Inverse Power-law Distribution Research Articles

Reliability investigation of exponentiated Weibull distribution using IPL through numerical and artificial neural network modeling

AbstractIn current investigation, a novel implementation of intelligent numerical computing solver depending on artificial neural networks (ANN) has been provided to interpret failure function (FF), reliability function (RF), hazard rate function (HRF), Mils ratio (MR), and mean time to failure (MTTF). This study investigates a reliability model centered on the exponentiated Weibull distribution (EWD) and the inverse power law (IPL) model employing the ANN model. The nonmonotonic failure rate can be modeled via this distribution. A data set for the proposed ANN has been generated for various scenarios of (Exponentiated Weibull Inverse Power Law Distribution) EWIPLD model by variation of involved pertinent parameters via the Galerkin weighted residual method (GWRM). Levenberg–Marquard training algorithm has been used in the multi‐layer perceptron (MLP) network model developed with 10 nodes in the hidden layer. The Coefficient of Determination (R) value for the ANN model has been obtained as 0.9999. The findings obtained, revealed that ANNs are an excellent technique that can be applied to predict reliability measures in conjunction with the right statistical model.

Effect of Environmental Change Distribution on Artificial Life Simulations

It is already well known that environmental variation has a big effect on real evolution, and similar effects have been found in evolutionary artificial life simulations. In particular, a lot of research has been carried out on how the various evolutionary outcomes depend on the noise distributions representing the environmental changes, and how important it is for models to use inverse power-law distributions with the right noise colour. However, there are two distinct factors of relevance-the average total magnitude of change per unit time and the distribution of individual change magnitudes-and misleading results may emerge if those factors are not properly separated. This article makes use of an existing agent-based artificial life modeling framework to explore this issue using models previously tried and tested for other purposes. It begins by demonstrating how the total magnitude and distribution effects can easily be confused, and goes on to show how it is possible to untangle the influence of these interacting factors by using correlation-based normalization. It then presents a series of simulation results demonstrating that interesting dependencies on the noise distribution remain after separating those factors, but many effects involving the noise colour of inverse power-law distributions disappear, and very similar results arise across restricted-range white-noise distributions. The average total magnitude of change per unit time is found to have a substantial effect on the simulation outcomes, but the distribution of individual changes has very little effect. A robust counterexample is thereby provided to the idea that it is always important to use accurate environmental change distributions in artificial life models.

Self‐organisation in dialogical patterns of a patient with borderline personality disorder and their therapist: A case study from nonlinear dynamics perspective

AbstractObjectiveThe purpose of this study was to identify self‐organised dynamics in sessions with dialogical patterns (i.e. reflective interactions of self‐states associated with psychological change) within long‐term psychotherapy. One of the hallmarks of self‐organisation is the presence of an Inverse Power Law (IPL) in the frequency distribution of relevant events, which reflect better adaptation to internal and environmental changes, produced by psychotherapy.MethodsA case study of successful psychotherapy (44 weekly sessions) with a patient diagnosed with borderline personality disorder (BPD) and the therapist was conducted. The expressions of the subjectivity (i.e. voices) in the patient and therapist were identified through the Model of Analysis of Discursive Positioning in Psychotherapy (MAPP). State Space Grids (SSG) identified sessions that presented dialogical patterns acting as attractors (8 sessions) and those with non‐dialogical patterns acting as attractors (12 sessions). Finally, Orbital Decomposition (ORBDE) produced statistics of self‐organisation in both groups of sessions and the degree of fit to an IPL.ResultsBoth groups of sessions presented an acceptable fit to the IPL distributions (R2 = .853 and .938), and statistics placed them in different self‐organised regions.ConclusionsAlthough dialogical patterns are an emergent property of therapy, their structure is similar to sessions of other types, in that dynamic processes occur in them that can be empowered in patients to consolidate psychological change. The nonlinear and emergent character of these self‐organised processes and their clinical implications for activating the patients' psychological resources are discussed.

New Probability Distributions in Astrophysics: VII. The Truncated Gamma-Pareto Distribution Applied to Cosmic Rays

Many astrophysical phenomena are modeled by an inverse power law distribution at high values of the random variable but often at low values of the random variable we have a departure from an inverse power law. In order to insert a continuous transition from low to high values of the random variable we analyse the truncated gamma-Pareto distribution in two versions by deriving the most important statistical parameters. The application of the results to the distribution in energy of cosmic rays allows deriving an analytical expression for the average energy, which is 2.6 GeV.

Burstiness and Stochasticity in the Malleability of Physical Activity.

This study examined whether patterns of self-organization in physical activity (PA) predicted long-term success in a yearlong PA intervention. Increased moderate to vigorous PA (MVPA) was targeted in insufficiently active adults (N = 512) via goal setting and financial reinforcement. The degree to which inverse power law distributions, which are reflective of self-organization, summarized (a)daily MVPA and (b)time elapsed between meeting daily goals (goal attainment interresponse times) was calculated. Goal attainment interresponse times were also used to calculate burstiness, the degree to which meeting daily goals clustered in time. Inverse power laws accurately summarized interresponse times, but not daily MVPA. For participants with higher levels of MVPA early in the study, burstiness in reaching goals was associated with long-term resistance to intervention, while stochasticity in meeting goals predicted receptiveness to intervention. These results suggest that burstiness may measure self-organizing resistance to change, while PA stochasticity could be a precondition for behavioral malleability.

Wall-attached structures over a traveling wavy boundary: Scalar transport

The wall-attached structures of scalar fluctuations over a traveling wavy boundary are investigated using large eddy simulations (LESs) at a moderate Reynolds number (Reτ≈1000) and various Prandtl numbers (Pr=0.2, 0.7, 1, and 2). The wave slope is ak=0.1, where a is the wave amplitude and k is the wavenumber. The prescribed wave age is c/u*=2, where c is the wave phase speed and u* is the friction velocity. For comparison, the results of an LES over a smooth wall are also discussed. The results provide evidence for the presence of self-similar wall-attached structures of scalar fluctuations in both turbulent flow over a traveling wavy boundary and that over a smooth wall. In particular, the population density of the attached structures exhibits an inverse power-law distribution, reminiscent of a hierarchy of attached structures. The conditionally averaged scalar variance reconstructed from the superposition of attached structures has a near-wall peak and a logarithmic variation in the log region. The magnitude of this peak varies logarithmically with the hierarchical length scale, and the variation slope decreases with increase in the Prandtl number. In addition, the correlations between the attached structures and vertical scalar transport are examined by direct observation of instantaneous snapshots. The ejection event of the fluid leads the negative wall-attached structures to contribute positive vertical scalar flux and the positive wall-attached structures to contribute negative vertical scalar flux.

Interacting faults in california and hindu kush

We study seismic fluctuations in California and Hindu Kush using Diffusion Entropy Analysis (DEA), a technique designed to detect the action of crucial events in time series generated by complex dynamical systems. The time distance between two consecutive crucial events is described by an inverse power law distribution density with a power index $\mu$ close to the value $\mu = 2$, corresponding to an ideal $1/f$ noise. DEA was used in the recent past to study neurophysiological processes that in the healthy condition are found to generate $1/f$ noise and $\mu$ close to $2$. In this paper we find that the seismic fluctuations in both California and Hindu-Kush of extended areas implying the action of many faults, $\mu \approx 2.1$, while in regions involving the action of only one fault, or of a very small number of faults, $\mu \approx 2.4$. This observation led us the conjectures that the seismic criticality is due to the interaction of many faults. To support this conjecture we adopted a dynamical model for fault dynamics proposed by Brown and Tosatti and we have extended it to describe the interaction between many faults. The DEA applied to surrogate sequences generated by this dynamical model, yields $\mu = 2.1$ for a single fault and $\mu = 2,4$ for many interacting faults, in a good agreement with the observation of real seismic fluctuations. This result supports our conjecture and suggests interesting applications to neurophysiological and sociological processes.

Landslide hazard probability and risk assessment at the community level: a case of western Hubei, China

Abstract. Small communities living in mountainous terrain in Hubei province are often affected by landslides. Previous studies by the China Geological Survey focused on the 1:100 000 scale. Therefore, a more detailed assessment, especially at the community level, is urgently required by local governments for risk management. In this study, we conducted a more detailed semiquantitative landslide and risk assessment at the community level using a scale of 1:10 000. We applied the probabilistic method to assess landslide spatial, temporal, and size probabilities, while the hazard and risk assessment were considered for four return periods (5, 10, 20, and 50 years) and two size scenarios (landslide volume). The spatial probability from susceptibility mapping with an accuracy of 84 % indicates that the major controlling factors are Quaternary deposits and weathered eluvium from Ordovician limestones. This study revealed that most building areas in hazard maps are at the foot of major slopes with very high hazard probabilities, and therefore we computed the potential loss of life and property for each slope. The results reveal that 1530 people and USD 18 million worth of property were at risk of landslides within a 50-year return period and a landslide volume of 50 000 m3. The longer the return period is, the higher the hazard probability is. Compared with the classic inverse gamma and power law distribution of landslide magnitude and frequency, the function by the ordinary least squares method is more suitable for landslide size probability analysis of the study area. According to these methods, the proposed procedure of landslide risk assessment proves more useful than the existing data from the 1:100 000 scale in western Hubei, China.

Intrinsic blinking characteristics of single colloidal CdSe-CdS/ZnS core-multishell quantum dots

Fluorescence blinking of single colloidal semiconductor quantum dots (QDs) has been extensively studied, and several sophisticated models have been proposed. In this work, we derive Heisenberg equations of motion to carefully study principal transition processes, i.e., photoexcitation, energy relaxation, impact ionization and Auger recombination, radiative and nonradiative recombinations, and tunneling between core states and surface states, of the electron-hole pair in single CdSe-CdS/ZnS core-multishell QDs and show that the on-state probability density distribution of the QD fluorescence obeys the random telegraph signal theory because of the random radiative recombination of the photoexcited electron-hole pair in the QD core, while the off-state probability density distribution obeys the inverse power law distribution due to the series of random walks of the photoexcited electron in the two-dimensional surface-state network after the electron tunnels from the QD core to the QD surface. These two different blinking characteristics of the single QD are resolved experimentally by properly adjusting the optical excitation power and the bin time.

Self-organizing Complex Networks: individual versus global rules.

We introduce a form of Self-Organized Criticality (SOC) inspired by the new generation of evolutionary game theory, which ranges from physiology to sociology. The single individuals are the nodes of a composite network, equivalent to two interacting subnetworks, one leading to strategy choices made by the individuals under the influence of the choices of their nearest neighbors and the other measuring the Prisoner's Dilemma Game payoffs of these choices. The interaction between the two networks is established by making the imitation strength K increase or decrease according to whether the last two payoffs increase or decrease upon increasing or decreasing K. Although each of these imitation strengths is selected selfishly, and independently of the others as well, the social system spontaneously evolves toward the state of cooperation. Criticality is signaled by temporal complexity, namely the occurrence of non-Poisson renewal events, the time intervals between two consecutive crucial events being given by an inverse power law index μ = 1.3 rather than by avalanches with an inverse power law distribution as in the original form of SOC. This new phenomenon is herein labeled self-organized temporal criticality (SOTC). We compare this bottom-up self-organization process to the adoption of a global choice rule based on assigning to all the units the same value K, with the time evolution of common K being determined by consciousness of the social benefit, a top-down process implying the action of a leader. In this case self-organization is impeded by large intensity fluctuations and the global social benefit turns out to be much weaker. We conclude that the SOTC model fits the requests of a manifesto recently proposed by a number of European social scientists.

Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall

Dynamical systems in nature exhibit self-similar fractal space–time fluctuations on all scales indicating long-range correlations and, therefore, the statistical normal distribution with implicit assumption of independence, fixed mean and standard deviation cannot be used for description and quantification of fractal data sets. The author has developed a general systems theory based on classical statistical physics for fractal fluctuations which predicts the following. (1) The fractal fluctuations signify an underlying eddy continuum, the larger eddies being the integrated mean of enclosed smaller-scale fluctuations. (2) The probability distribution of eddy amplitudes and the variance (square of eddy amplitude) spectrum of fractal fluctuations follow the universal Boltzmann inverse power law expressed as a function of the golden mean. (3) Fractal fluctuations are signatures of quantum-like chaos since the additive amplitudes of eddies when squared represent probability densities analogous to the sub-atomic dynamics of quantum systems such as the photon or electron. (4) The model predicted distribution is very close to statistical normal distribution for moderate events within two standard deviations from the mean but exhibits a fat long tail that are associated with hazardous extreme events. Continuous periodogram power spectral analyses of available GHCN annual total rainfall time series for the period 1900–2008 for Indian and USA stations show that the power spectra and the corresponding probability distributions follow model predicted universal inverse power law form signifying an eddy continuum structure underlying the observed inter-annual variability of rainfall. On a global scale, man-made greenhouse gas related atmospheric warming would result in intensification of natural climate variability, seen immediately in high frequency fluctuations such as QBO and ENSO and even shorter timescales. Model concepts and results of analyses are discussed with reference to possible prediction of climate change. Model concepts, if correct, rule out unambiguously, linear trends in climate. Climate change will only be manifested as increase or decrease in the natural variability. However, more stringent tests of model concepts and predictions are required before applications to such an important issue as climate change. Observations and simulations with climate models show that precipitation extremes intensify in response to a warming climate (O’Gorman in Curr Clim Change Rep 1:49–59, 2015).

Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

Universal characteristics of fractal fluctuations in prime number distribution

The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.

The volume-to-surface-area ratio constrains the rollover of the power law distribution for landslide size

The rollover observed in the landslide size distribution, i.e., the deviation from power law behavior for small landslides, has not been previously described by a widely accepted physical interpretation. In this paper, the volume-to-failure-surface-area ratio V/S is proposed as an essential constraining factor for the emergence of the rollover. We first demonstrate that, from a three-dimensional perspective, landslide failures require a minimum threshold V/S value for a given mechanical configuration. The heterogeneity of the slope properties presents each potential landslide with a probability that V/S will exceed the threshold value, and further, results in a decay of frequency but not a cutoff for small landslides in the landslide size distribution. Empirical data show that the variance of the landslide shape is one aspect of the heterogeneity that can cause the inverse power law distribution of the landslide volume to decrease for smaller sizes.

Universal inverse power-law distribution for temperature and rainfall in the UK region

Meteorological parameters, such as temperature, rainfall, pressure, etc., exhibit selfsimilar space–time fractal fluctuations generic to dynamical systems in nature such as fluid flows, spread of forest fires, earthquakes, etc. The power spectra of fractal fluctuations display inverse power-law form signifying long-range correlations. A general systems theory model predicts universal inverse power-law form incorporating the golden mean for the fractal fluctuations. The model predicted distribution was compared with observed distribution of fractal fluctuations of all size scales (small, large and extreme values) in the historic month-wise temperature (maximum and minimum) and total rainfall for the four stations Oxford, Armagh, Durham and Stornoway in the UK region, for data periods ranging from 92 years to 160 years. For each parameter, the two cumulative probability distributions, namely cmax and cmin starting from respectively maximum and minimum data value were used. The results of the study show that (i) temperature distributions (maximum and minimum) follow model predicted distribution except for Stornowy, minimum temperature cmin. (ii) Rainfall distribution for cmin follow model predicted distribution for all the four stations. (iii) Rainfall distribution for cmax follows model predicted distribution for the two stations Armagh and Stornoway. The present study suggests that fractal fluctuations result from the superimposition of eddy continuum fluctuations.

Walking across Wikipedia: a scale-free network model of semantic memory retrieval.

Semantic knowledge has been investigated using both online and offline methods. One common online method is category recall, in which members of a semantic category like “animals” are retrieved in a given period of time. The order, timing, and number of retrievals are used as assays of semantic memory processes. One common offline method is corpus analysis, in which the structure of semantic knowledge is extracted from texts using co-occurrence or encyclopedic methods. Online measures of semantic processing, as well as offline measures of semantic structure, have yielded data resembling inverse power law distributions. The aim of the present study is to investigate whether these patterns in data might be related. A semantic network model of animal knowledge is formulated on the basis of Wikipedia pages and their overlap in word probability distributions. The network is scale-free, in that node degree is related to node frequency as an inverse power law. A random walk over this network is shown to simulate a number of results from a category recall experiment, including power law-like distributions of inter-response intervals. Results are discussed in terms of theories of semantic structure and processing.

Network of echoes

The decision making model (DMM) previously developed [3,35] has been shown to generate phase transitions, to be topologically complex as manifest by inverse power-law (IPL) degree distributions, and to produce temporal complexity through IPL distributions in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the DMM imperfectly imitate one another, which is postulated herein as the echo response hypothesis; an echo being an imperfect copy of an original signal. Some implications of this hypothesis for the human sciences are explored.

Criticality and avalanches in neural networks

Experimental work, both in vitro and in vivo, reveals the occurrence of neural avalanches with an inverse power law distribution in size and time duration. These properties are interpreted as an evident manifestation of criticality, thereby suggesting that the brain is an operating near criticality complex system: an attractive theoretical perspective that according to Gerhard Werner may help to shed light on the origin of consciousness. However, a recent experimental observation shows no clear evidence for power-law scaling in awake and sleeping brain of mammals, casting doubts on the assumption that the brain works at criticality. This article rests on a model proposed by our group in earlier publications to generate neural avalanches with the time duration and size distribution matching the experimental results on neural networks. We now refine the analysis of the time distance between consecutive firing bursts and observe the deviation of the corresponding distribution from the Poisson statistics, as the system moves from the non-cooperative to the cooperative regime. In other words, we make the assumption that the genuine signature of criticality may emerge from temporal complexity rather than from the size and time duration of avalanches. We argue that the Mittag–Leffler (ML) exponential function is a satisfactory indicator of temporal complexity, namely of the occurrence of non-Poisson and renewal events. The assumption that the onset of criticality corresponds to the birth of renewal non-Poisson events establishes a neat distinction between the ML function and the power law avalanches generating regime. We find that temporal complexity uncontaminated by periodicity occurs in a narrow range of values of the control parameter, in correspondence of which the information transfer from one network to another becomes maximal. We argue that if this enhancement of information transport is interpreted as a signature of criticality, then the power law avalanches are a manifestation of supercriticality rather than criticality. This leads us to conclude that the recent experiment showing no evidence of power-law avalanches in the brain of mammals would not conflict with the hypothesis that the brain works at criticality.

An Inverse Power-Law Distribution of Molecular Bond Lifetimes Predicts Fractional Derivative Viscoelasticity in Biological Tissue

Viscoelastic characteristics of many materials falling under the category of soft glassy substances, including biological tissue, often exhibit a mechanical complex modulus Y(ω) well described by a fractional derivative model: Y(ω) = E(iω/ϕ)k, where E = a generalized viscoelastic stiffness; i = (−1)1/2; ω = angular frequency; ϕ = scaling factor; and k = an exponent valued between 0 and 1. The term “fractional derivative” refers to the value of k: when k = 0 the viscoelastic response is purely elastic, and when k = 1 the response is purely viscous. We provide an analytical derivation of the fractional derivative complex modulus based on the hypothesis that the viscoelastic response arises from many intermittent molecular crosslinks, whose lifetimes longer than a critical threshold lifetime, tcrit, are distributed with an inverse power law proportional to t-(k+2). We demonstrate that E is proportional to the number and stiffness of crosslinks formed at any moment; the scaling factor ϕ is equivalent to reciprocal of tcrit; and the relative mean lifetime of the attached crosslinks is inversely proportional to the parameter k. To test whether electrostatic molecular bonds could be responsible for the fractional derivative viscoelasticity, we used chemically skinned human skeletal muscle as a one-dimensional model of a soft glassy substance. A reduction in ionic strength from 175 to 110 mEq resulted in a larger E with no change in k, consistent with a higher probability of interfilament molecular interactions. Thick to thin filament spacing was reduced by applying 4% w/v of the osmolyte Dextran T500, which also resulted in a larger E, indicating a greater probability of crosslink formation in proportion to proximity. A 10°C increase in temperature resulted in an increase in k, which corresponded to a decrease in cross-bridge attachment lifetime expected with higher temperatures. These theoretical and experimental results suggest that the fractional derivative viscoelasticity observed in some biological tissue arises as a mechanical consequence of electrostatic interactions, whose longest lifetimes are distributed with an inverse power law.

Cooperation in neural systems: Bridging complexity and periodicity

Inverse power law distributions are generally interpreted as a manifestation of complexity, and waiting time distributions with power index μ<2 reflect the occurrence of ergodicity-breaking renewal events. In this paper we show how to combine these properties with the apparently foreign clocklike nature of biological processes. We use a two-dimensional regular network of leaky integrate-and-fire neurons, each of which is linked to its four nearest neighbors, to show that both complexity and periodicity are generated by locality breakdown: Links of increasing strength have the effect of turning local interactions into long-range interactions, thereby generating time complexity followed by time periodicity. Increasing the density of neuron firings reduces the influence of periodicity, thus creating a cooperation-induced renewal condition that is distinctly non-Poissonian.