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Abstract We consider line failure cascading in power networks where an initial random failure of a few lines leads to consecutive other line overloads and failures before the system settles in a steady state. Such cascades are rooted in non-obvious, long-range, and higher-order couplings among the lines’ flows induced by physical constraints on the network. Failure interaction graph encodes which and to what extent other lines in a networked system are affected after each line failure and can help to predict the final state after an initial disturbance. We perform data analytics on the final lines’ steady states of cascade trajectories to infer a specific line’s state given the states of others. We use a generative model to reconstruct possible steady states, and a predictive model aims to predict the probability of each line’s failures after the initial failure as a regression problem. The generative model uses regularized pseudolikelihood estimator to infer interaction weights by solving the inverse Ising problem and deploys Glauber dynamics to generate steady states. The discriminative model uses boosted trees to efficiently learn over training and predict over test data the state of each line as a target finding an appropriate subset of other lines’ states as explanatory variables. We analyze the degree distribution of the corresponding interaction graphs to study the number of other components affected by each line failure (out-degree) or the number of lines that affect the state of a given line (in-degree). Both models show that the in-degree follows a power-law distribution. Finally, we discuss the possible application of the interaction graph for early link removal to mitigate the failure-cascading consequences.

Abstract In the present work, we model the diffusion of scientific knowledge embodied in the textbooks of the Sphaera corpus. This corpus consists of more than 350 different editions of textbooks used for teaching astronomy in European universities during the early modern period. Connections between the editions are based on mutual semantic knowledge and are arranged in a multiplex network of four layers, with each layer representing a different semantic relation. The modeling aims for a better understanding (and possible prediction) of the process of knowledge accumulation in the various editions. We consider semantic text-parts as knowledge units transferred between the editions, and show how these units spread using both an SI model and its modified version, the Bass model. Both models include a parameter representing the rate of transfer, which is interpreted as the mechanism underlying the process of knowledge accumulation; the Bass model has an extra parameter that represents the rate of external influence and stands out as the “resistance to adoption from” and “resistance to be influenced by” other knowledge systems. The modeling has helped us to chart the path and mechanisms of knowledge transformation in the early modern period. Networks are identified by adding further layers whose graphs express socioeconomic relationships and conditional sub-networks. The comparison between the model and these layers enables us to conclude that the accumulation of knowledge was highly dependent on the institutional embedding of scientific production because the diffusion of knowledge was mostly determined by the economic constraints of early modern printers and publishers. It further suggests that geographic proximity played a role—although secondary—in the diffusion of knowledge but only under the condition that the book producers involved were still living. The transformation of early modern scientific knowledge is, therefore, highly dependent on the institutional and economic contexts of the book producers.

There is an increasing interest in determining if there exist observable patterns or structures within the digits of irrational numbers. We extend this search by investigating the interval in position between two consecutive occurrences of the same digit, a kind of waiting time statistics. We characterise these by the burstiness measure which distinguishes if the inter-event times are periodic, bursty, or Poisson processes. Furthermore, the complexity–entropy plane was used to determine if the intervals are stochastic or chaotic. We analyse sequences of the first 1 million digits of the numbers π, e, 2, and ϕ. We find that the intervals between single, double, and triple digits are Poisson processes with a burstiness measure in the range −0.05≤B≤0.05 for the four numbers studied. This result is supported by a complexity–entropy plane analysis, which shows that the time intervals have the same characteristics as Gaussian noise. The four irrational numbers have identical degrees of complexity and burstiness in their inter-event analysis.

In modern power grids, knowing the required electric power demand and its variations is necessary to balance demand and supply. The authors propose a data-driven approach to create high-resolution load profiles and characterize their fluctuations, based on recorded data of electricity consumption.

The energy system is rapidly changing to accommodate the increasing number of renewable generators and the general transition towards a more sustainable future. Simultaneously, business models and market designs evolve, affecting power-grid operation and power-grid frequency. Problems raised by this ongoing transition are increasingly addressed by transdisciplinary research approaches, ranging from purely mathematical modelling to applied case studies. These approaches require a stochastic description of consumer behaviour, fluctuations by renewables, market rules, and how they influence the stability of the power-grid frequency. Here, we introduce an easy-to-use, data-driven, stochastic model for the power-grid frequency and demonstrate how it reproduces key characteristics of the observed statistics of the Continental European and British power grids. Using data analysis tools and a Fokker-Planck approach, we estimate parameters of our deterministic and stochastic model. We offer executable code and guidelines on how to use the model on any power grid for various mathematical or engineering applications.

We present a method to infer the arbitrary space-dependent drift and diffusion of a nonlinear stochastic model driven by multiplicative fractional Gaussian noise from a single trajectory. Our method, fractional Onsager-Machlup optimisation (fOMo), introduces a maximum likelihood estimator by minimising a field-theoretic action which we construct from the observed time series. We successfully test fOMo for a wide range of Hurst exponents using artificial data with strong nonlinearities, and apply it to a data set of daily mean temperatures. We further highlight the significant systematic estimation errors when ignoring non-Markovianity, underlining the need for nonlinear fractional inference methods when studying real-world long-range (anti-)correlated systems.

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with β≤0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α→1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices α close to unity.

We introduce a method for reconstructing macroscopic models of one-dimensional stochastic processes with long-range correlations from sparsely sampled time series by combining fractional calculus and discrete-time Langevin equations. The method is illustrated for the ARFIMA(1,d,0) process and a nonlinear autoregressive toy model with multiplicative noise. We reconstruct a model for daily mean temperature data recorded at Potsdam, Germany and use it to predict the first-frost date by computing the mean first passage time of the reconstructed process and the 0^{∘}C temperature line, illustrating the potential of long-memory models for predictions in the subseasonal-to-seasonal range.

We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 10 ^17 steps and thereby also study finite-time crossover phenomena.

The present work investigates the process of emergence of new epistemic communities. The research is based on semantic, content-related data extracted from a corpus of 359 printed editions, mainly of textbooks used to teach cosmology at European universities between 1472 and 1650. Epistemic communities are identified as families of editions, grouped according to their content, that eventually came to shape knowledge within and by way of the European educational framework. First, a method of classifying the textual content of the books is introduced. Second, a directed, multiplex network is constructed in five layers whose tructures are defined specifically for the research question at hand. Then the network is analyzed, first by making use of the aggregated graph—which accounts for the connectivity between books when any of the potential semantic relations are indistinctly considered—and second by showing the contribution of each layer to the emergence of new families of editions. Finally, we interpret the results within a historical framework and identify an epistemic community that represents continuity with the medieval tradition, plus two new scientific and diverging communities that originated in the cultural context of the Reformed countries, which appear in the 1530s. The characteristics of the identified epistemic communities are further analyzed in order to draw general inferences concerning mechanisms of emergence of epistemic communities and their identification in corpora of historical sources. The work concludes by describing future research endeavors related to the corpus, also based on new series of data.

Using the Feynman-Kac formula, a work fluctuation theorem for a Brownian particle in a nonconfining potential, e.g., a potential well with finite depth, is derived. The theorem yields an inequality that puts a lower bound on the average work needed to change the potential in time. In comparison to the Jarzynski equality, which holds for confining potentials, an additional term describing a form of energy related to the never-ending diffusive expansion appears.

Extensive time-series encoding the position of particles such as viruses, vesicles, or individual proteins are routinely garnered in single-particle tracking experiments or supercomputing studies. They contain vital clues on how viruses spread or drugs may be delivered in biological cells. Similar time-series are being recorded of stock values in financial markets and of climate data. Such time-series are most typically evaluated in terms of time-averaged mean-squared displacements (TAMSDs), which remain random variables for finite measurement times. Their statistical properties are different for different physical stochastic processes, thus allowing us to extract valuable information on the stochastic process itself. To exploit the full potential of the statistical information encoded in measured time-series we here propose an easy-to-implement and computationally inexpensive new methodology, based on deviations of the TAMSD from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion (BM) to check the applicability of this approach to simulated and real data sets. By comparing the probability of deviations for different data sets, we demonstrate how the theoretical bound for BM reveals additional information about observed stochastic processes. We apply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracer beads measured in mucin hydrogels, and of geographic surface temperature anomalies. Our analysis shows how the large-deviation properties can be efficiently used as a simple yet effective routine test to reject the BM hypothesis and unveil relevant information on statistical properties such as ergodicity breaking and short-time correlations.

We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics. The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the ‘Joseph effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), fat-tails of the increment probability density lead to a ‘Noah effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), and non-stationarity, to the ‘Moses effect’ (Chen et al 2017 Phys. Rev. E 95 042141). After appropriate rescaling, based on the quantification of these effects, the increment distribution converges at increasing times to a time-invariant asymptotic shape. For different processes, this asymptotic limit can be an equilibrium state, an infinite-invariant, or an infinite-covariant density. We use numerical methods of time-series analysis to quantify the three effects in a model of a non-linearly coupled Lévy walk, compare our results to theoretical predictions, and discuss the generality of the method.

We discuss the motion of substance in a channel containing nodes of a network. Each node of the channel can exchange substance with: (i) neighboring nodes of the channel, (ii) network nodes which do not belong to the channel, and (iii) environment of the network. The new point in this study is that we assume possibility for exchange of substance among flows of substance between nodes of the channel and: (i) nodes that belong to the network but do not belong to the channel and (ii) environment of the network. This leads to an extension of the model of motion of substance and the extended model contains previous models as particular cases. We use a discrete-time model of motion of substance and consider a stationary regime of motion of substance in a channel containing a finite number of nodes. As results of the study, we obtain a class of probability distributions connected to the amount of substance in nodes of the channel. We prove that the obtained class of distributions contains all truncated discrete probability distributions of discrete random variable ω which can take values 0,1,⋯,N. Theory for the case of a channel containing infinite number of nodes is presented in Appendix A. The continuous version of the discussed discrete probability distributions is described in Appendix B. The discussed extended model and obtained results can be used for the study of phenomena that can be modeled by flows in networks: motion of resources, traffic flows, motion of migrants, etc.

The frequency constitutes a key state variable of electrical power grids. However, as the frequency is subject to several sources of fluctuations, ranging from renewable volatility to demand fluctuations and dispatch, it is strongly dynamic. Yet, the statistical and stochastic properties of the frequency fluctuation dynamics are far from fully understood. Here we analyze properties of power-grid frequency trajectories recorded from different synchronous regions. We highlight the non-Gaussian and still approximately Markovian nature of the frequency statistics. Furthermore, we find that the frequency displays significant fluctuations exactly at the time intervals of regulation and trading, confirming the need of having a regulatory and market design that respects the technical and dynamical constraints in future highly renewable power grids. Finally, employing a recently proposed synthetic model for the frequency dynamics, we combine our statistical and stochastic analysis and analyze in how far dynamically modeled frequency properties match the ones of real trajectories.

The onset of spring is crucial for planning agricultural practices and has influences on animal activities as well. It is no doubt beneficial to properly forecast spring onset in advance. In this work, we present a new method based on harmonic analysis of daily mean temperature to predict the date of spring onset in the next year. The algorithm of the proposed method considers the memory of the seasonal cycle and can be easily conducted using the local past records. This study is based on gridded observational surface air temperature (SAT) in Europe. The SAT data are first decomposed into harmonics by wavelet transform and each harmonic’s time-dependent amplitude and phase are extracted. Then the time evolution for amplitude and phase are modelled by an AR(2) process, where we employ a new method to fit its coefficients from the data. This provides a prediction of the time dependent amplitudes and phases in the next year, from which we compose the future seasonal cycle and identify the prediction of spring onset by threshold crossing. We compare our model with a classical climatological seasonal cycle forecast. While this benchmark forecast leads to only very small variations of the predicted onset date, our method yields predictions which vary by about 15 days from year to year. We verify the correctness of our predictions by a correlation measure and the root mean squared errors.

Exploiting recent progress in the theoretical understanding of detrended fluctuation analysis (DFA), we use the non-asymptotic properties of the fluctuation function in order to extract more information from time series data than just its Hurst exponent. In particular, we can identify exponential relaxation and oscillation periods and estimate their specific values. We illustrate the strength of this method through applications to climate data. Thereby, we determine the relaxation time of the atmospheric response to perturbations. We also find by DFA a period length of the dominant frequency mode of the El Niño Southern Oscillation to be 3.3 years.

We propose a dynamical mechanism for a strictly finite prediction horizon, i.e. a scenatio of chaotic motion where asymptotically a more precise knowledge of the initial condition does note translate into a longer closeness of the forecast to the truth. For this, we propose a class of hierarchical dynamical systems which possess a scale dependent error growth rate in the form of a power law. Actually, this is motivated by and consistent with well known hierarchies of patterns in atmospheric dynamics. This scale dependent error growth rate in form of a power law translates in power law error growth over time instead of exponential error growth as in conventional chaotic systems. The consequence is a strictly finite prediction horizon, since in the limit of infinitesimal errors of initial conditions, the error growth rate diverges and hence additional accuracy is not translated into longer prediction times. By re-analyzing data of the National Center for Environmental Protect Global Forecast System, a weather prediction model, published by Harlim et al (2005 Phys. Rev. Lett. 94 228501) we show that such a power law error growth rate can indeed be found in numerical weather forecast models and estimate it average maximal prediction horizon to about 15 d.

We decompose the anomalous diffusive behavior found in an aging system into its fundamental constitutive causes. The model process is a sum of increments that are iterates of a chaotic dynamical system, the Pomeau–Manneville map. The increments can have long-time correlations, fat-tailed distributions and be non-stationary. Each of these properties can cause anomalous diffusion through what is known as the Joseph, Noah and Moses effects, respectively. The model can have either sub- or super-diffusive behavior, which we find is generally due to a combination of the three effects. Scaling exponents quantifying each of the three constitutive effects are calculated using analytic methods and confirmed with numerical simulations. They are then related to the scaling of the distribution of the process through a scaling relation. Finally, the importance of the Moses effect in the anomalous diffusion of experimental systems is discussed.