Networks of recurrent events, a theory of records, and an application to finding causal signatures in seismicity

Estimating the Epicenter of an Impending Strong Earthquake by Combining the Seismicity Order Parameter Variability Analysis with Earthquake Networks and Nowcasting: Application in the Eastern Mediterranean

The variance κ1 of the natural time analysis of earthquake catalogs was proposed in 2005 as an order parameter for seismicity, whose fluctuations proved, in 2011, to be minimized a few months before the strongest mainshock when studying the earthquakes in a given area. After the introduction of earthquake networks based on similar activity patterns, in 2012, the study of their higher order cores revealed, in 2019, the selection of appropriate areas in which the precursory minima βmin of the fluctuations β of the seismicity order parameter κ1 could be observed up to six months before all strong earthquakes above a certain threshold. The eastern Mediterranean region was studied in 2019, where all earthquakes of magnitude M≥7.1 were found to be preceded by βmin without any false alarm. Combining these results with the method of nowcasting earthquakes, introduced in 2016, for seismic risk estimation, here, we show that the epicenter of an impending strong earthquake can be estimated. This is achieved by employing—at the time of observing the βmin—nowcasting earthquakes in a square lattice grid in the study area and by averaging, self-consistently, the results obtained for the earthquake potential score. This is understood in the following context: The minimum βmin is ascertained to almost coincide with the onset of Seismic Electric Signals activity, which is accompanied by the development of long range correlations between earthquake magnitudes in the area that is a candidate for a mainshock.

Acoustic Emission Spectroscopy: Applications in Geomaterials and Related Materials

As a non-destructive testing technology with fast response and high resolution, acoustic emission is widely used in material monitoring. The material deforms under stress and releases elastic waves. The wave signals are received by piezoelectric sensors and converted into electrical signals for rapid storage and analysis. Although the acoustic emission signal is not the original stress signal inside the material, the typical statistical distributions of acoustic emission energy and waiting time between signals are not affected by signal conversion. In this review, we first introduce acoustic emission technology and its main parameters. Then, the relationship between the exponents of power law distributed AE signals and material failure state is reviewed. The change of distribution exponent reflects the transition of the material’s internal failure from a random and uncorrelated state to an interrelated state, and this change can act as an early warning of material failure. The failure process of materials is often not a single mechanism, and the interaction of multiple mechanisms can be reflected in the probability density distribution of the AE energy. A large number of examples, including acoustic emission analysis of biocemented geological materials, hydroxyapatite (human teeth), sandstone creep, granite, and sugar lumps are introduced. Finally, some supplementary discussions are made on the applicability of Båth’s law.

Record statistics of bursts signals the onset of acceleration towards failure

Forecasting the imminent catastrophic failure has a high importance for a large variety of systems from the collapse of engineering constructions, through the emergence of landslides and earthquakes, to volcanic eruptions. Failure forecast methods predict the lifetime of the system based on the time-to-failure power law of observables describing the final acceleration towards failure. We show that the statistics of records of the event series of breaking bursts, accompanying the failure process, provides a powerful tool to detect the onset of acceleration, as an early warning of the impending catastrophe. We focus on the fracture of heterogeneous materials using a fiber bundle model, which exhibits transitions between perfectly brittle, quasi-brittle, and ductile behaviors as the amount of disorder is increased. Analyzing the lifetime of record size bursts, we demonstrate that the acceleration starts at a characteristic record rank, below which record breaking slows down due to the dominance of disorder in fracturing, while above it stress redistribution gives rise to an enhanced triggering of bursts and acceleration of the dynamics. The emergence of this signal depends on the degree of disorder making both highly brittle fracture of low disorder materials, and ductile fracture of strongly disordered ones, unpredictable.

Markov modeling via ordinal partitions: An alternative paradigm for network-based time-series analysis

Mapping time series to complex networks to analyze observables has recently become popular, both at the theoretical and the practitioner's level. The intent is to use network metrics to characterize the dynamics of the underlying system. Applications cover a wide range of problems, from geoscientific measurements to biomedical data and financial time series. It has been observed that different dynamics can produce networks with distinct topological characteristics under a variety of time-series-to-network transforms that have been proposed in the literature. The direct connection, however, remains unclear. Here, we investigate a network transform based on computing statistics of ordinal permutations in short subsequences of the time series, the so-called ordinal partition network. We propose a Markovian framework that allows the interpretation of the network using ergodic-theoretic ideas and demonstrate, via numerical experiments on an ensemble of time series, that this viewpoint renders this technique especially well-suited to nonlinear chaotic signals. The aim is to test the mapping's faithfulness as a representation of the dynamics and the extent to which it retains information from the input data. First, we show that generating networks by counting patterns of increasing length is essentially a mechanism for approximating the analog of the Perron-Frobenius operator in a topologically equivalent higher-dimensional space to the original state space. Then, we illustrate a connection between the connectivity patterns of the networks generated by this mapping and indicators of dynamics such as the hierarchy of unstable periodic orbits embedded within a chaotic attractor. The input is a scalar observable and any projection of a multidimensional flow suffices for reconstruction of the essential dynamics. Additionally, we create a detailed guide for parameter tuning. We argue that there is no optimal value of the pattern length m, rather it admits a scaling region akin to traditional embedding practice. In contrast, the embedding lag and overlap between successive patterns can be chosen exactly in an optimal way. Our analysis illustrates the potential of this transform as a complementary toolkit to traditional time-series methods.

PageRank: An alarming index of probable earthquake occurrence

Here, we introduce PageRank (PR) in a seismic network as an appropriate alarming clue before the occurrence of the event to be worthwhile in hazard probabilistic evaluation of earthquakes. Studying PR changes of two main shocks in Iran and Italy by means of temporal and spatial windows reveals that their PR values increase drastically before the event, while there is no trend for other considered locations and/or other time intervals. Therefore, the PR value seems to be an appropriate index of a place induction by previous events and its susceptibility for having a new earthquake. Moreover, summing over the PRs of areas close to the Italy event location and tracking this newly defined PR behavior show an increasing trend before the main shock implying that the close regions are influenced and become highly connected before the event as well as the earthquake location itself. It is also indicated that PR behavior is not necessarily correlated to the number of occurring earthquakes and is inherently the result of points connectivity and interactions.

Exactly Solvable Record Model for Rainfall

Daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days. Assuming that wet days follow a Bernoulli process with success probability p, we show that the presence of dry days induces negative correlations between record-breaking precipitation events. The resulting nonmonotonic behavior of the Fano factor of the record counting process is recovered in empirical data. We derive the full probability distribution P(R,n) of the number of records R_{n} up to time n, and show that for large n, it converges to a Poisson distribution with parameter ln(pn). We also study in detail the joint limit p→0, n→∞, which yields a random record model in continuous time t=pn.

Record-breaking statistics near second-order phase transitions

When a quantity reaches a value higher (or lower) than its value at any time before, it is said to have made a record. We numerically study the statistical properties of records in the time series of order parameters in different models near their critical points. Specifically, we choose the transversely driven Edwards-Wilkinson model for interface depinning in (1+1) dimensions and the Ising model in two dimensions, as paradigmatic and simple examples of nonequilibrium and equilibrium critical behaviors, respectively. The total number of record-breaking events in the time series of the order parameters of the models show maxima when the system is near criticality. The number of record-breaking events and associated quantities, such as the distribution of the waiting time between successive record events, show power-law scaling near the critical point. The exponent values are specific to the universality classes of the respective models. Such behaviors near criticality can be used as a precursor to imminent criticality, i.e., abrupt and catastrophic changes in the system. Due to the extreme nature of the records, its measurements are relatively free of detection errors and thus provide a clear signal regarding the state of the system in which they are measured.

Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors

The quest for predictive indicators for the collapse of coal mines has led to a robust criterion from scale-model tests in the laboratory. Mechanical collapse under uniaxial stress forms avalanches with a power-law probability distribution function of radiated energy P∼E^{-}^{ɛ}, with exponent ɛ=1.5. Impending major collapse is preceded by a reduction of the energy exponent to the mean-field value ɛ=1.32. Concurrently, the crackling noise increases in intensity and the waiting time between avalanches is reduced when the major collapse is approaching. These latter criteria were so-far deemed too unreliable for safety assessments in coal mines. We report a reassessment of previously collected extensive collapse data sets using "record-breaking analysis," based on the statistical appearance of "superjerks" within a smaller spectrum of collapse events. Superjerks are defined as avalanche signals with energies that surpass those of all previous events. The final major collapse is one such superjerk but other "near collapse" events equally qualify. In this way a very large data set of events is reduced to a sparse sequence of superjerks (21 in our coal sample). The main collapse can be anticipated from the sequence of energies and waiting times of superjerks, ignoring all weaker events. Superjerks are excellent indicators for the temporal evolution, and reveal clear nonstationarity of the crackling noise at constant loading rate, as well as self-similarity in the energy distribution of superjerks as a function of the number of events so far in the sequence E_{sj}∼n^{δ} with δ=1.79. They are less robust in identifying the precise time of the final collapse, however, than the shift of the energy exponents in the whole data set which occurs only over a short time interval just before the major event. Nevertheless, they provide additional diagnostics that may increase the reliability of such forecasts.

Records in fractal stochastic processes

The record statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions, the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes; otherwise, it has no impact on the record statistics.

Record-breaking events during the compressive failure of porous materials

An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uniaxial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. The distribution of the size and lifetime of records are power laws with relatively low exponents. We demonstrate the existence of a characteristic record rank k(*), which separates the two regimes of the time evolution. Up to this rank deceleration occurs due to the effect of random disorder. Record breaking then accelerates towards macroscopic failure, when physical interactions leading to spatial and temporal correlations dominate the location and timing of local ruptures. The size distribution of records of different ranks has a universal form independent of the record rank. Subsequences of events that occur between consecutive records are characterized by a power-law size distribution, with an exponent which decreases as failure is approached. High-rank records are preceded by smaller events of increasing size and waiting time between consecutive events and they are followed by a relaxation process. As a reference, surrogate time series are generated by reshuffling the event times. The record statistics of the uncorrelated surrogates agrees very well with the corresponding predictions of independent identically distributed random variables, which confirms that temporal and spatial correlation in the crackling noise is responsible for the observed unique behavior. In principle the results could be used to improve forecasting of catastrophic failure events, if they can be observed reliably in real time.

Disentangling different types of El Niño episodes by evolving climate network analysis

Complex network theory provides a powerful toolbox for studying the structure of statistical interrelationships between multiple time series in various scientific disciplines. In this work, we apply the recently proposed climate network approach for characterizing the evolving correlation structure of the Earth's climate system based on reanalysis data for surface air temperatures. We provide a detailed study of the temporal variability of several global climate network characteristics. Based on a simple conceptual view of red climate networks (i.e., networks with a comparably low number of edges), we give a thorough interpretation of our evolving climate network characteristics, which allows a functional discrimination between recently recognized different types of El Niño episodes. Our analysis provides deep insights into the Earth's climate system, particularly its global response to strong volcanic eruptions and large-scale impacts of different phases of the El Niño Southern Oscillation.

Record-breaking avalanches in driven threshold systems

Record-breaking avalanches generated by the dynamics of several driven nonlinear threshold models are studied. Such systems are characterized by intermittent behavior, where a slow buildup of energy is punctuated by an abrupt release of energy through avalanche events, which usually follow scale-invariant statistics. From the simulations of these systems it is possible to extract sequences of record-breaking avalanches, where each subsequent record-breaking event is larger in magnitude than all previous events. In the present work, several cellular automata are analyzed, among them the sandpile model, the Manna model, the Olami-Feder-Christensen (OFC) model, and the forest-fire model to investigate the record-breaking statistics of model avalanches that exhibit temporal and spatial correlations. Several statistical measures of record-breaking events are derived analytically and confirmed through numerical simulations. The statistics of record-breaking avalanches for the four models are compared to those of record-breaking events extracted from the sequences of independent and identically distributed (i.i.d.) random variables. It is found that the statistics of record-breaking avalanches for the above cellular automata exhibit behavior different from that observed for i.i.d. random variables, which in turn can be used to characterize complex spatiotemporal dynamics. The most pronounced deviations are observed in the case of the OFC model with a strong dependence on the conservation parameter of the model. This indicates that avalanches in the OFC model are not independent and exhibit spatiotemporal correlations.

Earthquake networks based on similar activity patterns

Earthquakes are a complex spatiotemporal phenomenon, the underlying mechanism for which is still not fully understood despite decades of research and analysis. We propose and develop a network approach to earthquake events. In this network, a node represents a spatial location while a link between two nodes represents similar activity patterns in the two different locations. The strength of a link is proportional to the strength of the cross correlation in activities of two nodes joined by the link. We apply our network approach to a Japanese earthquake catalog spanning the 14-year period 1985-1998. We find strong links representing large correlations between patterns in locations separated by more than 1000 kilometers, corroborating prior observations that earthquake interactions have no characteristic length scale. We find network characteristics not attributable to chance alone, including a large number of network links, high node assortativity, and strong stability over time.

No evidence of magnitude clustering in an aftershock sequence of nano- and picoseismicity

One of the hallmarks of our current understanding of seismicity as highlighted by the epidemic-type-aftershock sequence model is that the magnitudes of earthquakes are independent of one another and can be considered as randomly drawn from the Gutenberg-Richter distribution. This assumption forms the basis of many approaches for forecasting seismicity rates and hazard assessment. Recently, it has been suggested that the assumption of independent magnitudes is not valid. It was subsequently argued that this conclusion was not supported by the original earthquake data from California. One of the main challenges is the lack of completeness of earthquake catalogs. Here, we study an aftershock sequence of nano- and picoseismicity as observed at the Mponeng mine, for which the issue of incompleteness is much less pronounced. We show that this sequence does not exhibit any significant evidence of magnitude correlations.

Earthquake correlations and networks: a comparative study

We quantify the correlation between earthquakes and use the same to extract causally connected earthquake pairs. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski [M. Baiesi and M. Paczuski, Phys. Rev. E 69, 066106 (2004)]. A network of earthquakes is then constructed from the time-ordered catalog and with links between the more correlated ones. A list of recurrences to each of the earthquakes is identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. Data pertaining to three different seismic regions (viz., California, Japan, and the Himalayas) are comparatively analyzed using such a network model. The distribution of recurrence lengths and recurrence times are two of the key features analyzed to draw conclusions about the universal aspects of such a network model. We find that the unimodal feature of recurrence length distribution, which helps to associate typical rupture lengths with different magnitude earthquakes, is robust across the different seismic regions. The out-degree of the networks shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws, with two regimes having different exponents, are obtained with recurrence time distribution. The first regime confirms the Omori law for aftershocks while the second regime, with a faster falloff for the larger recurrence times, establishes that pure spatial recurrences also follow a power-law distribution. The crossover to the second power-law regime can be taken to be signaling the end of the aftershock regime in an objective fashion.

Nonextensivity and natural time: The case of seismicity

Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.

Edge direction and the structure of networks

Directed networks are ubiquitous and are necessary to represent complex systems with asymmetric interactions--from food webs to the World Wide Web. Despite the importance of edge direction for detecting local and community structure, it has been disregarded in studying a basic type of global diversity in networks: the tendency of nodes with similar numbers of edges to connect. This tendency, called assortativity, affects crucial structural and dynamic properties of real-world networks, such as error tolerance or epidemic spreading. Here we demonstrate that edge direction has profound effects on assortativity. We define a set of four directed assortativity measures and assign statistical significance by comparison to randomized networks. We apply these measures to three network classes--online/social networks, food webs, and word-adjacency networks. Our measures (i) reveal patterns common to each class, (ii) separate networks that have been previously classified together, and (iii) expose limitations of several existing theoretical models. We reject the standard classification of directed networks as purely assortative or disassortative. Many display a class-specific mixture, likely reflecting functional or historical constraints, contingencies, and forces guiding the system's evolution.

Ambiguities in recurrence-based complex network representations of time series

Recently, different approaches have been proposed for studying basic properties of time series from a complex network perspective. In this work, the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail. We discuss the main requirements that permit a feasible system-theoretic interpretation of network topology in terms of dynamically invariant phase-space properties. Possible artifacts induced by disregarding these requirements are pointed out and systematically studied. Finally, a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed.

Network of recurrent events for the Olami-Feder-Christensen model

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder, and Christensen (OFC) to mimic earthquakes and investigate to what extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicity. Following a recently proposed method to characterize such clustering by networks of recurrent events [J. Davidsen, P. Grassberger, and M. Paczuski, Geophys. Res. Lett. 33, L11304 (2006)], we find that for synthetic catalogs generated by the OFC model these networks have many nontrivial statistical properties. This includes characteristic degree distributions, very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs, indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity.