerical simulations. We stress the distinction between the "bare" Omori law describing the seismic rate activated directly by a mainshock and the "renormalized" Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the origin of seismic diffusion in the ETAS model. We present predictions and insights provided by the ETAS to CTRW mapping which suggest different ways for studying seismic catalogs. Finally, we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock, stressing the caveats and limitations of previous empirical works.

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PubMed2002

Occurrence of finite-time singularities in epidemic models of rupture, earthquakes, and starquakes.

We present a new kind of critical stochastic finite-time singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star, as well as of earthquake sequences.

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PubMed2002

Classification of possible finite-time singularities by functional renormalization.

Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable phi(t) as a function of the time variable in some interval 0 < or = t < or = T, we investigate how to extrapolate/forecast in some optimal stability sense the future evolution of phi(t) for time t>T. Using the functional renormalization of Yukalov and Gluzman, we offer a general classification of the possible regimes that can be defined based on the sole knowledge of the coefficients of a second-order polynomial representation of the dynamics. In particular, we investigate the conditions for the occurrence of finite-time singularities from the structure of the time series, and quantify the critical time and the functional nature of the singularity when present. We also describe the regimes when a smooth extremum replaces the singularity and determine its position and amplitude. This extends previous works by (1) quantifying the stability of the functional renormalization method more accurately, (2) introducing more global constraints in terms of moments, and (3) going beyond the "mean-field" approximation.

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