Double Power-law Distribution Research Articles

Increased extinction probability of the Madden-Julian oscillation after about 27 days.

The Madden-Julian oscillation (MJO) is a tropical weather system that has an important influence in the tropics and beyond; however, many of its characteristics are poorly understood, including their initiation and termination. Here we define Madden-Julian events as contiguous time periods with an active MJO, and we show that both the durations and the sizes of these events are well described by a double power-law distribution. Thus, small events have no characteristic scale, and the same for large events; nevertheless, both types of events are separated by a characteristic duration of about 27 days (this corresponds to half a cycle, roughly). Thus, after 27 days, there is a sharp increase in the probability that an event becomes extinct. We find that this effect is independent of the starting and ending phases of the events, which seems to point to an internal mechanism of exhaustion rather than to the effect of an external barrier. Our results would imply an important limitation of the MJO as a driver of subseasonal predictability.

Analysis of Acoustic Emission Energy Distribution and Avalanche Dynamics of Sandstone with Different Particle Sizes.

The mechanical properties of sandstone have an important impact on the stability of the coal mine roof and floor, sandstone gas mining, and underground engineering safety. In order to study the critical characteristics on the failure process of sandstone with different particle sizes under uniaxial compression conditions, avalanche dynamics theory and a critical model are used to analyze the distribution of acoustic emission (AE) parameters, and the maximum likelihood estimation is used to accurately estimate the critical parameters. The results showed that the AE phenomenon of sandstone can be divided into four stages: initial compaction period, quiet period, crack stable growth period and outbreak period. During the process of compression failure, the larger the particle size is, the more seriously the sandstone is damaged. The AE energy probability density distribution follows single power-law distribution, and the AE energy critical exponent is 1.20 and follows the characteristics of scale-free regarding the power-law distribution on the particle sizes. When the stress runs up to 90% of peak stress, the bifurcation ratio increases sharply and shows the characteristics of the critical state. The waiting time and the avalanche size distribution follow double power-law distribution, and the inflection points are 0.03 and 37. Before and after the inflection point, the waiting time critical exponent and the avalanche size critical exponent are 1.90, 0.40 and 2.40, 1.60. This shows that the dynamic evolution process of sandstone under uniaxial compression condition can be characterized well by the fiber bundle model.

Double power-law distribution in spatial network induced by cost constraints

We investigate the scaling laws of the degree distribution in an evolving spatial network where the long-range links of a node are subject to a cost constraint. The constraint can cause a discontinuous reduction in the length of the links to be attached to the node once the node reaches some critical degree k c . We show that this effect can result in an abrupt change in the attachment probability and consequently induces a double power-law degree distribution. We derive the distribution analytically for the homogeneous constraint and demonstrate a consistent result for the heterogeneous one. Our model finds a robust connection between the double power law and the spatial constraint and offers a plausible explanation of the common occurrence of the distribution in airline networks.

Multiple scaling law in networks with dynamic spatial constraint

We study a network model where the spatial constraint evolves with the network growth and demonstrate that the dynamic constraint can generally cause a non-stationary multiple scaling law in degree distribution. In specific, we show that the different power-law segments emerge successively during the network evolution, from a single power law at the beginning to a double one and to more scaling laws due to the particular model setting. Our findings indicate a plausible geometric origin of the double power-law distribution and predict the non-stationary nature accompanied by its emergence, both of which are supported by our empirical study on the Chinese airline network.

Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution

The power-law distribution is an important descriptive characteristic of scale-free complex supply chain networks (SCN). The power-law distribution and deviation phenomena of SCN nodes are explored in combination with complex network theory, so it is imporant to accurately characterize the dynamic characteristics of network evolution on a time scale. Based on the analysis of the topological structure and evolutionary characteristics of the small-world network and scale-free SCN, the single and double power-law distribution and evolutionary dynamic characteristics of the complex SCN are further analyzed, and the deviation phenomenon of the power-law distribution is analyzed. On the premise of setting three parameters of new nodes, new edges, and node reconnection in the process of network evolution, the power-law distribution deviation evolution model under a complex network environment is constructed, and then the parameters of the SCN evolution model are analyzed. Combined with numerical simulation and model simulation, the evolution of SCN with two kinds of power-law deviation is analyzed. The results show that the deviation of the two-stage power-law distribution is not caused by the process of adding nodes or connecting edges, while it has a certain influence on the change of the power index, and the deviation of the power-law distribution in SCN increases with the extension of the evolution time. When p1=0, the single power-law distribution of SCN tends to change to a δ distribution when the time step is large enough.

Lognormals, power laws and double power laws in the distribution of frequencies of harmonic codewords from classical music

Zipf’s law is a paradigm describing the importance of different elements in communication systems, especially in linguistics. Despite the complexity of the hierarchical structure of language, music has in some sense an even more complex structure, due to its multidimensional character (melody, harmony, rhythm, timbre, etc.). Thus, the relevance of Zipf’s law in music is still an open question. Using discrete codewords representing harmonic content obtained from a large-scale analysis of classical composers, we show that a nearly universal Zipf-like law holds at a qualitative level. However, in an in-depth quantitative analysis, where we introduce the double power-law distribution as a new player in the classical debate between the superiority of Zipf’s (power) law and that of the lognormal distribution, we conclude not only that universality does not hold, but also that there is not a unique probability distribution that best describes the usage of the different codewords by each composer.

Dynamic Study of Intelligent Traffic Behaviour Based on Multiple Traffic Modes

With the rapid development of society, the traffic problem has become increasingly severe, and the traditional methods can no longer effectively solve the current social traffic behaviour problems. Although studies on the dynamics of human traffic behaviour based on traffic modes can effectively reveal the anomalies in traffic behaviour, few studies integrate intelligent traffic behaviour with multiple traffic modes. Based on the numerous traffic data of bike-sharing and ride-hailing in a Chinese city, this paper reveals the dynamic characteristics of various traffic behaviours in the city by combining spatiotemporal characteristics index and urban spatial structure with human traffic behaviour patterns. The experimental results show that the traffic behaviour of the town presents a double logarithmic power-law distribution in time characteristics, and there is a close interdependent dynamic relationship with the city’s spatial structure. The research in this paper can reveal the relationship between bimodal power-law distribution and spatial characteristics in complex systems and help solve social traffic problems effectively in social reality. Further research results can provide practical planning guidance for the behavioural integration of multiple traffic in smart cities.

A-Train estimates of the sensitivity of the cloud-to-rainwater ratio to cloud size, relative humidity, and aerosols

Abstract. Precipitation efficiency has been found to play an important role in constraining the sensitivity of the climate through its role in controlling cloud cover, yet its controls are not fully understood. Here we use CloudSat observations to identify individual contiguous shallow cumulus cloud objects and compute the ratio of cloud water path to rainwater (WRR) path as a proxy for warm-rain efficiency. Cloud objects are then conditionally sampled by cloud-top height, relative humidity, and aerosol optical depth (AOD) to analyze changes in WRR as a function of cloud size (extent). For a fixed cloud-top height, WRR increases with extent and environmental humidity following a double power-law distribution, as a function of extent. Similarly, WRR increases, holding average relative humidity at or below 850 mb constant. There is little relationship between WRR and AOD when conditioned by cloud-top height, suggesting that, once rain drop formation begins, aerosols may not be as important for WRR as cloud size and depth. Consistent with prior studies, results show an increase in WRR with sea-surface temperature. However, for a given depth and SST, WRR is also dependent on cloud size and becomes larger as cloud size increases. Given that larger objects become more frequent with increasing SST, these results imply that increasing precipitation efficiencies with SST are due not only to deeper clouds with greater cloud water contents but also to the propensity for larger clouds which may have more protected updrafts.

Double Power-Law in Leucosome Width Distribution: Implications for Recognizing Melt Movement in Migmatites

Leucosome sizes in migmatites has been shown to follow power-law distribution, which is indicative of self-organized criticality governing accumulation and transport of partial melts in anatexis. In our measurements of leucosome widths in the Olkiluoto migmatite in western Finland, we found double power-law distributions in addition to single power-law distributions. We divided the double power-law behaviors into those with positive and negative kinks separating the two power-law regimes. We propose that double power-law distributions in migmatites result from impediments on the bottom-up self-organized criticality that melt transport processes commonly display. These impediments can be caused by some leucosomes solidifying before others, in which case the leucosomes were effectively formed in two distinct events or phases. We also consider the emergence of double power-laws as a result of melt accumulation in and sudden melt removal from conduits of a specific size.

Emergence of double power-law degree distribution by controlling the evolution of BA model

Recent empirical studies have witnessed the wide existence of double power-law distribution in complex systems. However its underlying origin remains inadequately disclosed. We show this distribution can arise from the proper control of BA model. Particularly, the control policy demands each new edge to determine the range(degree≤kc or >kc) of its target node by probability p before applying the preferential attachment. We analytically derive the degree distribution and demonstrate a critical control parameter pc so that normal and reversed double power-law degree distribution emerges for p>pc and p<pc respectively, and a single power law is recovered when p=pc. The control policy of our model can spontaneously result from various constraints, which provides possible explanation for some real systems and indicates the validity of our model.

Avalanche properties in striplike ferromagnetic systems.

We present numerical findings on the behavior of the athermal nonequilibrium random-field Ising model of spins at the thin striplike L_{1}×L_{2}×L_{3} cubic lattices with L_{1}<L_{2}<L_{3}. Changing of system sizes highly influences the evolution and shape of avalanches. The smallest avalanches [classified as three-dimension- (3D) like] are unaffected by the system boundaries, the larger are sandwiched between the top and bottom system faces so are 2D-like, while the largest are extended over the system lateral cross section and propagate along the length L_{3} like in 1D systems. Such a structure of avalanches causes double power-law distributions of their size, duration, and energy with larger effective critical exponent corresponding to 3D-like and smaller to 2D-like avalanches. The distributions scale with thickness L_{1} and are collapsible following the proposed scaling predictions which, together with the distributions' shape, might be important for analysis of the Barkhausen noise experimental data for striplike samples. Finally, the impact of system size on external field that triggers the largest avalanche for a given disorder is presented and discussed.

The Role of Cloud Size and Environmental Moisture in Shallow Cumulus Precipitation

AbstractCloud models show that precipitation is more likely to occur in larger shallow clouds and/or in an environment with more moisture, in part as a result of decreasing the impacts of entrainment mixing on the updrafts. However, the role of cloud size in shallow cloud precipitation onset from global satellite observations has mostly been examined with precipitation proxies from imagers and has not been systematically examined in active sensors, primarily because of sensitivity limitations of previous spaceborne active instruments. Here we use the more sensitive CloudSat/CALIPSO observations to identify and characterize the properties of individual contiguous shallow cumulus cloud objects. The objects are conditionally sampled by cloud-top height to determine the changes in precipitation likelihood with increasing cloud size and column water vapor. On average, raining shallow cumulus clouds are typically taller by a factor of 2 and have a greater horizontal extent than their nonraining counterparts. Results show that for a fixed cloud-top height the likelihood of precipitation increases with increasing cloud size and generally follows a double power-law distribution. This suggests that the smallest cloud objects are able to grow freely within the boundary layer but the largest cloud objects are limited by environmental moisture. This is supported by our results showing that, for a fixed cloud-top height and cloud size, the precipitation likelihood also increases as environmental moisture increases. These results are consistent with the hypothesis that larger clouds occurring in a wetter environment may be better able to protect their updrafts from entrainment effects, increasing their chances of raining.

Temporal Patterns Underlying Domestic Departure Passengers Behavior in the Airport

Air travelers’ behavior is closely related to the operational performance of any airport terminal. Much of previous research has focused on how airport operators balance the number of facilities in a terminal and the Level of Service (LOS), while the behavior of passengers is less considered. Not much is known, however, about passenger’s behavior during the entire departure process in an airport. In this study, we analyze empirical departure passenger’s data to gain an insight into the regular patterns of their activities in an airport. We find that there exist two distinguished temporal patterns during two discretionary periods— post check-in and pre-security check , post security check and pre-boarding . The time that departure passengers spend in these two periods is well approximated by a double power-law distribution and an exponential truncated power-law distribution respectively. The two distinguished distributions suggest that there may be different mechanisms underlying passengers’ behavior as indicated by previous studies on human mobility. We introduce a stochastic model that considers traveling experience and time pressure to capture the decision dynamics of human behavior. Simulation results suggest that traveling experience and time pressure dominate passenger’s decisions before and after security respectively. Our findings contribute to a better understanding of human dynamics, and also offer the potential for optimizing and simulation of airport terminal operation.

Exponentially decayed double power-law distribution of Bitcoin trade sizes

Bitcoin is the most important cryptocurrency that leads the cryptocurrency market. The 776,629,356 Bitcoin trades from 3 January 2009 to 31 December 2017 are retrieved. It is found that the medium and large trade sizes comply with the exponentially decayed double power-law distribution. The tail exponent α1 of the medium trade sizes fluctuates in 2009, increases during the period from 2011 to 2014, and then converges to a stable level at α1=0.266±0.127. The tail exponent α2 of the large trade sizes fluctuates more and we have α2=1.752±1.688.

Avalanche dynamics of ferroelectric phase transitions in BaTiO3 and 0.7Pb(Mg2∕3Nb1∕3)O3-0.3PbTiO3 single crystals

The motion of phase fronts during a ferroelectric phase transition is intermittent and follows avalanche dynamics. In the present study, we show that an intermittent propagation mode generates spikes of depolarization currents at an extremely slow heating rate of 0.05 K/min in BaTiO3 (BTO) and 0.7Pb(Mg2∕3Nb1∕3)O3-0.3PbTiO3 (PMN-PT) single crystals. Such “jerks” are indicative of avalanche dynamics, and their energy exhibits a power law distribution with exponents of ε = 1.3 ± 0.10 and ε = 1.5 ± 0.10 for BTO and PMN-PT, respectively. The rate of aftershocks after big events decays as an Omori-like power-law and interevent times are characterized by a universal double power-law distribution, indicating the critical temporal correlations between the avalanche events.

Finding patterns in the degree distribution of real-world complex networks: going beyond power law

The most important structural characteristics in the study of large-scale real-world complex networks in pattern analysis are degree distribution. Empirical observations on the pattern of the real-world networks have led to the claim that their degree distributions follow, in general, a single power law. However, a closer observation, while fitting, shows that the single power-law distribution is often inadequate to meet the data characteristics properly. Since the degree distribution in the log–log scale actually displays, under inspection, two different slopes unlike what happens while fitting with the single power law. These two slopes with a transition in between closely resemble the pattern of the sigmoid function. This motivates us to derive a novel double power-law distribution for accurately modeling the real-world networks based on the sigmoid function. The proposed modeling approach further leads to the identification of a transition degree which, it has been demonstrated, may have a significant implication in analyzing the complex networks. The applicability, as well as effectiveness of the proposed methodology, is shown using rigorous experiments and also validated using statistical tests.

Ferroelectric switching and scale invariant avalanches in BaTiO3

Ferroelectric-field switching in $\mathrm{BaTi}{\mathrm{O}}_{3}$ generates ``Barkhausen noise'' when domain walls are displaced. We show by acoustic emission spectroscopy that electric-field switching of ${90}^{\ensuremath{\circ}}$ boundaries generates large strain fields, which emit acoustic phonons during ferroelectric hysteresis measurements. We use highly sensitive receivers (microphones) to measure the time sequences of noise in close analogy to noise patterns in ferroelastic and magnetic materials. Domain-wall interactions and jamming generate the ``crackling noise'' that follows scale invariant avalanche dynamics: the avalanche energy and amplitude probability distribution functions follow power laws with exponents $\ensuremath{\varepsilon}=1.65$ (energy) and \ensuremath{\tau}\ensuremath{'} = 2.25 (amplitudes). Aftershocks are very common and follow Omori law with probability $\ensuremath{\sim}{t}^{\ensuremath{-}p}$ where $t$ is the time elapsed after the main shock and $p$ is the Omori exponent $p\ensuremath{\sim}1$. The interevent times follow a double power-law distribution with exponents 0.9 for small times and 2.2 for the larger times. The scaling behavior is consistent with predictions of mean field theory.

Numerical computation of gravitational field of infinitely-thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33

AbstractWe developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.

The Structure of Titan’s N2 and CH4 Coronae

Abstract In this study, we analyze the structures of Titan’s N2 and CH4 coronae using a large data set acquired by the Ion Neutral Mass Spectrometer (INMS) instrument on board Cassini. The N2 and CH4 densities measured from the exobase up to 2000 km imply a mean exobase temperature of 146 K and 143 K, respectively, which is lower than the mean upper atmospheric temperature by 4 and 7 K. This indicates that on average, Titan possesses a subthermal rather than suprathermal corona. A careful examination reveals that the variability in corona structure is not very likely to be solar driven. Within the framework of the collisionless kinetic model, we investigate how the CH4 energy distribution near the exobase could be constrained if strong CH4 escape occurs on Titan. Several functional forms for the CH4 energy distribution are attempted, assuming two representative CH4 escape rates of s−1 and s−1. We find that the double Maxwellian and power-law distributions can reproduce the shape of the CH4 corona structure as well as the imposed CH4 escape rate. In both cases, the escape rate is contributed by a suprathermal CH4 population on the high-energy tail, with a number fraction below 5% and a characteristic energy of 0.1–0.6 eV per suprathermal CH4 molecule. The coexistence of the subthermal CH4 corona revealed by the INMS data and substantial CH4 escape suggested by some previous works could be reconciled by a significant departure in the exobase CH4 energy distribution from ideal Maxwellian that enhances escape and causes a noticeable redistribution of the corona structure.

A Direct Imaging Survey of Spitzer-detected Debris Disks: Occurrence of Giant Planets in Dusty Systems*

Abstract We describe a joint high-contrast imaging survey for planets at the Keck and Very Large Telescope of the last large sample of debris disks identified by the Spitzer Space Telescope. No new substellar companions were discovered in our survey of 30 Spitzer-selected targets. We combine our observations with data from four published surveys to place constraints on the frequency of planets around 130 debris disk single stars, the largest sample to date. For a control sample, we assembled contrast curves from several published surveys targeting 277 stars that do not show infrared excesses. We assumed a double power-law distribution in mass and semimajor axis (SMA) of the form , where we adopted power-law values and logarithmically flat values for the mass and SMA of planets. We find that the frequency of giant planets with masses 5–20 M Jup and separations 10–1000 au around stars with debris disks is 6.27% (68% confidence interval 3.68%–9.76%), compared to 0.73% (68% confidence interval 0.20%–1.80%) for the control sample of stars without disks. These distributions differ at the 88% confidence level, tentatively suggesting distinctness of these samples.