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We consider trends in the m seasonal subrecords of a record. To determine the statistical significance of the m trends, one usually determines the p value of each season either numerically or analytically and compares it with a significance level $${{\tilde{\alpha }}}$$ α ~ . We show in great detail for short- and long-term persistent records that this procedure, which is standard in climate science, is inadequate since it produces too many false positives (false discoveries). We specify, on the basis of the family wise error rate and by adapting ideas from multiple testing correction approaches, how the procedure must be changed to obtain more suitable significance criteria for the m trends. Our analysis is valid for data with all kinds of persistence. Specifically for long-term persistent data, we derive simple analytical expressions for the quantities of interest, which allow to determine easily the statistical significance of a trend in a seasonal record. As an application, we focus on 17 Antarctic station data. We show that only four trends in the seasonal temperature data are outside the bounds of natural variability, in marked contrast to earlier conclusions.

Tree-ring chronologies are the main source for annually resolved and absolutely dated temperature reconstructions of the last millennia and thus for studying the intriguing problem of climate impacts. Here we focus on central Europe and compare the tree-ring based temperature reconstruction with reconstructions from harvest dates, long meteorological measurements, and historical model data. We find that all data are long-term persistent, but in the tree-ring based reconstruction the strength of the persistence quantified by the Hurst exponent is remarkably larger ($$h\cong 1.02$$ h ≅ 1.02 ) than in the other data ($$h=$$ h = 0.52–0.69), indicating an unrealistic exaggeration of the historical temperature variations.We show how to correct the tree-ring based reconstruction by a mathematical transformation that adjusts the persistence and leads to reduced amplitudes of the warm and cold periods. The new transformed record agrees well with both the observational data and the harvest dates-based reconstructions and allows more realistic studies of climate impacts. It confirms that the present warming is unprecedented.

The El Niño Southern Oscillation (ENSO) is the most important driver of interannual global climate variability and affects weather and climate in large parts of the world. Recently, we have developed a dynamical network approach for predicting the onset of El Niño events well before the spring predictability barrier. In the regarded climate network, the nodes are grid points in the Pacific, and the strengths of the links (teleconnections) between them are characterized by the cross-correlations of the atmospheric surface temperatures at the grid points. In the year before an El Niño event, the links between the eastern equatorial Pacific and the rest of the Pacific tend to strengthen such that the average link strength exceeds a certain threshold. This feature can be used to predict the onset of an El Niño with 73% probability and its absence with 90% probability. The p-value of the hindcasting and forecasting phase (1981-2021) for this performance based on random guessing with the climatological average is 4.6*10-5.To assess whether this predictive feature is also present in coupled general circulation models, we apply our algorithm to historical and control runs of CMIP5 and CMIP6. We find that the predictive performance present in observational data is absent or very low in GCMs. The lack of this feature may explain the difficulties of GCMs to overcome the spring barrier.

In the past decades, the Antarctic sea ice extent (SIE) has been steadily increasing, but recently showed a sharp decline. Here we address the questions whether (1) the observed changes in the Antarctic SIE can be fully explained by natural variability and (2) whether the recent unprecedented decline in the SIE can serve as an indication that the long-term positive trend has reached a turning point entailing further decline. To study these questions, we extended the analysis period of previous studies (until 2013) by considering data until May 2018 and applied a statistical model which accurately reflects the natural variability of the SIE. Contrary to earlier detection studies we find that none of the annual trends of the SIE in whole Antarctica and its five sectors are statistically significant. When studying the seasonal changes, we find that the only trends in the Antarctic SIE that cannot be explained by natural variability and are probably tied to the warming of the Antarctic Peninsula, are the negative trends of the SIE in austral autumn ( $$p=0.043$$ ) and February ( $$p=0.012$$ ) in the Bellinghausen and Amundsen Seas (BellAm). In contrast, when the recent decline is omitted from the analysis and only data until 2015 are included, the (annual and seasonal) increases of the SIE in whole Antarctica and the Ross Sea become significant, while the significance of the decreasing trends in BellAm is slightly decreased. We consider this as a first indication that the Antarctic SIE may have reached a turning point towards a further decrease.

The El Niño Southern Oscillation (ENSO) is one of the most prominent interannual climate phenomena. An early and reliable ENSO forecasting remains a crucial goal, due to its serious implications for economy, society, and ecosystem. Despite the development of various dynamical and statistical prediction models in the recent decades, the “spring predictability barrier” (SPB) remains a great challenge for long (over 6-month) lead-time forecasting. To overcome this barrier, here we develop an analysis tool, the System Sample Entropy (SysSampEn), to measure the complexity (disorder) of the system composed of temperature anomaly time series in the Niño 3.4 region. When applying this tool to several near surface air temperature and sea surface temperature datasets, we find that in all datasets a strong positive correlation exists between the magnitude of El Niño and the previous calendar year’s SysSampEn (complexity). We show that this correlation allows to forecast the magnitude of an El Niño with a prediction horizon of 1 year and high accuracy (i.e., Root Mean Square Error = 0.23°C for the average of the individual datasets forecasts). For the 2018 El Niño event, our method forecasts a weak El Niño with a magnitude of 1.11±0.23°C. Our framework presented here not only facilitates a long–term forecasting of the El Niño magnitude but can potentially also be used as a measure for the complexity of other natural or engineering complex systems.

The El Nino Southern Oscillation (ENSO) is one of the most prominent interannual climate phenomena. An early and reliable ENSO forecasting remains a crucial goal, due to its serious implications for economy, society, and ecosystem. Despite the development of various dynamical and statistical prediction models in the recent decades, the ``spring predictability barrier'' (SPB) remains a great challenge for long (over 6-month) lead-time forecasting. To overcome this barrier, here we develop an analysis tool, the System Sample Entropy (SysSampEn), to measure the complexity (disorder) of the system composed of temperature anomaly time series in the Nino 3.4 region. When applying this tool to several near surface air-temperature and sea surface temperature datasets, we find that in all datasets a strong positive correlation exists between the magnitude of El Nino and the previous calendar year's SysSampEn (complexity). We show that this correlation allows to forecast the magnitude of an El Nino with a prediction horizon of 1 year and high accuracy (i.e., Root Mean Square Error $=0.23^\circ C$ for the average of the individual datasets forecasts). For the on-going 2018 El Nino event, our method forecasts a weak El Nino with a magnitude of $1.11\pm 0.23^\circ C$. Our framework presented here not only facilitates a long--term forecasting of the El Nino magnitude but can potentially also be used as a measure for the complexity of other natural or engineering complex systems.

The rate of mass transfer is among the key numbers determining the efficiency of nanoporous materials in their use for matter upgrading by heterogeneous catalysis or mass separation. Transport enhancement by pore space optimization is, correspondingly, among the main strategies of efficiency promotion. Any such activity involves probing and testing of the appropriate routes of material synthesis and post-synthesis modification just as the exploration of the transport characteristics of the generated material. Modelling and molecular simulation is known to serve as a most helpful tool for correlating these two types of activities and their results. The present paper reports about a concerted research activity comprising these three types of activities. Recent progress in producing pore space replicas enabled focusing, in these studies, on “complementary” pore spaces, i.e. on pairs of material, where the pore space of one sample did just coincide with the solid space of the other. We report about the correlations in mass transfer as observable, in this type of material, by pulsed field gradient NMR diffusion studies, with reference to the prediction as resulting from a quite general, theoretical treatment of mass transfer in complementary pore spaces.

We present the toolbox ClimateLearn to tackle problems in climate prediction using machine learning techniques and climate network analysis. The package allows basic operations of data mining, i.e. reading, merging, and cleaning data, and running machine learning algorithms such as multilayer artificial neural networks and symbolic regression with genetic programming. Because spatial temporal information on climate variability can be efficiently represented by complex network measures, such data are considered here as input to the machine-learning algorithms. As an example, the toolbox is applied to the prediction of the occurrence and the development of El Niño in the equatorial Pacific, first concentrating on the occurrence of El Niño events one year ahead and second on the evolution of sea surface temperature anomalies with a lead time of three months.

In addition to the respective pore space geometries, transport enhancement in hierarchical nanoporous materials may most decisively depend on the nature of the probe molecule under consideration. As a function of these two influences, mass transfer in hierarchical materials may follow quite different scenarios. Considering mass transfer enhancement by a network of mesopores within a microporous continuum, this work presents an approach to assess these influences quantitatively. The validity of this approach is tested by dynamic Monte Carlo simulations and by comparison with the results of diffusion measurements.

Previous estimates of the strength and the uncertainty of the observed Antarctic temperature trends assumed that the natural annual temperature fluctuations can be represented by an auto-regressive process of first order [AR(1)]. Here we find that this hypothesis is inadequate. We consider the longest observational temperature records in Antarctica and show that their variability is better represented by a long-term persistent process that has a propensity of large and enduring natural excursions from the mean. As a consequence, the statistical significance of the recent (presumably anthropogenic) Antarctic warming trend is lower than hitherto reported, while the uncertainty about its magnitude is enhanced. Indeed, all records except for one (Faraday/Vernadsky) fail to show a significant trend. When increasing the signal-to-noise ratio by considering appropriate averages of the local temperature series, we find that the warming trend is still not significant in East Antarctica and the Antarctic Peninsula. In West Antarctica, however, the significance of the trend is above $$97.4 \,\%$$ , and its magnitude is between 0.08 and 0.96 °C per decade. We argue that the persistent temperature fluctuations not only have a larger impact on regional warming uncertainties than previously thought but also may provide a potential mechanism for understanding the transient weakening (“hiatus”) of the regional and global temperature trends.

The chapter introduces into the manifold occurrences in our world, which are associated with spreading phenomena. It starts with the presentation of those features which are common to essentially all spreading phenomena and refers, subsequently, to the peculiarities in each individual field of their occurrence. All these features are discussed in correspondence with the spreading phenomena the book is dealing with and which are reviewed in this introductory chapter. The chapter has been drafted with the intention that the potential reader, by looking into it, will be motivated to look into also the subsequent chapters and to become, eventually, “infected” by the desire for tracing spreading phenomena anywhere in his/her field.

Scientific Reports 7: Article number: 43034; published online: 22 February 2017; updated: 22 December 2017. This Article contains errors. An improper normalization factor was inadvertently applied, resulting in an incorrect form of Eq. (4). In the Introduction section, “In this setting, according tothe law of total probability, the macroscopic energy distribution is then given by where P(β) is the distribution of β over all local cells in the macroscopic system and Z(β) is a normalization factor for e−βE at specified β 41.

Nanoporous silicon consisting of tubular pores imbedded in a silicon matrix has found many technological applications and provides a useful model system for studying phase transitions under confinement. Recently, a model for mass transfer in these materials has been elaborated [Kondrashova et al., Sci. Rep. 7, 40207 (2017)], which assumes that adjacent channels can be connected by “bridges” (with probability p bridge) which allows diffusion perpendicular to the channels. Along the channels, diffusion can be slowed down by “necks” which occur with probability p neck. In this paper we use Monte-Carlo simulations to study diffusion along the channels and perpendicular to them, as a function of p bridge and p neck, and find remarkable correlations between the diffusivities in longitudinal and radial directions. For clarifying the diffusivity in radial direction, which is governed by the concentration of bridges, we applied percolation theory. We determine analytically how the critical concentration of bridges depends on the size of the system and show that it approaches zero in the thermodynamic limit. Our analysis suggests that the critical properties of the model, including the diffusivity in radial direction, are in the universality class of two-dimensional lattice percolation, which is confirmed by our numerical study.

The question whether a seasonal climate trend (e.g., the increase of summer temperatures in Antarctica in the last decades) is of anthropogenic or natural origin is of great importance for mitigation and adaption measures alike. The conventional significance analysis assumes that (i) the seasonal climate trends can be quantified by linear regression, (ii) the different seasonal records can be treated as independent records, and (iii) the persistence in each of these seasonal records can be characterized by short-term memory described by an autoregressive process of first order. Here we show that assumption ii is not valid, due to strong intraannual correlations by which different seasons are correlated. We also show that, even in the absence of correlations, for Gaussian white noise, the conventional analysis leads to a strong overestimation of the significance of the seasonal trends, because multiple testing has not been taken into account. In addition, when the data exhibit long-term memory (which is the case in most climate records), assumption iii leads to a further overestimation of the trend significance. Combining Monte Carlo simulations with the Holm–Bonferroni method, we demonstrate how to obtain reliable estimates of the significance of the seasonal climate trends in long-term correlated records. For an illustration, we apply our method to representative temperature records from West Antarctica, which is one of the fastest-warming places on Earth and belongs to the crucial tipping elements in the Earth system.

Understanding the physical principles that govern the complex DNA structural organization as well as its mechanical and thermodynamical properties is essential for the advancement in both life sciences and genetic engineering. Recently we have discovered that the complex DNA organization is explicitly reflected in the arrangement of nucleotides depicted by the universal power law tailed internucleotide interval distribution that is valid for complete genomes of various prokaryotic and eukaryotic organisms. Here we suggest a superstatistical model that represents a long DNA molecule by a series of consecutive ~150 bp DNA segments with the alternation of the local nucleotide composition between segments exhibiting long-range correlations. We show that the superstatistical model and the corresponding DNA generation algorithm explicitly reproduce the laws governing the empirical nucleotide arrangement properties of the DNA sequences for various global GC contents and optimal living temperatures. Finally, we discuss the relevance of our model in terms of the DNA mechanical properties. As an outlook, we focus on finding the DNA sequences that encode a given protein while simultaneously reproducing the nucleotide arrangement laws observed from empirical genomes, that may be of interest in the optimization of genetic engineering of long DNA molecules.

In the context of global warming, the question of why Antarctic sea ice extent (SIE) has increased is one of the most fundamental unsolved mysteries. Although many mechanisms have been proposed, it is still unclear whether the increasing trend is anthropogenically originated or only caused by internal natural variability. In this study, we employ a new method where the underlying natural persistence in the Antarctic SIE can be correctly accounted for. We find that the Antarctic SIE is not simply short-term persistent as assumed in the standard significance analysis, but actually characterized by a combination of both short- and long-term persistence. By generating surrogate data with the same persistence properties, the SIE trends over Antarctica (as well as five sub-regions) are evaluated using Monte-Carlo simulations. It is found that the SIE trends over most sub-regions of Antarctica are not statistically significant. Only the SIE over Ross Sea has experienced a highly significant increasing trend (p = 0.008) which cannot be explained by natural variability. Influenced by the positive SIE trend over Ross Sea, the SIE over the entire Antarctica also increased over the past decades, but the trend is only at the edge of being significant (p = 0.034).

In this article we review the occurrence and consequences of long-term memory in geophysical records like climate and seismic records, and describe similarities with financial data sets. We review several methods to detect linear and nonlinear long-term correlations, also in the presence of external trends, and show how external trends can be detected in data with long-term memory. We show as well that long-term correlations lead to a natural clustering of extreme events and discuss the implications for several geophysical data sets.

Using the example of SiO2 dispersions in LiClO4/polyethylene glycol electrolytes, the conduction mechanism of “soggy sand” electrolytes is discussed. The study is essentially based on zeta potential, impedance and transference number measurements as well as on modeling. All the results can be explained by anion adsorption by the oxide particles and increased concentration of free Li+ in the double layer. The initially colloidal dispersion quickly forms fractal networks by cluster–cluster aggregation. Once they percolate, an interfacially dominated Li+ conductance is observed. The subsequent coarsening of the network is self-decelerating leading to a steady state conductivity that is, for low volume fractions, enhanced compared to SiO2 free electrolytes. At higher values, blocking and inhomogeneity effects (e.g., salt trapping) lead to decreased values of the overall conductivity.

We study the performance of multifractal detrended fluctuation analysis (MF-DFA) applied to long-term correlated and multifractal data records in the presence of additive white noise, short-term memory and periodicities. Such additions and disturbances that can be typically found in the observational records of various complex systems ranging from climate dynamics to physiology, network traffic, and finance. In monofractal records, we find that (i) additive white noise hardly results in spurious multifractality, but causes underestimated generalized Hurst exponents h ( q ) for all q values; (ii) short-range correlations lead to pronounced crossovers in the generalized fluctuation functions F q ( s ) at positions that decrease with increasing moment q , thus causing significantly overestimated h ( q ) for small q and spurious multifractality; (iii) periodicities like seasonal trends (with standard deviations comparable with the one of the studied process) result in spurious “reversed” multifractality where h ( q ) increases with increasing q (except for very short time windows). We also show that in multifractal cascades moderate additions of noise, short-range memory, or periodic trends cause flawed results for h ( q ) with q 2 , while h ( q ) with q > 2 remains nearly unchanged.

We study the predictability of extreme events in records with linear and nonlinear long-range memory in the presence of additive white noise using two different approaches: (i) the precursory pattern recognition technique (PRT) that exploits solely the information about short-term precursors, and (ii) the return interval approach (RIA) that exploits long-range memory incorporated in the elapsed time after the last extreme event. We find that the PRT always performs better when only linear memory is present. In the presence of nonlinear memory, both methods demonstrate comparable efficiency in the absence of white noise. When additional white noise is present in the record (which is the case in most observational records), the efficiency of the PRT decreases monotonously with increasing noise level. In contrast, the RIA shows an abrupt transition between a phase of low level noise where the prediction is as good as in the absence of noise, and a phase of high level noise where the prediction becomes poor. In the phase of low and intermediate noise the RIA predicts considerably better than the PRT, which explains our recent findings in physiological and financial records.

SUMMARY We consider sequences of earthquakes from northern and southern California. We study these data sets for long-term persistence using the concept of natural time and fluctuation analyses. We also construct a simulation model for regional seismicity using random background seismicity and the BASS model for aftershock occurrence. We include in the simulation model corrections for long-term persistence in the background seismicity and for missing data early in aftershock sequences. We find excellent agreement between the California data and the simulations indicating significant long-term correlations in earthquake magnitudes.

It is well known that while daily price returns of financial markets are uncorrelated, their absolute values (‘volatility’) are long-term correlated. Here we provide evidence that certain subsequences of the returns themselves also exhibit long-term memory. These subsequences consist of maxima (or minima) of returns in consecutive time windows of R days. Our analysis shows that for both stocks and currency exchange rates, long-term correlations are significant for R ≥ 4 . We argue that this long-term memory which is similar to that observed in volatility clustering sheds further insight on price dynamics that might be used for risk estimation.

We use the Detrended Fluctuation Analysis (DFA) to quantify underlying trends in long-term correlated records. Our approach is based on the fact that different orders of DFA are affected differently by trends. For a given instrumental record of length N , we compare the fluctuation exponent α 0 of DFA0 where trends are not being eliminated, with the fluctuation exponent α 2 of DFA2 where possible linear trends in the instrumental record are being eliminated. From this we deduce numerically the probability density p ( A ) that in the considered long-term correlated record, a linear trend with a slope between A and A + d A occurs. Without loss of generality we focus on Gaussian distributed data. As an example, we apply our analysis to several long temperature records (Melbourne, Oxford, Prague, Pusan, Uppsala, and Vienna), where we discuss the trends within the last 90 years, which may originate from both, urban and global warming.

The diffusion of gas molecules in pores is determined by the collisions between the molecules as well as by the collisions of the molecules with the pore walls. In many applications the so-called Knudsen regime is of particular interest. In this regime the collisions of the molecules with the pore walls play the crucial role, while the inter-molecular collisions can be neglected. Here we study the influence of surface roughness on the coefficients of self (or tracer) diffusion and transport diffusion. Considering the first four iterations of a generalised fractal Koch surface, we construct pore models of different roughness. For these model pores we have performed detailed simulations of both diffusion coefficients using a cube-based algorithm. The molecular trajectories can be mapped onto L´evy walks to determine the diffusion properties. In linear two-dimensional (2d) channels we observe anomalous diffusion, which can also be induced in smooth and rough three-dimensional (3d) pores by anomalous reflection laws. Normal diffusion is found in convoluted 2d pores and in all 3d pores when a diffuse reflection law is applied.

We study the statistics of the return intervals in multifractal data sets with and without linear correlations. In the absence of linear correlations, we find that the nonlinear correlations inherent in multifractal data yield (i) a power-law decay of the autocorrelation function of the return intervals, (ii) a power-law increase of the conditional return period as function of the previous return interval, and (iii) a power-law decay of the probability density function of the return intervals. These features remain unchanged in the presence of linear long-term correlations. Deviations observed in the asymptotic behaviour are probably due to finite size effects. We compare our results with those obtained for uncorrelated and for monofractal long-term correlated data, and demonstrate significant differences. Applications can be found in studying the dynamics of several processes characterised by multifractality, such as turbulence, climate dynamics, heartbeat dynamics, stock market dynamics, and tele-traffic in large networks.

We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.

We propose a model for the localization of risky information in social (scale free) networks where we assume that risky information can propagate only between “mutually trusted nodes” (MTN). We propose an algorithm to construct the MTN network and show that there is a critical value m ¯ c > 2 of trusted nodes below which information localizes. This critical value increases drastically if a fraction p of nodes does not transfer information at all. We study the fraction of initial messengers needed to inform a desired fraction of the network as a function of the average number of trusted nodes m ¯ and discuss possible applications of the model also to marketing and to the spreading of a disease with very short incubation time.

Zusammenfassung: Analyse langjahriger Zeitreihen der Lufttemperatur in Zentralasien Verschiedene Phanomene wie die verstarkte Schrumpfung der Gletscher im Tien Schan und Pamir-Alaj-Gebirge deuten darauf hin, dass sich in der jungeren Vergangenheit in Zentralasien eine Klimaerwarmung vollzogen hat. Deshalb wird gefragt, seit wann eine Klimaerwarmung eingetreten ist und mit welcher Intensitat sie sich vollzogen hat. Insbesondere wird der Frage nachgegangen, ob die Klimaerwarmung der jungeren Vergangenheit Ausdruck einer trendhaften oder zyklischen Entwicklung der Jahresmitteltemperaturen ist. Zu diesem Zweck wurden auf der Basis von Monats- und Jahresmittelwerten der Lufttemperatur fur ausgewahlte Klimastationen, die in Zentralasien typische Standorte reprasentieren, Analysen langfristiger Zeitreihen durchgefuhrt. Eingesetzt wurden Verfahren der Regressions- und Korrelationsanalyse sowie Verfahren der „Detrended Fluctuation Analysis (DFA)”, um in den Zeitreihen Langzeitkorrelationen und Trends zu identifizieren. Zur Quantifizierung auftretender Trends wurde eine Weiterentwicklung der DFA vorgenommen. Summary: Different phenomena, as the enhanced shrinking of glaciers in the Tien Shan and Pamiro Alay, indicate that a climate warming has occurred in Central Asia in the recent past. Thus the questions are since when the climate warming has been taking place and what its intensity has been. A question, which is especially investigated, is whether climate warming in the recent past expresses a trend-like or cyclic development of annual temperature averages. For this purpose analyses of long time series were performed, consisting of monthly and annual air temperature averages at selected climate stations representing typical locations in Central Asia. Applied were methods of the Regression and Correlation Analysis, in particular the Detrended Fluctuation Analysis (DFA), in order to identify long-term correlations and trends in the time series. For the purpose of quantification of occurring trends, the DFA was further extended.

We show that the scaling properties of river runoff records represent a useful tool for evaluating precipitation-runoff models that are widely used in hydrology for assessment of the water balance in a given river catchment. In this respect, it is important that the model maps the processes that control the water balance. The main field of application is therefore water management in a given area over a long time scale (at least several years). Here, we compare the temporal scaling properties of the runoff of three Bavarian rivers (Naab, Regnitz, and Vils) with the corresponding ASGi model records. In the evaluation, we use: (i) detrended fluctuation analysis (DFA); (ii) multifractal analysis; (iii) periodic volatility analysis; and (iv) long-term volatility analysis. Our study generally shows close similarity between real and simulated data for the main statistical parameters (e.g., correlation and multifractal exponents). Therefore, the ASGi model output seems to adequately describe real basin processes and might be useful for hydrological purposes, such as a posteriori estimation of water balance in a river catchment.

We use Monte Carlo simulations and arguments from percolation theory in order to determine how “confidential” information propagates or localizes on a scale-free network. The basic assumption of our models is that this type of information propagates through the subnetwork of “best friends” which constitute a persons “circle of trust”. We find that there is a sharp percolation transition between a phase where “confidential” information localizes and a phase where “confidential” information propagates. This transition is controlled by the number of best friends m 0 that a person is willing to have, and occurs for m 0 values higher than intuitively expected from the “small world” property of random networks.

Structural, localization and functional properties of unknown proteins are often being predicted from their primary polypeptide chains using sequence alignment with already characterized proteins and consequent molecular modeling. Here we suggest an approach to predict various structural and structure-associated properties of proteins directly from the mass distributions of their proteolytic cleavage fragments. For amino-acid-specific cleavages, the distributions of fragment masses are determined by the distributions of inter-amino-acid intervals in the protein, that in turn apparently reflect its structural and structure-related features. Large-scale computer simulations revealed that for transmembrane proteins, either α-helical or β -barrel secondary structure could be predicted with about 90% accuracy after thermolysin cleavage. Moreover, 3/4 intrinsically disordered proteins could be correctly distinguished from proteins with fixed three-dimensional structure belonging to all four SCOP structural classes by combining 3–4 different cleavages. Additionally, in some cases the protein cellular localization (cytosolic or membrane-associated) and its host organism (Firmicute or Proteobacteria) could be predicted with around 80% accuracy. In contrast to cytosolic proteins, for membrane-associated proteins exhibiting specific structural conformations, their monotopic or transmembrane localization and functional group (ATP-binding, transporters, sensors and so on) could be also predicted with high accuracy and particular robustness against missing cleavages.

We study temporal correlations and multifractal properties of long river discharge records from 41 hydrological stations around the globe. To detect long-term correlations and multifractal behaviour in the presence of trends, we apply several recently developed methods [detrended fluctuation analysis (DFA), wavelet analysis, and multifractal DFA] that can systematically detect and overcome non-stationarities in the data at all time scales. We find that above some crossover time that usually is several weeks, the daily runoffs are long-term correlated, being characterized by a correlation function C(s) that decays as C(s)∼s−γ. The exponent γ varies from river to river in a wide range between 0.1 and 0.9. The power–law decay of C(s) corresponds to a power–law increase of the related fluctuation function F2(s)∼sH where H=1−γ/2. We also find that in most records, for large times, weak multifractality occurs. The Renyi exponent τ(q) for q between −10 and +10 can be fitted to the remarkably simple form τ ( q ) = − ln ( a q + b q ) / ln 2 , with solely two parameters a and b between 0 and 1 with a+b≥1. This type of multifractality is obtained from a generalization of the multiplicative cascade model.

For both stock and currency markets, we study the return intervals τ between the daily volatilities of the price changes that are above a certain threshold q . We find that the distribution function P q (τ) scales with the mean return interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\bar {{\tau}}}\end{equation*}\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}P_{q}({\tau})={\bar {{\tau}}}^{-1}f({\tau}/{\bar {{\tau}}})\end{equation*}\end{document} . The scaling function f ( x ) is similar in form for all seven stocks and for all seven currency databases analyzed, and f ( x ) is consistent with a power-law form, f ( x ) ∼ x -γ with γ ≈ 2. We also quantify how the conditional distribution P q (τ|τ 0 ) depends on the previous return interval τ 0 and find that small (or large) return intervals are more likely to be followed by small (or large) return intervals. This “clustering” of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long-term correlations known to be present in the volatility.