Power-law Patterns Research Articles

Productivity or Equity? Tradeoffs in Volunteer Microtasking in Humanitarian OpenStreetMap

Microtasking, the decomposition of tasks into small units of work, is prolific in human computation and crowdsourcing. Some peer production systems are beginning to leverage this same technique in volunteer contribution-based settings. While early research suggests that focusing volunteer work in this way using microtasking may be fruitful, the effects of microtasking on contributor behavior in volunteer peer production settings, like OpenStreetMap, remain unclear. This paper takes advantage of a natural experiment facilitated by the Humanitarian OpenStreetMap's Tasking Manager, a microtasking system, and employs causal inference analysis to evaluate the effects of a microtasking intervention on contributor dynamics. Our study systematically leverages a global dataset to analyze peer production dynamics, building on prior research to address a gap in peer production literature and informing the design of the microtasking interfaces. We causally show that, indeed, microtasking can be an effective intervention in peer production settings, but it may exacerbate power-law patterns that are common in such settings. By further analyzing project design decisions and characteristics, we develop implications for platform practitioners, with a focus on addressing engagement and contribution inequity issues prevalent in settings like OpenStreetMap.

The Multiscale Principle in Nature (Principium luxuriæ): Linking Multiscale Thermodynamics to Living and Non-Living Complex Systems

Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. However, a proposed interpretation could shed light on the inherent principle behind the creation of fractals. This is the multiscale thermodynamic perspective, which states that an increase in external energy could initiate energy transport mechanisms that facilitate the dissipation or release of excess energy at different scales. Within this framework, it is revealed that power law patterns, and to a lesser extent, fractals, can emerge as a geometric manifestation to dissipate energy in response to external forces. In this context, the exponent of these power law patterns (thermodynamic fractal dimension D) serves as an indicator of the balance between entropy production at small and large scales. Thus, when a system is more efficient at releasing excess energy at the microscopic (macroscopic) level, D tends to increase (decrease). While this principle, known as Principium luxuriæ, may sound promising for describing both multiscale and complex systems, there is still uncertainty about its true applicability. Thus, this work explores different physical, astrophysical, sociological, and biological systems to attempt to describe and interpret them through the lens of the Principium luxuriæ. The analyzed physical systems correspond to emergent behaviors, chaos theory, and turbulence. To a lesser extent, the cosmic evolution of the universe and geomorphology are examined. Biological systems such as the geometry of human organs, aging, human brain development and cognition, moral evolution, Natural Selection, and biological death are also analyzed. It is found that these systems can be reinterpreted and described through the thermodynamic fractal dimension. Therefore, it is proposed that the physical principle that could be behind the creation of fractals is the Principium luxuriæ, which can be defined as “Systems that interact with each other can trigger responses at multiple scales as a manner to dissipate the excess energy that comes from this interaction”. That is why this framework has the potential to uncover new discoveries in various fields. For example, it is suggested that the reduction in D in the universe could generate emergent behavior and the proliferation of complexity in numerous fields or the reinterpretation of Natural Selection.

Power-law patterns in the Phanerozoic sedimentary records of carbon, oxygen, sulfur, and strontium isotopes

Power-law patterns appear in a variety of natural systems on the modern Earth; nevertheless, whether such behaviors appeared in the deep-time environment has rarely been studied. Isotopic records in sedimentary rocks, which are widely used to reconstruct the geological/geochemical conditions in paleoenvironments and the evolutionary trajectories of biogeochemical cycles, offer an opportunity to investigate power laws in ancient geological systems. In this study, I focus on the Phanerozoic sedimentary records of carbon, oxygen, sulfur, and strontium isotopes, which have well documented and extraordinarily comprehensive datasets. I perform statistical analyses on these datasets and show that the variations in the sedimentary records of the four isotopes exhibit power-law behaviors. The exponents of these power laws range between 2.2 and 2.9; this narrow interval indicates that the variations in carbon, oxygen, sulfur, and strontium isotopes likely belong to the same universality class, suggesting that these systematic power-law patterns are governed by universal, scale-free mechanisms. I then derive a general form for these power laws from a minimalistic model based on basic physical principles and geosystem-specific assumptions, which provides an interpretation for the power-law patterns from the perspective of thermodynamics. The fundamental mechanisms regulating such patterns might have been ubiquitous in paleoenvironments, implying that similar power-law behaviors may exist in the sedimentary records of other isotopes.

Remaining popular: power-law regularities in network dynamics

The structure of networks has been a focal research topic over the past few decades. These research efforts have enabled the discovery of numerous structural patterns and regularities, bringing forth advancements in many fields. In particular, the ubiquitous power-law patterns evident in degree distributions, graph eigenvalues and human mobility patterns have provided the opportunity to model many different complex systems. However, regularities in the dynamical patterns of networks remain a considerably less explored terrain. In this study we examine the dynamics of networks, focusing on stability characteristics of node popularity, and present our results using various empirical datasets. Specifically, we address several intriguing questions – for how long are popular nodes expected to remain so? How much time is expected to pass between two consecutive popularity periods? What characterizes nodes which manage to maintain their popularity for long periods of time? Surprisingly, we find that such temporal aspects are governed by a power-law regime, and that these power-law regularities are equally likely across all node ages.

Engineering quasi-steady-state correlations in uncorrelated thermal states using stochastic driving

Nonequilibrium quantum dynamics can give rise to the emergence of novel steady states. We propose a scheme for driving an initially uncorrelated thermal state to generate customized correlation functions by determining and reverse engineering the steady-state two-point functions for a class of Markov processes. We also extend the formalism to the calculation of four-point functions. We then apply our method to generating power-law correlated fermionic Green's functions. Furthermore, we find that the power-law patterns emerge at much shorter times than the convergence to the steady state, at which point the disorder in the two-point correlations disappears. On the other hand, the density-density correlations exhibit steady-state disorder while following a power-law trend line. These ideal steady states appear as intermediate-time quasi-steady states in the presence of perturbations.

Entropy scaling of high resolution systems—Example from football

This comment on the article “Hidden power law patterns in the top European football leagues” written by Da Silva et al. derives a universal logarithmic scaling behaviour of Shannon entropy with respect to the system size that holds for sufficiently smooth distributions. Taking this scaling behaviour into account allows to compare the entropy of ensembles with different sizes, e.g. if they have the same underlying distribution, but are sampled at different resolutions. As an example, these findings help to improve Da Silva’s method that uses entropy as a measure of competitiveness to compare European football leagues.

Continuous time random walk and diffusion with generalized fractional Poisson process

A non-Markovian counting process, the ‘generalized fractional Poisson process’ (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters 0<β≤1, α>0 and a time scale parameter. Generalizations to Laskin’s fractional Poisson distribution and to the fractional Kolmogorov–Feller equation are derived. We develop a continuous time random walk subordinated to a GFPP in the infinite integer lattice Zd. For this stochastic motion, we deduce a ‘generalized fractional diffusion equation’. For long observations, the generalized fractional diffusion exhibits the same power laws as fractional diffusion with fat-tailed waiting time densities and subdiffusive tβ-power law for the expected number of arrivals. However, in short observation times, the GFPP exhibits distinct power-law patterns, namely tαβ−1-scaling of the waiting time density and a tαβ-pattern for the expected number of arrivals. The latter exhibits for αβ>1 superdiffusive behavior when the observation time is short. In the special cases α=1 with 0<β<1 the equations of the Laskin fractional Poisson process and for α=1 with β=1 the classical equations of the standard Poisson process are recovered. The remarkably rich dynamics introduced by the GFPP opens a wide field of applications in anomalous transport and in the dynamics of complex systems.

Power-law distributions in binned empirical data

Many man-made and natural phenomena, including the intensity of earthquakes, population of cities and size of international wars, are believed to follow power-law distributions. The accurate identification of power-law patterns has significant consequences for correctly understanding and modeling complex systems. However, statistical evidence for or against the power-law hypothesis is complicated by large fluctuations in the empirical distribution’s tail, and these are worsened when information is lost from binning the data. We adapt the statistically principled framework for testing the power-law hypothesis, developed by Clauset, Shalizi and Newman, to the case of binned data. This approach includes maximum-likelihood fitting, a hypothesis test based on the Kolmogorov–Smirnov goodness-of-fit statistic and likelihood ratio tests for comparing against alternative explanations. We evaluate the effectiveness of these methods on synthetic binned data with known structure, quantify the loss of statistical power due to binning, and apply the methods to twelve real-world binned data sets with heavy-tailed patterns.

Failure mechanisms of load-sharing complex systems

We investigate the failure mechanisms of load-sharing complex systems. The system is composed of multiple nodes or components whose failures are determined based on the interaction of their respective strengths and loads (or capacity and demand, respectively) as well as the ability of a component to share its load with its neighbors when needed. We focus on two distinct mechanisms to model the interaction between components' strengths and loads. The failure mechanisms of these two models demonstrate temporal scaling phenomena, phase transitions, and multiple distinct failure modes excited by extremal dynamics. For critical ranges of parameters the models demonstrate power-law and exponential failure patterns. We identify the similarities and differences between the two mechanisms and the implications of our results for the failure mechanisms of complex systems in the real world.

Can log-periodic power law structures arise from random fluctuations?

Recent research has established log-periodic power law (LPPL) patterns prior to the detonation of the German stock index (DAX) bubble in 1998. The purpose of this article is to explore whether a Langevin equation extracted from real world data can generate synthetic time series with comparable LPPL structures. To this end, we first estimate the stochastic process underlying the DAX log-returns during the period from mid-1997 until end-2003. The employed data set contains about 3.93⋅106 intraday DAX quotes at a sampling rate of 15 s. Our results indicate that the DAX log-returns can be described as a Markov process. As a consequence, a Langevin equation is derived. Based on this model equation, we run extensive simulations in order to generate 100 synthetic DAX trajectories each covering 3000 trading days. We find LPPL behavior in ten artificial time series. Moreover, we can establish a link between LPPL patterns and ensuing bubble bursts in seven synthetic 600-week windows. However, the LPPL components in most synthetic trajectories differ fundamentally from those LPPL structures that have previously been detected in real financial time series. Summarized, this paper demonstrates that LPPL structures are not necessarily the signature of imitative behavior among investors but can also stem from noise, even though the likelihood of this is extremely low. Thus, our findings confirm with high statistical confidence that the LPPL structures in the DAX development are rooted deeper than only in the random fluctuations of the German stock market.

Why credit risk markets are predestined for exhibiting log-periodic power law structures

Recent research has established the existence of log-periodic power law (LPPL) patterns in financial institutions’ credit default swap (CDS) spreads. The main purpose of this paper is to clarify why credit risk markets are predestined for exhibiting LPPL structures. To this end, the credit risk prediction of two variants of logistic regression, i.e. polynomial logistic regression (PLR) and kernel logistic regression (KLR), are firstly compared to the standard logistic regression (SLR). In doing so, the question whether the performances of rating systems based on balance sheet ratios can be improved by nonlinear transformations of the explanatory variables is resolved. Building on the result that nonlinear balance sheet ratio transformations hardly improve the SLR’s predictive power in our case, we secondly compare the classification performance of a multivariate SLR to the discriminative powers of probabilities of default derived from three different capital market data, namely bonds, CDSs, and stocks. Benefiting from the prompt inclusion of relevant information, the capital market data in general and CDSs in particular increasingly outperform the SLR while approaching the time of the credit event. Due to the higher classification performances, it seems plausible for creditors to align their investment decisions with capital market-based default indicators, i.e., to imitate the aggregate opinion of the market participants. Since imitation is considered to be the source of LPPL structures in financial time series, it is highly plausible to scan CDS spread developments for LPPL patterns. By establishing LPPL patterns in governmental CDS spread trajectories of some European crisis countries, the LPPL’s application to credit risk markets is extended. This novel piece of evidence further strengthens the claim that credit risk markets are adequate breeding grounds for LPPL patterns.

Hidden power law patterns in the top European football leagues

Because sports are stylized combat, sports may follow power laws similar to those found for wars, individual clashes, and acts of terrorism. We show this fact for football (soccer) by adjusting power laws that show a close relationship between rank and points won by the clubs participating in the latest seasons of the top fifteen European football leagues. In addition, we use Shannon entropy for gauging league competitive balance. As a result, we are able to rank the leagues according to competitiveness.

Analysis of the Respiratory Dynamics During Normal Breathing by Means of Pseudophase Plots and Pressure–Volume Loops

This paper reports on the analysis of tidal breathing patterns measured during noninvasive forced oscillation lung function tests in six individual groups. The three adult groups were healthy, with prediagnosed chronic obstructive pulmonary disease, and with prediagnosed kyphoscoliosis, respectively. The three children groups were healthy, with prediagnosed asthma, and with prediagnosed cystic fibrosis, respectively. The analysis is applied to the pressure-volume curves and the pseudophase-plane loop by means of the box-counting method, which gives a measure of the area within each loop. The objective was to verify if there exists a link between the area of the loops, power-law patterns, and alterations in the respiratory structure with disease. We obtained statistically significant variations between the data sets corresponding to the six groups of patients, showing also the existence of power-law patterns. Our findings support the idea that the respiratory system changes with disease in terms of airway geometry and tissue parameters, leading, in turn, to variations in the fractal dimension of the respiratory tree and its dynamics.

The “S” curve relationship between export diversity and economic size of countries

The highly detailed international trade data among all countries in the world during 1971–2000 shows that the kinds of export goods and the logarithmic GDP (gross domestic production) of a country have an S-shaped relationship. This indicates that all countries can be divided into three stages accordingly. First, the small economies always export very few kinds of products as we expect. Second, once the economic size magnitude (log(GDP)) of a country is beyond a threshold, its export diversity may increase dramatically. However, this is not the case for large economies because a ceiling on the export diversity is observed when their logarithmic GDPs are higher than another threshold. This pattern is very stable for different years although the concrete parameters of the fitting sigmoid functions may change with time. In addition, we also discussed other relationships such as import diversity with respect to logarithmic GDP, diversity of exporters with respect to the number of export goods etc.; all of these relationships show S-shaped or power law patterns which can be derived by the “S” curve relations. Although this paper does not explain the origin of the S-shaped curve, it may provide a basic empirical fact and insights for economic diversity.

Magnetic perturbations in the events of broadband ELF turbulence observed by FAST

[1] Magnetic perturbations in the topside auroral ionosphere observed by the FAST satellite in the events of broadband extra low frequency (BB ELF) turbulence are investigated at frequencies 0.5–8 Hz (scales from 14 km down to 0.9 km). The power-law scaling and peculiar polarization patterns of the perturbations are highlighted. By comparing with the magnetic fields simulated according to the Chang et al. (2004) scenario of coarse-graining process development, we demonstrate that the magnetic perturbations observed in the events of the BB ELF turbulence can be reasonably understood in terms of non-linearly interacting multiscale field-aligned currents traversed by the spacecraft.

Assessing Lévy walks as models of animal foraging

The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.

The spatial distribution of soils across Europe: A fractal approach

In this work we investigate the applicability of fractal modeling to the soil surface system in Europe, as described by the European Soil Database (V2.0) and the soil classification of the World Reference Base [FAO, 1999. World Reference Base for Soil Resources. ISSS–ISRIC–FAO, Rome, World Soil Resources Report No. 84]. Firstly, we observe the power law patterns of the distribution of pedotaxa as well as their pedorichness–area relationships. After this, we consider the 45 most abundant soils that cover 92% of the surface of Europe and estimate the box-counting fractal dimension of the areas covered by each of the pedotaxa within three orders of magnitude. Our analysis shows a remarkable scaling behavior of soil distribution and strongly suggests the fractal nature of pedotaxa distribution across Europe. These findings lead us to propose that considering the pedotaxa fractal dimension could play an important role in the analysis of the complexity of European soil systems.

More on patterns in Mie scattering

The powerlaw patterns in Mie scattering (the normalized light intensity I( θ)/ I(0) vs. the dimensionless qR where q=4 πλ −1 sin θ 2 is the magnitude of the wave vector transfer at the scattering angle θ for wavelength λ, and R is the radius of the nonabsorbing sphere with a relative refractive index m>1) are analyzed using the geometrical optics approximation for particles of a large size parameter. The ( qR) −4 powerlaw regime is shown to be present only in Mie scattering of soft particles. The ( qR) −2 powerlaw regime occurs at the scattering angles of the p=1 geometrical ray (refracted without internal reflections) from the portion of the incident beam with an incidence angle around π/4 upon the particle. The ( qR) −2 powerlaw regimes from particles sharing one common relative refractive index but differing in size parameters are collinear. Simple analytical expressions are derived to describe these powerlaw regimes of Mie scattering.

Quantitative Evaluation of Fractures on Monkshood Anticline, a Detachment Fold in the Foothills of&amp;nbsp; Western Canada

Monkshood anticline is a well-exposed surface anticline located in the Foothills in northeastern British Columbia. Extension fractures are well developed in the Prophet Formation (a carbonate-shale-chert sequence) throughout this fold. Virtually all of these fractures are filled with some combination of calcite and quartz cements. The fractures formed and the mineral cements were deposited during the fold development. Most fractures on Monkshood anticline have formed at high angles to bedding, and they commonly fall into two to three distinct orientation sets. The dominant fracture trend is aligned with the fold axis through the backlimb of the fold, but there is considerable variance in the dominant orientation in the forelimb. An order of magnitude variation is found in fracture densities across this fold, and over two orders of magnitude variation occur in mean fracture aperture. These variations in density and aperture do not correlate with particular structural positions. Fracture trace lengths exhibit power-law distribution patterns, suggesting fractal character. Fracture aperture displays a roughly linear correlation to volumetric fracture strain, but shows no consistent association with either density or trace length. Stochastic modeling of the fracture networks on Monkshood anticline suggests density is the primary factor affecting fracture network connectivity, but fracture size also can play an important supporting role. The occurrence of a dominant fracture orientation set can impart a significant directional bias to connectivity. Measurement stations on Monkshood anticline that have both high fracture densities (for connectivity) and substantial fracture apertures (for conductance) occur primarily in midlimb positions.